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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/6512/#Item_18" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <h2 id="BasicNotionsOfHomotopyTheory">Basic notions of homotopy theory</h2> <p>Traditionally, <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> and <a class="existingWikiWord" href="/nlab/show/physics">physics</a> have been <a class="existingWikiWord" href="/nlab/show/foundations">founded</a> on <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>, whose concept of <em><a class="existingWikiWord" href="/nlab/show/sets">sets</a></em> is that of “bags of distinguishable points”.</p> <p>But fundamental <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is governed by the <em><a class="existingWikiWord" href="/nlab/show/gauge+principle">gauge principle</a></em>. This says that given any two “things”, such as two <a class="existingWikiWord" href="/nlab/show/field+history">field histories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, it is in general wrong to ask whether they are <a class="existingWikiWord" href="/nlab/show/equality">equal</a> or not, instead one has to ask where there is a <em><a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mover><mo>⟶</mo><mi>γ</mi></mover><mi>y</mi></mrow><annotation encoding="application/x-tex"> x \stackrel{\gamma}{\longrightarrow} y </annotation></semantics></math></div> <p>between them. In mathematics this is called a <em><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a></em>.</p> <p>This principle applies also to <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a>/<a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> themselves, and thus leads to <em><a class="existingWikiWord" href="/nlab/show/gauge-of-gauge+transformations">gauge-of-gauge transformations</a></em> or <em><a class="existingWikiWord" href="/nlab/show/homotopies+of+homotopies">homotopies of homotopies</a></em></p> <center><img src="https://ncatlab.org/nlab/files/2Cell.jpg" width="200" /></center> <p>and so on to ever <em><a class="existingWikiWord" href="/nlab/show/higher+gauge+transformations">higher gauge transformations</a></em> or <em><a class="existingWikiWord" href="/nlab/show/higher+homotopies">higher homotopies</a></em>:</p> <center> <img src="https://ncatlab.org/nlab/files/3Cell.jpg" width="200" /> </center> <p>This shows that what <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> here are <a class="existingWikiWord" href="/nlab/show/elements">elements</a> of is not really a <a class="existingWikiWord" href="/nlab/show/set">set</a> in the sense of <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>. Instead, such a collection of <a class="existingWikiWord" href="/nlab/show/elements">elements</a> with <a class="existingWikiWord" href="/nlab/show/higher+gauge+transformations">higher gauge transformations</a>/<a class="existingWikiWord" href="/nlab/show/higher+homotopies">higher homotopies</a> between them is called a <em><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></em>.</p> <p>Hence the theory of <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> – <em><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em> – is much like <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>, but with the concept of <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>/<a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> built right into its <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a>. Homotopy theory is gauged mathematics.</p> <p>A <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model</a> for <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> are simply <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>: Their points represent the elements, the <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous</a> <a class="existingWikiWord" href="/nlab/show/paths">paths</a> between points represent the <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a>, and continuous deformations of paths represent <a class="existingWikiWord" href="/nlab/show/higher+gauge+transformations">higher gauge transformations</a>. A central result of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> is the <a class="existingWikiWord" href="/nlab/show/proof">proof</a> of the <em><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a></em>, which says that under this identification <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> are <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">equivalent</a> to <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> viewed, in turn, up to “<a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>”.</p> <p>In the special case of a <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> with a single <a class="existingWikiWord" href="/nlab/show/element">element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a> necessarily go from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to itself and hence form a <em><a class="existingWikiWord" href="/nlab/show/group">group</a> of <a class="existingWikiWord" href="/nlab/show/symmetries">symmetries</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> <center> <img src="https://ncatlab.org/nlab/files/GroupActing.jpg" width="80" /> </center> <p>This way <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> subsumes <a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a>.</p> <p>If there are higher order <a class="existingWikiWord" href="/nlab/show/gauge-of-gauge+transformations">gauge-of-gauge transformations</a>/<a class="existingWikiWord" href="/nlab/show/homotopies+of+homotopies">homotopies of homotopies</a> between these <a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a> <a class="existingWikiWord" href="/nlab/show/group">group</a>-elements, then one speaks of <em><a class="existingWikiWord" href="/nlab/show/2-groups">2-groups</a></em>, <em><a class="existingWikiWord" href="/nlab/show/3-groups">3-groups</a></em>, … <em><a class="existingWikiWord" href="/nlab/show/n-groups">n-groups</a></em>, and eventually of <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a></em>. When <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> are represented by <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, then <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a> are represented by <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a>.</p> <p>This way <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> subsumes parts of <a class="existingWikiWord" href="/nlab/show/topological+group">topological</a> <a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a>.</p> <p>Since, generally, there is more than one element in a <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>, these are like “groups with several elements”, and as such they are called <em><a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a></em> (Def. <a class="maruku-ref" href="#Groupoid"></a>).</p> <p>If there are higher order <a class="existingWikiWord" href="/nlab/show/gauge-of-gauge+transformations">gauge-of-gauge transformations</a>/<a class="existingWikiWord" href="/nlab/show/homotopies+of+homotopies">homotopies of homotopies</a> between the transformations in such a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, one speaks of <em><a class="existingWikiWord" href="/nlab/show/2-groupoids">2-groupoids</a></em>, <em><a class="existingWikiWord" href="/nlab/show/3-groupoids">3-groupoids</a></em>, … <em><a class="existingWikiWord" href="/nlab/show/n-groupoids">n-groupoids</a></em>, and eventually of <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a></em>. The plain <a class="existingWikiWord" href="/nlab/show/sets">sets</a> are recovered as the special case of <a class="existingWikiWord" href="/nlab/show/0-groupoids">0-groupoids</a>.</p> <p>Due to the higher orders <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> appearing here, <a class="existingWikiWord" href="/nlab/show/mathematical+structures">mathematical structures</a> based not on <a class="existingWikiWord" href="/nlab/show/sets">sets</a> but on <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> are also called <em><a class="existingWikiWord" href="/schreiber/show/Higher+Structures">higher structures</a></em>.</p> <p>Hence <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> are equivalently <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a></em>. This perspective makes explicit that <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> are the unification of plain <a class="existingWikiWord" href="/nlab/show/sets">sets</a> with the concept of <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge</a>-<a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a> <a class="existingWikiWord" href="/nlab/show/groups">groups</a>.</p> <p>An efficient way of handling <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> is in their explicit guise as <em><a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a></em> (Def. <a class="maruku-ref" href="#KanComplexe"></a> below); these are the non-abelian generalization of the <em><a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a></em> used in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>. Indeed, <em><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></em> is a special case of the general concept of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, and hence <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> forms but a special abelian corner within <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>. Conversely, <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> may be understood as the non-abelian generalization of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>.</p> <p>Hence, in a self-reflective manner, there are many different but <em><a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalent</a></em> incarnations of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>. Below we discuss in turn:</p> <ul> <li> <p><em><a href="#TopologicalHomotopyTheory">Topological homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> modeled by <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. This is the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> familiar from traditional <a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, such as <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a>-theory.</p> </li> <li> <p><em><a href="#SimplicialHomotopyTheory">Simplicial homotopy theory</a></em>.</p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> modeled on <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>, whose <a class="existingWikiWord" href="/nlab/show/fibrant+objects">fibrant objects</a> are the <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>. This <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a> is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a> (the “<a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>”), which makes explicit that <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> is not really about <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, but about the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> that these represent.</p> </li> </ul> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>Ideally, abstract homotopy theory would simply be a complete replacement of <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>, obtained by <em>removing</em> the assumption of strict <a class="existingWikiWord" href="/nlab/show/equality">equality</a>, relaxing it to <a class="existingWikiWord" href="/nlab/show/gauge+equivalence">gauge equivalence</a>/<a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>. As such, abstract homotopy theory would be part and parcel of the <a class="existingWikiWord" href="/nlab/show/foundations+of+mathematics">foundations of mathematics</a> themselves, not requiring any further discussion. This ideal perspective is the promise of <em><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></em> and may become full practical reality in the next decades.</p> <p>Until then, abstract homotopy theory has to be formulated on top of the traditional <a class="existingWikiWord" href="/nlab/show/foundations+of+mathematics">foundations of mathematics</a> provided by <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>, much like one may have to run a Linux emulator on a Windows machine, if one does happen to be stuck with the latter.</p> <p>A very convenient and powerful such emulator for homotopy theory within set theory is <em><a class="existingWikiWord" href="/nlab/show/model+categories">model category theory</a></em>, originally due to <a href="model+category#Quillen67">Quillen 67</a> and highly developed since. This we introduce here.</p> <p>The idea is to consider ordinary <a class="existingWikiWord" href="/nlab/show/categories">categories</a> (Def. <a class="maruku-ref" href="#Categories"></a>) but with the understanding that some of their <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex"> X \overset{f}{\longrightarrow} Y </annotation></semantics></math></div> <p>should be <em><a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a></em> (Def. <a class="maruku-ref" href="#HomotopyEquivalence"></a>), namely similar to <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> (Def. <a class="maruku-ref" href="#Isomorphism"></a>), but not necessarily satisfying the two <a class="existingWikiWord" href="/nlab/show/equations">equations</a> defining an actual isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>f</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>X</mi></msub><mphantom><mi>AAAA</mi></mphantom><mi>f</mi><mo>∘</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex"> f^{-1} \circ f \;=\; id_{X} \phantom{AAAA} f \circ f^{-1} \;=\; id_Y </annotation></semantics></math></div> <p>but intended to satisfy this only with <a class="existingWikiWord" href="/nlab/show/equality">equality</a> relaxed to <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>/<a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>:</p> <div class="maruku-equation" id="eq:HomotopyEquivalenceCondition"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>f</mi><mspace width="thickmathspace"></mspace><mover><mo>⇒</mo><mi>gauge</mi></mover><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>X</mi></msub><mphantom><mi>AAAA</mi></mphantom><mi>f</mi><mo>∘</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mover><mo>⇒</mo><mi>gauge</mi></mover><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>Y</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^{-1} \circ f \;\overset{gauge}{\Rightarrow}\; id_{X} \phantom{AAAA} f \circ f^{-1} \;\overset{gauge}{\Rightarrow}\; id_Y \,. </annotation></semantics></math></div> <p>Such <em>would-be homotopy equivalences</em> are called <em><a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a></em> (Def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a> below).</p> <p>In principle, this information already defines a <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> by a construction called <em><a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></em>, which turns <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> into actual <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> in a suitable way.</p> <p>However, without further tools this construction is unwieldy. The extra structure of a <em><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></em> (Def. <a class="maruku-ref" href="#ModelCategory"></a> below) on top of a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> provides a set of tools.</p> <p>The idea here is to abstract (in Def. <a class="maruku-ref" href="#LeftAndRightHomotopyInAModelCategory"></a> below) from the evident concepts in <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a> of <em><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></em> (Def. <a class="maruku-ref" href="#LeftHomotopy"></a>) and <em><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></em> (Def. <a class="maruku-ref" href="#RightHomotopy"></a>) between <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a>: These are provided by continuous functions out of a <a class="existingWikiWord" href="/nlab/show/cylinder">cylinder space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Cyl(X) = X \times [0,1]</annotation></semantics></math> or into a <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>X</mi> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Path(X) = X^{[0,1]}</annotation></semantics></math>, respectively, where in both cases the <a class="existingWikiWord" href="/nlab/show/interval">interval space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> serves to parameterize the relevant <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>/<a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>.</p> <p>Now a little reflection shows (this was the seminal insight of <a href="model+category#Quillen67">Quillen 67</a>) that what really matters in this construction of homotopies is that the <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> factors the <a class="existingWikiWord" href="/nlab/show/diagonal+morphism">diagonal morphism</a> from a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>diag</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><munderover><mo>⟶</mo><mtext>weak equiv.</mtext><mtext> cofibration </mtext></munderover><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext> fibration </mtext></mover><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> diag_X \;\colon\; X \underoverset{\text{weak equiv.}}{\text{ cofibration }}{\longrightarrow} Path(X) \overset{\text{ fibration }}{\longrightarrow} X \times X </annotation></semantics></math></div> <p>while the cylinder serves to factor the <a class="existingWikiWord" href="/nlab/show/codiagonal+morphism">codiagonal morphism</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>codiag</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mtext>cofibration</mtext></mover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mtext>weak equiv</mtext><mtext>fibration</mtext></munderover><mi>X</mi></mrow><annotation encoding="application/x-tex"> codiag_X \;\colon\; X \sqcup X \overset{ \text{cofibration} }{\longrightarrow} Cyl(X) \underoverset{ \text{weak equiv} }{ \text{fibration} }{\longrightarrow} X </annotation></semantics></math></div> <p>where in both cases “<a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>” means something like <em>well behaved <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a></em>, while “<a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a>” means something like <em>satisfying the <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> (Def. <a class="maruku-ref" href="#LiftingAndExtension"></a> below) against fibrations that are also weak equivalences</em>.</p> <p>Such factorizations subject to lifting properties is what the definition of <em><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></em> axiomatizes, in some generality. That this indeed provides a good toolbox for handling <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> is shown by the <em><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a> in <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a></em> (Lemma <a class="maruku-ref" href="#WhiteheadTheoremInModelCategories"></a> below), which exhibits all <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> as actual <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> after passage to “good representatives” of objects (fibrant/cofibrant <a class="existingWikiWord" href="/nlab/show/resolutions">resolutions</a>, Def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a> below). Accordingly, the first theorem of model category theory (<a href="model+category#Quillen67">Quillen 67, I.1 theorem 1</a>, reproduced as Theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a> below), provides a tractable expression for the <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> modulo <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> of the underlying <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> in terms of actual morphisms out of <a class="existingWikiWord" href="/nlab/show/cofibrant+resolutions">cofibrant resolutions</a> into <a class="existingWikiWord" href="/nlab/show/fibrant+resolutions">fibrant resolutions</a> (Lemma <a class="maruku-ref" href="#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory"></a> below).</p> <p>This is then generally how <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>-theory serves as a model for <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>: All homotopy-theoretic constructions, such as that of <a class="existingWikiWord" href="/nlab/show/long+homotopy+fiber+sequences">long homotopy fiber sequences</a> (Prop. <a class="maruku-ref" href="#LongFiberSequence"></a> below), are reflected via constructions of ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> but applied to suitably <a class="existingWikiWord" href="/nlab/show/resolution">resolved objects</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p><strong>Literature</strong> (<a href="model+category#DwyerSpalinski95">Dwyer-Spalinski 95</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_defn" id="ModelCategory"> <h6 id="definition">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/model+category">model category</a>)</strong></p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></strong> is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#Categories"></a>) with all <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> (Def. <a class="maruku-ref" href="#Limits"></a>);</p> </li> <li> <p>three sub-<a class="existingWikiWord" href="/nlab/show/classes">classes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>,</mo><mi>Fib</mi><mo>,</mo><mi>Cof</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W, Fib, Cof \subset Mor(\mathcal{C})</annotation></semantics></math> of its class of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>;</p> </li> </ol> <p>such that</p> <ol> <li> <p>the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> into a <strong><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></strong>, def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>;</p> </li> <li> <p>The pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Fib</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(W \cap Cof\;,\; Fib)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Cap</mi><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>W</mi><mo>∩</mo><mi>Fib</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Cap\;,\; W\cap Fib)</annotation></semantics></math> are both <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a>, def. <a class="maruku-ref" href="#WeakFactorizationSystem"></a>.</p> </li> </ol> <p>One says:</p> <ul> <li> <p>elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> are <em><a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a></em>,</p> </li> <li> <p>elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cof</mi></mrow><annotation encoding="application/x-tex">Cof</annotation></semantics></math> are <em><a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a></em>,</p> </li> <li> <p>elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fib</mi></mrow><annotation encoding="application/x-tex">Fib</annotation></semantics></math> are <em><a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a></em>,</p> </li> <li> <p>elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow><annotation encoding="application/x-tex">W\cap Cof</annotation></semantics></math> are <em><a class="existingWikiWord" href="/nlab/show/acyclic+cofibrations">acyclic cofibrations</a></em>,</p> </li> <li> <p>elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow><annotation encoding="application/x-tex">W \cap Fib</annotation></semantics></math> are <em><a class="existingWikiWord" href="/nlab/show/acyclic+fibrations">acyclic fibrations</a></em>.</p> </li> </ul> </div> <p>The form of def. <a class="maruku-ref" href="#ModelCategory"></a> is due to (<a href="model+category#Joyal">Joyal, def. E.1.2</a>). It implies various other conditions that (<a href="model+category#Quillen67">Quillen 67</a>) demands explicitly, see prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a> and prop. <a class="maruku-ref" href="#WeakEquivalencesAreClosedUnderRetracts"></a> below.</p> <p>We now dicuss the concept of <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a> (Def. <a class="maruku-ref" href="#WeakFactorizationSystem"></a> below) appearing in def. <a class="maruku-ref" href="#ModelCategory"></a>.</p> <h3 id="WeakFactorizationSystems">Factorization systems</h3> <div class="num_defn" id="LiftingAndExtension"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/lift">lift</a> and <a class="existingWikiWord" href="/nlab/show/extension">extension</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/category">category</a>. Given a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>p</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{f}{\longrightarrow}& Y \\ {}^{\mathllap{p}}\downarrow \\ B } </annotation></semantics></math></div> <p>then an <em><a class="existingWikiWord" href="/nlab/show/extension">extension</a></em> of the <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> along the <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a completion to a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>p</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mpadded></msub></mtd></mtr> <mtr><mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{f}{\longrightarrow}& Y \\ {}^{\mathllap{p}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ B } \,. </annotation></semantics></math></div> <p>Dually, given a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && A \\ && \downarrow^{\mathrlap{p}} \\ X &\stackrel{f}{\longrightarrow}& Y } </annotation></semantics></math></div> <p>then a <strong><a class="existingWikiWord" href="/nlab/show/lift">lift</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a completion to a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && A \\ &{}^{\mathllap{\tilde f}}\nearrow& \downarrow^{\mathrlap{p}} \\ X &\stackrel{f}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>Combining these cases: given a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mi>l</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>r</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ {}^{\mathllap{p_l}}\downarrow && \downarrow^{\mathrlap{p_r}} \\ X_2 &\stackrel{f_1}{\longrightarrow}& Y_2 } </annotation></semantics></math></div> <p>then a <strong><a class="existingWikiWord" href="/nlab/show/lifting">lifting</a></strong> in the diagram is a completion to a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mi>l</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>r</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ {}^{\mathllap{p_l}}\downarrow &\nearrow& \downarrow^{\mathrlap{p_r}} \\ X_2 &\stackrel{f_1}{\longrightarrow}& Y_2 } \,. </annotation></semantics></math></div> <p>Given a sub-<a class="existingWikiWord" href="/nlab/show/class">class</a> of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K \subset Mor(\mathcal{C})</annotation></semantics></math>, then</p> <ul> <li>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">p_r</annotation></semantics></math> as above is said to have the <strong><a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></strong> or to be a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphism">injective morphism</a></strong> if in all square diagrams with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">p_r</annotation></semantics></math> on the right and any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>l</mi></msub><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">p_l \in K</annotation></semantics></math> on the left a lift exists.</li> </ul> <p>dually:</p> <ul> <li>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>l</mi></msub></mrow><annotation encoding="application/x-tex">p_l</annotation></semantics></math> is said to have the <strong><a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></strong> or to be a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/projective+morphism">projective morphism</a></strong> if in all square diagrams with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>l</mi></msub></mrow><annotation encoding="application/x-tex">p_l</annotation></semantics></math> on the left and any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>r</mi></msub><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">p_r \in K</annotation></semantics></math> on the left a lift exists.</li> </ul> </div> <div class="num_defn" id="WeakFactorizationSystem"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a>)</strong></p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></strong> (WFS) on a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Proj</mi><mo>,</mo><mi>Inj</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Proj,Inj)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> such that</p> <ol> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X\to Y</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> may be factored as the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Proj</mi></mrow><annotation encoding="application/x-tex">Proj</annotation></semantics></math> followed by one in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Inj</mi></mrow><annotation encoding="application/x-tex">Inj</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Proj</mi></mrow></mover><mi>Z</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Inj</mi></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f\;\colon\; X \overset{\in Proj}{\longrightarrow} Z \overset{\in Inj}{\longrightarrow} Y \,. </annotation></semantics></math></div></li> <li> <p>The classes are closed under having the <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a>, def. <a class="maruku-ref" href="#LiftingAndExtension"></a>, against each other:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Proj</mi></mrow><annotation encoding="application/x-tex">Proj</annotation></semantics></math> is precisely the class of morphisms having the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against every morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Inj</mi></mrow><annotation encoding="application/x-tex">Inj</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Inj</mi></mrow><annotation encoding="application/x-tex">Inj</annotation></semantics></math> is precisely the class of morphisms having the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against every morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Proj</mi></mrow><annotation encoding="application/x-tex">Proj</annotation></semantics></math>.</p> </li> </ol> </li> </ol> </div> <div class="num_defn" id="FunctorialFactorization"> <h6 id="definition_4">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/functorial+factorization">functorial factorization</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category">category</a>, a <strong><a class="existingWikiWord" href="/nlab/show/functorial+factorization">functorial factorization</a></strong> of the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>fact</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo>⟶</mo><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> fact \;\colon\; \mathcal{C}^{\Delta[1]} \longrightarrow \mathcal{C}^{\Delta[2]} </annotation></semantics></math></div> <p>which is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup><mo>→</mo><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">d_1 \;\colon \;\mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In def. <a class="maruku-ref" href="#FunctorialFactorization"></a> we are using the following standard notation, see at <em><a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/nerve+of+a+category">nerve of a category</a></em>:</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[1] = \{0 \to 1\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo>→</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[2] = \{0 \to 1 \to 2\}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/ordinal+numbers">ordinal numbers</a>, regarded as <a class="existingWikiWord" href="/nlab/show/posets">posets</a> and hence as <a class="existingWikiWord" href="/nlab/show/categories">categories</a>. The <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Arr</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr(\mathcal{C})</annotation></semantics></math> is equivalently the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo>≔</mo><mi>Funct</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C})</annotation></semantics></math>, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup><mo>≔</mo><mi>Funct</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>,</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C})</annotation></semantics></math> has as objects pairs of composable morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. There are three injective functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\delta_i \colon [1] \rightarrow [2]</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\delta_i</annotation></semantics></math> omits the index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> in its image. By precomposition, this induces <a class="existingWikiWord" href="/nlab/show/functors">functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup><mo>⟶</mo><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">d_i \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}</annotation></semantics></math>. Here</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">d_1</annotation></semantics></math> sends a pair of composable morphisms to their <a class="existingWikiWord" href="/nlab/show/composition">composition</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">d_2</annotation></semantics></math> sends a pair of composable morphisms to the first morphisms;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">d_0</annotation></semantics></math> sends a pair of composable morphisms to the second morphisms.</p> </li> </ul> </div> <div class="num_defn" id="FunctorialWeakFactorizationSystem"> <h6 id="definition_5">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a>, def. <a class="maruku-ref" href="#WeakFactorizationSystem"></a>, is called a <strong>functorial weak factorization system</strong> if the factorization of morphisms may be chosen to be a <a class="existingWikiWord" href="/nlab/show/functorial+factorization">functorial factorization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>fact</mi></mrow><annotation encoding="application/x-tex">fact</annotation></semantics></math>, def. <a class="maruku-ref" href="#FunctorialFactorization"></a>, i.e. such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>2</mn></msub><mo>∘</mo><mi>fact</mi></mrow><annotation encoding="application/x-tex">d_2 \circ fact</annotation></semantics></math> lands in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Proj</mi></mrow><annotation encoding="application/x-tex">Proj</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo>∘</mo><mi>fact</mi></mrow><annotation encoding="application/x-tex">d_0\circ fact</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Inj</mi></mrow><annotation encoding="application/x-tex">Inj</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Not all weak factorization systems are functorial, def. <a class="maruku-ref" href="#FunctorialWeakFactorizationSystem"></a>, although most (including those produced by the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> (prop. <a class="maruku-ref" href="#SmallObjectArgument"></a> below), with due care) are.</p> </div> <div class="num_prop" id="ClosurePropertiesOfInjectiveAndProjectiveMorphisms"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K\subset Mor(\mathcal{C})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/class">class</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">K Proj</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Inj</mi></mrow><annotation encoding="application/x-tex">K Inj</annotation></semantics></math>, respectively, for the sub-classes of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/projective+morphisms">projective morphisms</a> and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphisms">injective morphisms</a>, def. <a class="maruku-ref" href="#LiftingAndExtension"></a>. Then:</p> <ol> <li> <p>Both classes contain the class of <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> <li> <p>Both classes are closed under <a class="existingWikiWord" href="/nlab/show/composition">composition</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">K Proj</annotation></semantics></math> is also closed under <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a>.</p> </li> <li> <p>Both classes are closed under forming <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> in the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math> (see remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">K Proj</annotation></semantics></math> is closed under forming <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a> of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>”).</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Inj</mi></mrow><annotation encoding="application/x-tex">K Inj</annotation></semantics></math> is closed under forming <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/base+change">base change</a>”).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">K Proj</annotation></semantics></math> is closed under forming <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Inj</mi></mrow><annotation encoding="application/x-tex">K Inj</annotation></semantics></math> is closed under forming <a class="existingWikiWord" href="/nlab/show/products">products</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We go through each item in turn.</p> <p><strong>containing isomorphisms</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Iso</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mi>g</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{f}{\rightarrow}& X \\ {}_{\mathllap{\in Iso}}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{p}} \\ B &\underset{g}{\longrightarrow}& Y } </annotation></semantics></math></div> <p>with the left morphism an isomorphism, then a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> is given by using the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> of this isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mrow><mi>f</mi><mo>∘</mo><msup><mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msup><mo>↗</mo></mrow><annotation encoding="application/x-tex">{}^{{f \circ i^{-1}}}\nearrow</annotation></semantics></math>. Hence in particular there is a lift when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">p \in K</annotation></semantics></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">i \in K Proj</annotation></semantics></math>. The other case is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> <p><strong>closure under composition</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi><mi>Inj</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi><mi>Inj</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y } </annotation></semantics></math></div> <p>consider its <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> decomposition as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi><mi>Inj</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi><mi>Inj</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ \downarrow &\searrow& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>Now the bottom commuting square has a lift, by assumption. This yields another <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> decomposition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi><mi>Inj</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi><mi>Inj</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ \downarrow &\nearrow& \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y } </annotation></semantics></math></div> <p>and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1\circ p_1</annotation></semantics></math> has the right lifting property against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and is hence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Inj</mi></mrow><annotation encoding="application/x-tex">K Inj</annotation></semantics></math>. The case of composing two morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">K Proj</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>. From this the closure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">K Proj</annotation></semantics></math> under <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> follows since the latter is given by <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> of sequential composition and successive lifts against the underlying sequence as above constitutes a <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a>, whence the extension of the lift to the colimit follows by its <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a>.</p> <p><strong>closure under retracts</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">i \in K Proj</annotation></semantics></math>, i.e. let there be a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>id</mi> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi><mi>Proj</mi></mrow></mpadded> <mpadded width="0"><mi>i</mi></mpadded></msubsup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>B</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>D</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ id_A \colon & A &\longrightarrow& C &\longrightarrow& A \\ & \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}} \\ id_B \colon & B &\longrightarrow& D &\longrightarrow& B } \,. </annotation></semantics></math></div> <p>Then for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>j</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ {}^{\mathllap{j}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& Y } </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a>, it is equivalent to its <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> composite with that retract diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi><mi>Proj</mi></mrow></mpadded> <mpadded width="0"><mi>i</mi></mpadded></msubsup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>D</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,. </annotation></semantics></math></div> <p>Here the pasting composite of the two squares on the right has a lift, by assumption:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mrow></mrow> <mpadded width="0"><mi>i</mi></mpadded></msubsup></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>D</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{} && \nearrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,. </annotation></semantics></math></div> <p>By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> has the left lifting property against all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">p \in K</annotation></semantics></math> and hence is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">K Proj</annotation></semantics></math>. The other case is <a class="existingWikiWord" href="/nlab/show/formal+duality">formally dual</a>.</p> <p><strong>closure under pushout and pullback</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>K</mi><mi>Inj</mi></mrow><annotation encoding="application/x-tex">p \in K Inj</annotation></semantics></math> and and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Z</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{{f^* p}}}\downarrow && \downarrow^{\mathrlap{p}} \\ Z &\stackrel{f}{\longrightarrow} & Y } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. We need to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow><annotation encoding="application/x-tex">f^* p</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> with respect to all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">i \in K</annotation></semantics></math>. So let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow></mpadded></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& Z \times_f X \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{\mathrlap{f^* p}}} \\ B &\stackrel{g}{\longrightarrow}& Z } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a>. We need to construct a diagonal lift of that square. To that end, first consider the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> composite with the pullback square from above to obtain the commuting diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Z</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>By the right lifting property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, there is a diagonal lift of the total outer diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd> <mtd><msup><mrow></mrow> <mover><mrow><mo stretchy="false">(</mo><mi>f</mi><mi>g</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">^</mo></mover></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>f</mi><mi>g</mi></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ \downarrow^{\mathrlap{i}} &{}^{\hat {(f g)}}\nearrow& \downarrow^{\mathrlap{p}} \\ B &\stackrel{f g}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> this gives rise to the lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat g</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mover><mi>g</mi><mo stretchy="false">^</mo></mover></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Z</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Z \times_f X &\longrightarrow& X \\ &{}^{\hat g} \nearrow& \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>In order for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat g</annotation></semantics></math> to qualify as the intended lift of the total diagram, it remains to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd> <mtd><msup><mrow></mrow> <mover><mi>g</mi><mo stretchy="false">^</mo></mover></msup><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& Z \times_f X \\ \downarrow^{\mathrlap{i}} & {}^{\hat g}\nearrow \\ B } </annotation></semantics></math></div> <p>commutes. To do so we notice that we obtain two <a class="existingWikiWord" href="/nlab/show/cones">cones</a> with tip <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>:</p> <ul> <li> <p>one is given by the morphisms</p> <ol> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \to Z \times_f X \to X</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} B \stackrel{g}{\to} Z</annotation></semantics></math></li> </ol> <p>with universal morphism into the pullback being</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">A \to Z \times_f X</annotation></semantics></math></li> </ul> </li> <li> <p>the other by</p> <ol> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mover><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} B \stackrel{g}{\to} Z</annotation></semantics></math>.</li> </ol> <p>with universal morphism into the pullback being</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mover><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X</annotation></semantics></math>.</li> </ul> </li> </ul> <p>The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is <em>unique</em> this implies the required identity of morphisms.</p> <p>The other case is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> <p><strong>closure under (co-)products</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>s</mi></msub><mover><mo>→</mo><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow></mover><msub><mi>B</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>K</mi><mi>Proj</mi><msub><mo stretchy="false">}</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{(A_s \overset{i_s}{\to} B_s) \in K Proj\}_{s \in S}</annotation></semantics></math> be a set of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">K Proj</annotation></semantics></math>. Since <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> in the <a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math> are computed componentwise, their <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in this <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> is the universal morphism out of the coproduct of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>A</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\underset{s \in S}{\coprod} A_s</annotation></semantics></math> induced via its <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> by the set of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">i_s</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>A</mi> <mi>s</mi></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow></mover><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>B</mi> <mi>s</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{s \in S}{\sqcup} A_s \overset{(i_s)_{s\in S}}{\longrightarrow} \underset{s \in S}{\sqcup} B_s \,. </annotation></semantics></math></div> <p>Now let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>A</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>B</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a>. This is in particular a <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> under the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of objects, hence by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the coproduct, this is equivalent to a set of commuting diagrams</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>K</mi><mi>Proj</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>}</mo></mrow> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\{ \;\;\;\;\;\;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in K Proj}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S} \,. </annotation></semantics></math></div> <p>By assumption, each of these has a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\ell_s</annotation></semantics></math>. The collection of these lifts</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Proj</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mi>ℓ</mi> <mi>s</mi></msub></mrow></msup><mo>↗</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>}</mo></mrow> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \left\{ \;\;\;\;\;\;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in Proj}}\downarrow &{}^{\ell_s}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S} </annotation></semantics></math></div> <p>is now itself a compatible <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a>, and so once more by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the coproduct, this is equivalent to a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ℓ</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\ell_s)_{s\in S}</annotation></semantics></math> in the original square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>A</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><msub><mi>ℓ</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow></msup><mo>↗</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>K</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>B</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow &{}^{(\ell_s)_{s\in S}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>This shows that the coproduct of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">i_s</annotation></semantics></math> has the left lifting property against all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">f\in K</annotation></semantics></math> and is hence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Proj</mi></mrow><annotation encoding="application/x-tex">K Proj</annotation></semantics></math>. The other case is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> </div> <p>An immediate consequence of prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a> is this:</p> <div class="num_prop" id="SaturationOfGeneratingCofibrations"> <h6 id="corollary">Corollary</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> with all small <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K\subset Mor(\mathcal{C})</annotation></semantics></math> be a sub-<a class="existingWikiWord" href="/nlab/show/class">class</a> of its morphisms. Then every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphism">injective morphism</a>, def. <a class="maruku-ref" href="#LiftingAndExtension"></a>, has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a>, def. <a class="maruku-ref" href="#LiftingAndExtension"></a>, against all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a>, def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a> and their <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a>, remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>.</p> </div> <div class="num_remark" id="RetractsOfMorphisms"> <h6 id="remark_3">Remark</h6> <p>By a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\longrightarrow} Y</annotation></semantics></math> in some <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> we mean a retract of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as an object in the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math>, hence a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>⟶</mo><mi>g</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \stackrel{g}{\longrightarrow} B</annotation></semantics></math> such that in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math> there is a factorization of the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>g</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>g</mi><mo>⟶</mo><mi>f</mi><mo>⟶</mo><mi>g</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> id_g \;\colon\; g \longrightarrow f \longrightarrow g \,. </annotation></semantics></math></div> <p>This means equivalently that in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>id</mi> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>B</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ id_A \colon & A &\longrightarrow& X &\longrightarrow& A \\ & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ id_B \colon & B &\longrightarrow& Y &\longrightarrow& B } \,. </annotation></semantics></math></div></div> <div class="num_lemma" id="RetractPreservesIsomorphism"> <h6 id="lemma">Lemma</h6> <p>In every <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> the class of <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> is preserved under retracts in the sense of remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>For</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>id</mi> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>B</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ id_A \colon & A &\longrightarrow& X &\longrightarrow& A \\ & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ id_B \colon & B &\longrightarrow& Y &\longrightarrow& B } \,. </annotation></semantics></math></div> <p>a retract diagram and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \overset{f}{\to} Y</annotation></semantics></math> an isomorphism, the inverse to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>g</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \overset{g}{\to} B</annotation></semantics></math> is given by the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & & & X & \longrightarrow & A \\ & && \uparrow^{\mathrlap{f^{-1}}} && \\ & B & \longrightarrow& Y&& } \,. </annotation></semantics></math></div></div> <p>More generally:</p> <div class="num_prop" id="WeakEquivalencesAreClosedUnderRetracts"> <h6 id="proposition_2">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> in the sense of def. <a class="maruku-ref" href="#ModelCategory"></a>, then its class of weak equivalences is closed under forming <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> (in the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a>, see remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>).</p> </div> <p>(<a href="model+category#Joyal">Joyal, prop. E.1.3</a>)</p> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>w</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ id \colon & A &\longrightarrow& X &\longrightarrow& A \\ & {}^{\mathllap{f}} \downarrow && \downarrow^{\mathrlap{w}} && \downarrow^{\mathrlap{f}} \\ id \colon & B &\longrightarrow& Y &\longrightarrow& B } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in the given model category, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">w \in W</annotation></semantics></math> a weak equivalence. We need to show that then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f \in W</annotation></semantics></math>.</p> <p>First consider the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>Fib</mi></mrow><annotation encoding="application/x-tex">f \in Fib</annotation></semantics></math>.</p> <p>In this case, factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> as a cofibration followed by an acyclic fibration. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">w \in W</annotation></semantics></math> and by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) this is even a factorization through an acyclic cofibration followed by an acyclic fibration. Hence we obtain a commuting diagram of the following form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mi>s</mi></mover></mtd> <mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>t</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>A</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{id}} \\ id \colon & A' &\overset{s}{\longrightarrow}& X' &\overset{\phantom{AA}t\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{f}}_{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> is uniquely defined and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> is any lift of the top middle vertical acyclic cofibration against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. This now exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as a retract of an acyclic fibration. These are closed under retract by prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a>.</p> <p>Now consider the general case. Factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as an acyclic cofibration followed by a fibration and form the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> in the top left square of the following diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>A</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{\in W \cap Cof}}\downarrow &(po)& \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{\in W \cap Cof}} \\ id \colon & A' &\overset{}{\longrightarrow}& X' &\overset{\phantom{AA}\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W }} && \downarrow^{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,, </annotation></semantics></math></div> <p>where the other three squares are induced by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the pushout, as is the identification of the middle horizontal composite as the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">A'</annotation></semantics></math>. Since acyclic cofibrations are closed under forming pushouts by prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a>, the top middle vertical morphism is now an acyclic fibration, and hence by assumption and by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> so is the middle bottom vertical morphism.</p> <p>Thus the previous case now gives that the bottom left vertical morphism is a weak equivalence, and hence the total left vertical composite is.</p> </div> <div class="num_lemma" id="RetractArgument"> <h6 id="lemma_2">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/retract+argument">retract argument</a>)</strong></p> <p>Consider a <a class="existingWikiWord" href="/nlab/show/composition">composite</a> morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mi>i</mi></mover><mi>A</mi><mover><mo>⟶</mo><mi>p</mi></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} Y \,. </annotation></semantics></math></div> <ol> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>We discuss the first statement, the second is <a class="existingWikiWord" href="/nlab/show/formal+duality">formally dual</a>.</p> <p>Write the factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{i}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{p}} \\ Y &= & Y } \,. </annotation></semantics></math></div> <p>By the assumed <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> there exists a diagonal filler <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> making a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{i}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow &{}^{\mathllap{g}}\nearrow& \downarrow^{\mathrlap{p}} \\ Y &= & Y } \,. </annotation></semantics></math></div> <p>By rearranging this diagram a little, it is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>Y</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mi>g</mi></munder></mtd> <mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mi>p</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & X &=& X \\ & {}^{\mathllap{f}}\downarrow && {}^{\mathllap{i}}\downarrow \\ id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>Completing this to the right, this yields a diagram exhibiting the required retract according to remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>id</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>Y</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mi>g</mi></munder></mtd> <mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mi>p</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ id_X \colon & X &=& X &=& X \\ & {}^{\mathllap{f}}\downarrow && {}^{\mathllap{i}}\downarrow && {}^{\mathllap{f}}\downarrow \\ id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}& Y } \,. </annotation></semantics></math></div></div> <p><strong>Small object argument</strong></p> <p>Given a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \subset Mor(\mathcal{C})</annotation></semantics></math> of morphisms in some <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, a natural question is how to factor any given morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \longrightarrow Y</annotation></semantics></math> through a relative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-cell complex, def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a>, followed by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphism">injective morphism</a>, def. <a class="maruku-ref" href="#RightLiftingProperty"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>C</mi><mi>cell</mi></mrow></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>C</mi><mi>inj</mi></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \stackrel{\in C cell}{\longrightarrow} \hat X \stackrel{\in C inj}{\longrightarrow} Y \,. </annotation></semantics></math></div> <p>A first approximation to such a factorization turns out to be given simply by forming <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>=</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\hat X = X_1</annotation></semantics></math> by attaching <strong>all</strong> possible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-cells to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Namely let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>dom</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>cod</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> (C/f) \coloneqq \left\{ \array{ dom(c) &\stackrel{}{\longrightarrow}& X \\ {}^{\mathllap{c\in C}}\downarrow && \downarrow^{\mathrlap{f}} \\ cod(c) &\longrightarrow& Y } \right\} </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <strong>all</strong> ways to find a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-cell attachment in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, and consider the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat X</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> over all these:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>dom</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>c</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>cod</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow &(po)& \downarrow^{\mathrlap{}} \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& X_1 } \,. </annotation></semantics></math></div> <p>This gets already close to producing the intended factorization:</p> <p>First of all the resulting map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X \to X_1</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-relative cell complex, by construction.</p> <p>Second, by the fact that the coproduct is over all commuting squres to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> itself makes a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>dom</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>c</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>cod</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow && \downarrow^{\mathrlap{f}} \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& Y } </annotation></semantics></math></div> <p>and hence the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is indeed factored through that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-cell complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1</annotation></semantics></math>; we may suggestively arrange that factorizing diagram like so:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>dom</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>dom</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>c</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>c</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder><mi>cod</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{}} \\ \underset{c\in(C/f)}{\coprod} dom(c) && X_1 \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow &\nearrow& \downarrow \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>This shows that, finally, the colimiting <a class="existingWikiWord" href="/nlab/show/co-cone">co-cone</a> map – the one that now appears diagonally – <strong>almost</strong> exhibits the desired right lifting of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X_1 \to Y</annotation></semantics></math> against the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c\in C</annotation></semantics></math>. The failure of that to hold on the nose is only the fact that a horizontal map in the middle of the above diagram is missing: the diagonal map obtained above lifts not all commuting diagrams of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c\in C</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, but only those where the top morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dom</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">dom(c) \to X_1</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X \to X_1</annotation></semantics></math>.</p> <p>The idea of the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> now is to fix this only remaining problem by iterating the construction: next factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X_1 \to Y</annotation></semantics></math> in the same way into</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> X_1 \longrightarrow X_2 \longrightarrow Y </annotation></semantics></math></div> <p>and so forth. Since relative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-cell complexes are closed under composition, at stage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> the resulting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X \longrightarrow X_n</annotation></semantics></math> is still a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-cell complex, getting bigger and bigger. But accordingly, the failure of the accompanying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X_n \longrightarrow Y</annotation></semantics></math> to be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphism">injective morphism</a> becomes smaller and smaller, for it now lifts against all diagrams where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dom</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">dom(c) \longrightarrow X_n</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⟶</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_{n-1}\longrightarrow X_n</annotation></semantics></math>, which intuitively is less and less of a condition as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n-1}</annotation></semantics></math> grow larger and larger.</p> <p>The concept of <em><a class="existingWikiWord" href="/nlab/show/small+object">small object</a></em> is just what makes this intuition precise and finishes the small object argument. For the present purpose we just need the following simple version:</p> <div class="num_defn" id="ClassOfMorphismsWithSmallDomains"> <h6 id="definition_6">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category">category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \subset Mor(\mathcal{C})</annotation></semantics></math> a sub-<a class="existingWikiWord" href="/nlab/show/set">set</a> of its morphisms, say that these have <em>small <a class="existingWikiWord" href="/nlab/show/domains">domains</a></em> if there is an <a class="existingWikiWord" href="/nlab/show/ordinal">ordinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> (def. <a class="maruku-ref" href="#PosetsWosetTosetsAndOrdinals"></a>) such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c\in C</annotation></semantics></math> and for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> given by a <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> (def. <a class="maruku-ref" href="#TransfiniteComposition"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>→</mo><mi>⋯</mi><mo>→</mo><msub><mi>X</mi> <mi>β</mi></msub><mo>→</mo><mi>⋯</mi><mo>⟶</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \to X_1 \to X_2 \to \cdots \to X_\beta \to \cdots \longrightarrow \hat X </annotation></semantics></math></div> <p>every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dom</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">dom(c)\longrightarrow \hat X</annotation></semantics></math> factors through a stage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>β</mi></msub><mo>→</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">X_\beta \to \hat X</annotation></semantics></math> of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo><</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">\beta \lt \alpha</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mi>β</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>dom</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X_\beta \\ & \nearrow & \downarrow \\ dom(c) &\longrightarrow& \hat X } \,. </annotation></semantics></math></div></div> <p>The above discussion proves the following:</p> <div class="num_prop" id="SmallObjectArgument"> <h6 id="proposition_3">Proposition</h6> <p><strong>(small object argument)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> with all small <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. If a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C\subset Mor(\mathcal{C})</annotation></semantics></math> of morphisms has all small domains in the sense of def. <a class="maruku-ref" href="#ClassOfMorphismsWithSmallDomains"></a>, then every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo></mrow><annotation encoding="application/x-tex">f\colon X\longrightarrow </annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> factors through a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a>, def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a>, followed by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphism">injective morphism</a>, def. <a class="maruku-ref" href="#RightLiftingProperty"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>C</mi><mi>cell</mi></mrow></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>C</mi><mi>inj</mi></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \stackrel{\in C cell}{\longrightarrow} \hat X \stackrel{\in C inj}{\longrightarrow} Y \,. </annotation></semantics></math></div></div> <p>(<a href="model+category#Quillen67">Quillen 67, II.3 lemma</a>)</p> <h3 id="homotopy">Homotopy</h3> <p>We discuss how the concept of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> is abstractly realized in <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>, def. <a class="maruku-ref" href="#ModelCategory"></a>.</p> <div class="num_defn" id="PathAndCylinderObjectsInAModelCategory"> <h6 id="definition_7">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, def. <a class="maruku-ref" href="#ModelCategory"></a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/object">object</a>.</p> <ul> <li>A <strong><a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a factorization of the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Delta_X \;\colon\; X \to X \times X</annotation></semantics></math> as</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mi>i</mi></munderover><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></munderover><mi>X</mi><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \underoverset{\in Fib}{(p_0,p_1)}{\longrightarrow} X \times X \,. </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\to Path(X)</annotation></semantics></math> is a weak equivalence and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Path(X) \to X \times X</annotation></semantics></math> is a fibration.</p> <ul> <li>A <strong><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(X)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a factorization of the <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</a> (or “fold map”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⊔</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\nabla_X \;\colon\; X \sqcup X \to X</annotation></semantics></math> as</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⊔</mo><mi>X</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Cof</mi></mrow><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></munderover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mi>p</mi></munderover><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla_X \;\colon\; X \sqcup X \underoverset{\in Cof}{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{\in W}{p}{\longrightarrow} X \,. </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Cyl(X) \to X</annotation></semantics></math> is a weak equivalence. and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi><mo>→</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sqcup X \to Cyl(X)</annotation></semantics></math> is a cofibration.</p> </div> <div class="num_remark" id="RemarkOnChoicesOfNonGoodPathAndCylinderObjects"> <h6 id="remark_4">Remark</h6> <p>For every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> in a model category, a cylinder object and a path space object according to def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a> exist: the factorization axioms guarantee that there exists</p> <ol> <li> <p>a factorization of the <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Cof</mi></mrow></mover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> \nabla_X \;\colon\; X \sqcup X \overset{\in Cof}{\longrightarrow} Cyl(X) \overset{\in W \cap Fib}{\longrightarrow} X </annotation></semantics></math></div></li> <li> <p>a factorization of the diagonal as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mover><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_X \;\colon\; X \overset{\in W \cap Cof}{\longrightarrow} Path(X) \overset{\in Fib}{\longrightarrow} X \times X \,. </annotation></semantics></math></div></li> </ol> <p>The cylinder and path space objects obtained this way are actually better than required by def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>: in addition to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Cyl(X)\to X</annotation></semantics></math> being just a weak equivalence, for these this is actually an acyclic fibration, and dually in addition to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\to Path(X)</annotation></semantics></math> being a weak equivalence, for these it is actually an acyclic cofibrations.</p> <p>Some authors call cylinder/path-space objects with this extra property “very good” cylinder/path-space objects, respectively.</p> <p>One may also consider dropping a condition in def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>: what mainly matters is the weak equivalence, hence some authors take cylinder/path-space objects to be defined as in def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a> but without the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi><mo>→</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sqcup X\to Cyl(X)</annotation></semantics></math> is a cofibration and without the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Path(X) \to X</annotation></semantics></math> is a fibration. Such authors would then refer to the concept in def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a> as “good” cylinder/path-space objects.</p> <p>The terminology in def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a> follows the original (<a href="model+category#Quillen67">Quillen 67, I.1 def. 4</a>). With the induced concept of left/right homotopy below in def. <a class="maruku-ref" href="#LeftAndRightHomotopyInAModelCategory"></a>, this admits a quick derivation of the key facts in the following, as we spell out below.</p> </div> <div class="num_lemma" id="ComponentMapsOfCylinderAndPathSpaceInGoodSituation"> <h6 id="lemma_3">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> is cofibrant, then for every <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, not only is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⊔</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(i_0,i_1) \colon X \sqcup X \to X</annotation></semantics></math> a cofibration, but each</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i_0, i_1 \colon X \longrightarrow Cyl(X) </annotation></semantics></math></div> <p>is an acyclic cofibration separately.</p> <p>Dually, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> is fibrant, then for every <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, not only is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(p_0,p_1) \colon Path(X)\to X \times X</annotation></semantics></math> a cofibration, but each</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> p_0, p_1 \colon Path(X) \longrightarrow X </annotation></semantics></math></div> <p>is an acyclic fibration separately.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>We discuss the case of the path space object. The other case is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> <p>First, that the component maps are weak equivalences follows generally: by definition they have a <a class="existingWikiWord" href="/nlab/show/right+inverse">right inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Path(X) \to X</annotation></semantics></math> and so this follows by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>).</p> <p>But if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is fibrant, then also the two projection maps out of the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times X \to X</annotation></semantics></math> are fibrations, because they are both pullbacks of the fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to \ast</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X\times X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow \\ X &\longrightarrow& \ast } \,. </annotation></semantics></math></div> <p>hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p_i \colon Path(X)\to X \times X \to X</annotation></semantics></math> is the composite of two fibrations, and hence itself a fibration, by prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a>.</p> </div> <p>Path space objects are very non-unique as objects up to isomorphism:</p> <div class="num_example" id="ComposedPathSpaceObjects"> <h6 id="example">Example</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> is a fibrant object in a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, def. <a class="maruku-ref" href="#ModelCategory"></a>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path_1(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path_2(X)</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/path+space+objects">path space objects</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, then the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>Path</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path_1(X) \times_X Path_2(X)</annotation></semantics></math> is another path space object for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: the pullback square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>Δ</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Path</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mi>X</mi></munder><msub><mi>Path</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>Path</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>Path</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>Δ</mi> <mi>X</mi></msub><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>pr</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>pr</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msubsup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>4</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{\Delta_X}{\longrightarrow}& X \times X \\ \downarrow && \downarrow \\ Path_1(X) \underset{X}{\times} Path_2(X) &\longrightarrow& Path_1(X)\times Path_2(X) \\ {}^{\mathllap{\in Fib}}\downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ X \times X \times X &\overset{(id,\Delta_X,id)}{\longrightarrow}& X \times X\times X \times X \\ \downarrow^{\mathrlap{(pr_1,pr_3)}}_{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{(p_1, p_4)}} \\ X\times X &=& X \times X } </annotation></semantics></math></div> <p>gives that the induced projection is again a fibration. Moreover, using lemma <a class="maruku-ref" href="#ComponentMapsOfCylinderAndPathSpaceInGoodSituation"></a> and <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>Path</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>Path</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to Path_1(X) \times_X Path_2(X)</annotation></semantics></math> is a weak equivalence.</p> <p>For the case of the canonical topological path space objects of def <a class="maruku-ref" href="#TopologicalPathSpace"></a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Path</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>X</mi> <mi>I</mi></msup><mo>=</mo><msup><mi>X</mi> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Path_1(X) = Path_2(X) = X^I = X^{[0,1]}</annotation></semantics></math> then this new path space object is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mi>I</mi><mo>∨</mo><mi>I</mi></mrow></msup><mo>=</mo><msup><mi>X</mi> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{I \vee I} = X^{[0,2]}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> out of the standard interval of length 2 instead of length 1.</p> </div> <div class="num_defn" id="LeftAndRightHomotopyInAModelCategory"> <h6 id="definition_8">Definition</h6> <p><strong>(abstract <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> and abstract <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g \colon X \longrightarrow Y</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> in a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>.</p> <ul> <li>A <strong>left homotopy</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><msub><mo>⇒</mo> <mi>L</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">\eta \colon f \Rightarrow_L g</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\eta \colon Cyl(X) \longrightarrow Y</annotation></semantics></math> from a <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, such that it makes this <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commute</a>:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟵</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>η</mi></mpadded></msup></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& Cyl(X) &\longleftarrow& X \\ & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}} \\ && Y } \,. </annotation></semantics></math></div> <ul> <li>A <strong>right homotopy</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><msub><mo>⇒</mo> <mi>R</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">\eta \colon f \Rightarrow_R g</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta \colon X \to Path(Y)</annotation></semantics></math> to some <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, such that this <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a>:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo>↙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>η</mi></mpadded></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟵</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X \\ & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}} \\ Y &\longleftarrow& Path(Y) &\longrightarrow& Y } \,. </annotation></semantics></math></div></div> <div class="num_lemma" id="LeftHomotopyWithCofibrantDomainImpliesRightHomotopyAndDually"> <h6 id="lemma_4">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g \colon X \to Y</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> in a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>.</p> <ol> <li> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be cofibrant. If there is a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo>⇒</mo> <mi>L</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow_L g</annotation></semantics></math> then there is also a <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo>⇒</mo> <mi>R</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow_R g</annotation></semantics></math> (def. <a class="maruku-ref" href="#LeftAndRightHomotopyInAModelCategory"></a>) with respect to any chosen path space object.</p> </li> <li> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be fibrant. If there is a <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo>⇒</mo> <mi>R</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow_R g</annotation></semantics></math> then there is also a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo>⇒</mo> <mi>L</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow_L g</annotation></semantics></math> with respect to any chosen cylinder object.</p> </li> </ol> <p>In particular if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is cofibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is fibrant, then by going back and forth it follows that every left homotopy is exhibited by every cylinder object, and every right homotopy is exhibited by every path space object.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>We discuss the first case, the second is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\eta \colon Cyl(X) \longrightarrow Y</annotation></semantics></math> be the given left homotopy. Lemma <a class="maruku-ref" href="#ComponentMapsOfCylinderAndPathSpaceInGoodSituation"></a> implies that we have a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> in the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>i</mi><mo>∘</mo><mi>f</mi></mrow></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>h</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>p</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>Y</mi><mo>×</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{i \circ f}{\longrightarrow}& Path(Y) \\ {}^{\mathllap{i_0}}_{\mathllap{\in W \cap Cof}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}} \\ Cyl(X) &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y } \,, </annotation></semantics></math></div> <p>where on the right we have the chosen path space object. Now the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo stretchy="false">˜</mo></mover><mo>≔</mo><mi>h</mi><mo>∘</mo><msub><mi>i</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tilde \eta \coloneqq h \circ i_1</annotation></semantics></math> is a right homotopy as required:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>h</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>p</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>Y</mi><mo>×</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && && Path(Y) \\ && &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}} \\ X &\overset{i_1}{\longrightarrow}& Cyl(X) &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y } \,. </annotation></semantics></math></div></div> <div class="num_prop" id="BetweenCofibFibLeftAndRightHomotopyAreEquivalentEquivalenceRelations"> <h6 id="proposition_4">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a cofibrant object in a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a>, then the <a class="existingWikiWord" href="/nlab/show/relations">relations</a> of <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo>⇒</mo> <mi>L</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow_L g</annotation></semantics></math> and of <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo>⇒</mo> <mi>R</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow_R g</annotation></semantics></math> (def. <a class="maruku-ref" href="#LeftAndRightHomotopyInAModelCategory"></a>) on the <a class="existingWikiWord" href="/nlab/show/hom+set">hom set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(X,Y)</annotation></semantics></math> coincide and are both <a class="existingWikiWord" href="/nlab/show/equivalence+relations">equivalence relations</a>.</p> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>That both relations coincide under the (co-)fibrancy assumption follows directly from lemma <a class="maruku-ref" href="#LeftHomotopyWithCofibrantDomainImpliesRightHomotopyAndDually"></a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/symmetric+relation">symmetry</a> and <a class="existingWikiWord" href="/nlab/show/reflexive+relation">reflexivity</a> of the relation is obvious.</p> <p>That right homotopy (hence also left homotopy) with domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/transitive+relation">transitive relation</a> follows from using example <a class="maruku-ref" href="#ComposedPathSpaceObjects"></a> to compose path space objects.</p> </div> <h3 id="TheHomotopyCategory">The homotopy category</h3> <p>We discuss the construction that takes a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, def. <a class="maruku-ref" href="#ModelCategory"></a>, and then universally forces all its <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> into actual <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>.</p> <div class="num_defn" id="HomotopyCategoryOfAModelCategory"> <h6 id="definition_9">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, def. <a class="maruku-ref" href="#ModelCategory"></a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are those objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> which are both <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant</a> and <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are the <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, hence the <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of morphism under the equivalence relation of prop. <a class="maruku-ref" href="#BetweenCofibFibLeftAndRightHomotopyAreEquivalentEquivalenceRelations"></a>;</p> </li> </ul> <p>and whose <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation is given on representatives by composition in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p>This is, up to <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>, the <strong><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of the model category</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>Def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a> is well defined, in that composition of morphisms between fibrant-cofibrant objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> indeed passes to <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a>.</p> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>Fix any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \overset{f}{\to} Y</annotation></semantics></math> between fibrant-cofibrant objects. Then for precomposition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> (-) \circ [f] \;\colon\; Hom_{Ho(\mathcal{C})}(Y,Z) \to Hom_{Ho(\mathcal{C}(X,Z))} </annotation></semantics></math></div> <p>to be well defined, we need that with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>∼</mo><mi>h</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">(g\sim h)\;\colon\; Y \to Z</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mi>g</mi><mo>∼</mo><mi>f</mi><mi>h</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">(f g \sim f h)\;\colon\; X \to Z</annotation></semantics></math>. But by prop <a class="maruku-ref" href="#BetweenCofibFibLeftAndRightHomotopyAreEquivalentEquivalenceRelations"></a> we may take the homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> to be exhibited by a right homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta \colon Y \to Path(Z)</annotation></semantics></math>, for which case the statement is evident from this diagram:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>h</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && && Z \\ && & {}^{\mathllap{g}}\nearrow & \uparrow^{\mathrlap{p_1}} \\ X &\overset{f}{\longrightarrow} & Y &\overset{\eta}{\longrightarrow}& Path(Z) \\ && & {}_{\mathllap{h}}\searrow & \downarrow_{\mathrlap{p_0}} \\ && && Z } \,. </annotation></semantics></math></div> <p>For postcomposition we may choose to exhibit homotopy by left homotopy and argue <a class="existingWikiWord" href="/nlab/show/formal+dual">dually</a>.</p> </div> <p>We now spell out that def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a> indeed satisfies the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> that defines the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> of a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> at its weak equivalences.</p> <div class="num_lemma" id="WhiteheadTheoremInModelCategories"> <h6 id="lemma_5">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a> in <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>. A <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> between two objects which are both <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant</a> and <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant</a> is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> <a class="maruku-eqref" href="#eq:HomotopyEquivalenceCondition">(1)</a>.</p> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>By the factorization axioms in the model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>), every weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \longrightarrow Y</annotation></semantics></math> factors through an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> as an acyclic cofibration followed by an acyclic fibration. In particular it follows that with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> both fibrant and cofibrant, so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math>, and hence it is sufficient to prove that acyclic (co-)fibrations between such objects are homotopy equivalences.</p> <p>So let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> be an acyclic fibration between fibrant-cofibrant objects, the case of acyclic cofibrations is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>. Then in fact it has a genuine <a class="existingWikiWord" href="/nlab/show/right+inverse">right inverse</a> given by a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f^{-1}</annotation></semantics></math> in the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>∅</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msup><mo>↗</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi><mo>∩</mo><mi>W</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \emptyset &\rightarrow& X \\ {}^{\mathllap{\in cof}}\downarrow &{}^{{f^{-1}}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib \cap W}} \\ X &=& X } \,. </annotation></semantics></math></div> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f^{-1}</annotation></semantics></math> is also a <a class="existingWikiWord" href="/nlab/show/left+inverse">left inverse</a> up to <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(X)</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>), hence a factorization of the <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as a cofibration followed by a an acyclic fibration</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow></mover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>p</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \sqcup X \stackrel{\iota_X}{\longrightarrow} Cyl(X) \stackrel{p}{\longrightarrow} X </annotation></semantics></math></div> <p>and consider the <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>⊔</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow></mpadded></msup><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Cof</mi></mrow></mpadded></msub><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>f</mi><mo>∘</mo><mi>p</mi></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X \sqcup X &\stackrel{(f^{-1}\circ f, id)}{\longrightarrow}& X \\ {}^{\mathllap{\iota_X}}{}_{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(X) &\underset{f\circ p}{\longrightarrow}& Y } \,, </annotation></semantics></math></div> <p>which <a class="existingWikiWord" href="/nlab/show/commuting+square">commutes</a> due to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f^{-1}</annotation></semantics></math> being a genuine right inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. By construction, this <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> now admits a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>, and that constitutes a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>f</mi><msub><mo>⇒</mo> <mi>L</mi></msub><mi>id</mi></mrow><annotation encoding="application/x-tex">\eta \colon f^{-1}\circ f \Rightarrow_L id</annotation></semantics></math>.</p> </div> <div class="num_defn" id="FibrantCofibrantReplacementFunctorToHomotopyCategory"> <h6 id="definition_10">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/fibrant+resolution">fibrant resolution</a> and <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, consider a <em>choice</em> for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> of</p> <ol> <li> <p>a factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>∅</mi><munderover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mo>∈</mo><mi>Cof</mi><mphantom><mi>A</mi></mphantom></mrow><mrow><msub><mi>i</mi> <mi>X</mi></msub></mrow></munderover><mi>Q</mi><mi>X</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow><mrow><msub><mi>p</mi> <mi>X</mi></msub></mrow></munderover><mi>X</mi></mrow><annotation encoding="application/x-tex"> \emptyset \underoverset{\phantom{A}\in Cof\phantom{A}}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_X}{\longrightarrow} X </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/initial+object">initial morphism</a> (Def. <a class="maruku-ref" href="#InitialObject"></a>), such that when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is already cofibrant then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>X</mi></msub><mo>=</mo><msub><mi>id</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">p_X = id_X</annotation></semantics></math>;</p> </li> <li> <p>a factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow><mrow><msub><mi>j</mi> <mi>X</mi></msub></mrow></munderover><mi>P</mi><mi>X</mi><munderover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mo>∈</mo><mi>Fib</mi><mphantom><mi>A</mi></mphantom></mrow><mrow><msub><mi>q</mi> <mi>X</mi></msub></mrow></munderover><mo>*</mo></mrow><annotation encoding="application/x-tex"> X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} P X \underoverset{\phantom{A} \in Fib \phantom{A}}{q_X}{\longrightarrow} \ast </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal morphism</a> (Def. <a class="maruku-ref" href="#InitialObject"></a>), such that when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is already fibrant then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>X</mi></msub><mo>=</mo><msub><mi>id</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">j_X = id_X</annotation></semantics></math>.</p> </li> </ol> <p>Write then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \gamma_{P,Q} \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C}) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> to the homotopy category, def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>, which sends an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Q</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">P Q X</annotation></semantics></math> and sends a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/homotopy+class">homotopy class</a> of the result of first lifting in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>∅</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Q</mi><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>X</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mi>Q</mi><mi>f</mi></mrow></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>Y</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Q</mi><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>f</mi><mo>∘</mo><msub><mi>p</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \emptyset &\longrightarrow& Q Y \\ {}^{\mathllap{i_X}}\downarrow &{}^{Q f}\nearrow& \downarrow^{\mathrlap{p_Y}} \\ Q X &\underset{f\circ p_X}{\longrightarrow}& Y } </annotation></semantics></math></div> <p>and then lifting (here: <a class="existingWikiWord" href="/nlab/show/extension">extending</a>) in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Q</mi><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>Y</mi></mrow></msub><mo>∘</mo><mi>Q</mi><mi>f</mi></mrow></mover></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>X</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mi>P</mi><mi>Q</mi><mi>f</mi></mrow></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>q</mi> <mrow><mi>Q</mi><mi>Y</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi><mi>Q</mi><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Q X &\overset{j_{Q Y} \circ Q f}{\longrightarrow}& P Q Y \\ {}^{\mathllap{j_{Q X}}}\downarrow &{}^{P Q f}\nearrow& \downarrow^{\mathrlap{q_{Q Y}}} \\ P Q X &\longrightarrow& \ast } \,. </annotation></semantics></math></div></div> <div class="num_lemma" id="ConstructionOfLocalizationFunctorForModelCategoryIsWellDefined"> <h6 id="lemma_6">Lemma</h6> <p>The construction in def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a> is indeed well defined.</p> </div> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>First of all, the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Q</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">P Q X</annotation></semantics></math> is indeed both fibrant and cofibrant (as well as related by a <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> of weak equivalences to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>∅</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Cof</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Cof</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Q</mi><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></munder></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></munder></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \emptyset \\ {}^{\mathllap{\in Cof}}\downarrow & \searrow^{\mathrlap{\in Cof}} \\ Q X &\underset{\in W \cap Cof}{\longrightarrow}& P Q X &\underset{\in Fib}{\longrightarrow}& \ast \\ {}^{\mathllap{\in W}}\downarrow \\ X } \,. </annotation></semantics></math></div> <p>Now to see that the image on morphisms is well defined. First observe that any two choices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Q</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">(Q f)_{i}</annotation></semantics></math> of the first lift in the definition are left homotopic to each other, exhibited by lifting in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Q</mi><mi>X</mi><mo>⊔</mo><mi>Q</mi><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Q</mi><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Cof</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>Y</mi></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>f</mi><mo>∘</mo><msub><mi>p</mi> <mi>X</mi></msub><mo>∘</mo><msub><mi>σ</mi> <mrow><mi>Q</mi><mi>X</mi></mrow></msub></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Q X \sqcup Q X &\stackrel{((Q f)_1, (Q f)_2 )}{\longrightarrow}& Q Y \\ {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_{Y}}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(Q X) &\underset{f \circ p_{X} \circ \sigma_{Q X}}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>Hence also the composites <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>Y</mi></mrow></msub><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>Q</mi> <mi>f</mi></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">j_{Q Y}\circ (Q_f)_i </annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopic</a> to each other, and since their domain is cofibrant, then by lemma <a class="maruku-ref" href="#LeftHomotopyWithCofibrantDomainImpliesRightHomotopyAndDually"></a> they are also <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopic</a> by a right homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>. This implies finally, by lifting in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Q</mi><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>κ</mi></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi><mi>Q</mi><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>,</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>Y</mi><mo>×</mo><mi>P</mi><mi>Q</mi><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Q X &\overset{\kappa}{\longrightarrow}& Path(P Q Y) \\ {}^{\mathllap{\in W \cap Cof}}\downarrow && \downarrow^{\mathrlap{\in Fib}} \\ P Q X &\underset{(R (Q f)_1, P (Q f)_2)}{\longrightarrow}& P Q Y \times P Q Y } </annotation></semantics></math></div> <p>that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">P (Q f)_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">P (Q f)_2</annotation></semantics></math> are right homotopic, hence that indeed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Q</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">P Q f</annotation></semantics></math> represents a well-defined <a class="existingWikiWord" href="/nlab/show/homotopy+class">homotopy class</a>.</p> <p>Finally to see that the assignment is indeed <a class="existingWikiWord" href="/nlab/show/functor">functorial</a>, observe that the commutativity of the lifting diagrams for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">Q f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Q</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">P Q f</annotation></semantics></math> imply that also the following diagram commutes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟵</mo><mrow><msub><mi>p</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>Q</mi><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>X</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>Q</mi><mi>f</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>P</mi><mi>Q</mi><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟵</mo><mrow><msub><mi>p</mi> <mi>y</mi></msub></mrow></munder></mtd> <mtd><mi>Q</mi><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>Y</mi></mrow></msub></mrow></munder></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& P Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{P Q f}} \\ Y &\underset{p_y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& P Q Y } \,. </annotation></semantics></math></div> <p>Now from the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟵</mo><mrow><msub><mi>p</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>Q</mi><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>X</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>Q</mi><mi>f</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>P</mi><mi>Q</mi><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟵</mo><mrow><msub><mi>p</mi> <mi>Y</mi></msub></mrow></munder></mtd> <mtd><mi>Q</mi><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>Y</mi></mrow></msub></mrow></munder></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>Q</mi><mi>g</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>P</mi><mi>Q</mi><mi>g</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Z</mi></mtd> <mtd><munder><mo>⟵</mo><mrow><msub><mi>p</mi> <mi>Z</mi></msub></mrow></munder></mtd> <mtd><mi>Q</mi><mi>Z</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>Z</mi></mrow></msub></mrow></munder></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& P Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{P Q f}} \\ Y &\underset{p_Y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& P Q Y \\ {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{Q g}} && \downarrow^{\mathrlap{P Q g}} \\ Z &\underset{p_Z}{\longleftarrow}& Q Z &\underset{j_{Q Z}}{\longrightarrow}& P Q Z } </annotation></semantics></math></div> <p>one sees that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P Q g)\circ (P Q f)</annotation></semantics></math> is a lift of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">g \circ f</annotation></semantics></math> and hence the same argument as above gives that it is homotopic to the chosen <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Q</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P Q(g \circ f)</annotation></semantics></math>.</p> </div> <p>For the following, recall the concept of <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> between <a class="existingWikiWord" href="/nlab/show/functors">functors</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>,</mo><mi>G</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F, G \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}</annotation></semantics></math> two functors, then a <em><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo>⇒</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\eta \colon F \Rightarrow G</annotation></semantics></math> is for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in Obj(\mathcal{C})</annotation></semantics></math> a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_c \colon F(c) \longrightarrow G(c)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>, such that for each morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f \colon c_1 \to c_2</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> the following is a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub></mrow></msub></mrow></mover></mtd> <mtd><mi>G</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub></mrow></munder></mtd> <mtd><mi>G</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F(c_1) &\overset{\eta_{c_1}}{\longrightarrow}& G(c_1) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} \\ F(c_2) &\underset{\eta_{c_2}}{\longrightarrow}& G(c_2) } \,. </annotation></semantics></math></div> <p>Such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is called a <em><a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></em> if its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\eta_c</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>.</p> <div class="num_defn" id="HomotopyCategoryOfACategoryWithWeakEquivalences"> <h6 id="definition_11">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/localization+of+a+category">localization of a category</a> <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>, its <strong><a class="existingWikiWord" href="/nlab/show/localization">localization</a> at the weak equivalences</strong> is, if it exists,</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/category">category</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}[W^{-1}]</annotation></semantics></math></p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>𝒞</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}] </annotation></semantics></math></div></li> </ol> <p>such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> sends weak equivalences to <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/universal+property">universal with this property</a>, in that:</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>⟶</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C} \longrightarrow D</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/functor">functor</a> out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> into any <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> takes weak equivalences to <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, it factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> up to a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>F</mi></mover></mtd> <mtd></mtd> <mtd><mi>D</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>γ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>ρ</mi></msup></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mover><mi>F</mi><mo stretchy="false">˜</mo></mover></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C} && \overset{F}{\longrightarrow} && D \\ & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}} \\ && Ho(\mathcal{C}) } </annotation></semantics></math></div> <p>and this factorization is unique up to unique isomorphism, in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\tilde F_1, \rho_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\tilde F_2, \rho_2)</annotation></semantics></math> two such factorizations, then there is a unique <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo lspace="verythinmathspace">:</mo><msub><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub><mo>⇒</mo><msub><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\kappa \colon \tilde F_1 \Rightarrow \tilde F_2</annotation></semantics></math> making the evident diagram of natural isomorphisms commute.</p> </li> </ol> </div> <div class="num_theorem" id="UniversalPropertyOfHomotopyCategoryOfAModelCategory"> <h6 id="theorem">Theorem</h6> <p><strong>(convenient <a class="existingWikiWord" href="/nlab/show/localization">localization</a> of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\gamma_{P,Q}</annotation></semantics></math> in def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a> (for any choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>) exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math> as indeed being the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> of the underlying <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> at its weak equivalences, in the sense of def. <a class="maruku-ref" href="#HomotopyCategoryOfACategoryWithWeakEquivalences"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>𝒞</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>γ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>𝒞</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C} &=& \mathcal{C} \\ {}^{\mathllap{\gamma_{P,Q}}}\downarrow && \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}) &\simeq& \mathcal{C}[W^{-1}] } \,. </annotation></semantics></math></div></div> <p>(<a href="model+category#Quillen67">Quillen 67, I.1 theorem 1</a>)</p> <div class="proof" id="ProofOfUniversalPropertyOfHomotopyCategoryOfAModelCategory"> <h6 id="proof_11">Proof</h6> <p>First, to see that that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\gamma_{P,Q}</annotation></semantics></math> indeed takes weak equivalences to isomorphisms: By <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) applied to the <a class="existingWikiWord" href="/nlab/show/commuting+diagrams">commuting diagrams</a> shown in the proof of lemma <a class="maruku-ref" href="#ConstructionOfLocalizationFunctorForModelCategoryIsWellDefined"></a>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Q</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">P Q f</annotation></semantics></math> is a weak equivalence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><munderover><mo>⟵</mo><mo>≃</mo><mrow><msub><mi>p</mi> <mi>X</mi></msub></mrow></munderover></mtd> <mtd><mi>Q</mi><mi>X</mi></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>X</mi></mrow></msub></mrow></munderover></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>Q</mi><mi>f</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>P</mi><mi>Q</mi><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><munderover><mo>⟵</mo><mrow><msub><mi>p</mi> <mi>y</mi></msub></mrow><mo>≃</mo></munderover></mtd> <mtd><mi>Q</mi><mi>Y</mi></mtd> <mtd><munderover><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>Q</mi><mi>Y</mi></mrow></msub></mrow><mo>≃</mo></munderover></mtd> <mtd><mi>P</mi><mi>Q</mi><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\underoverset{\simeq}{p_X}{\longleftarrow}& Q X &\underoverset{\simeq}{j_{Q X}}{\longrightarrow}& P Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{P Q f}} \\ Y &\underoverset{p_y}{\simeq}{\longleftarrow}& Q Y &\underoverset{j_{Q Y}}{\simeq}{\longrightarrow}& P Q Y } </annotation></semantics></math></div> <p>With this the “Whitehead theorem for model categories”, lemma <a class="maruku-ref" href="#WhiteheadTheoremInModelCategories"></a>, implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>Q</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">P Q f</annotation></semantics></math> represents an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math>.</p> <p>Now let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>⟶</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C}\longrightarrow D</annotation></semantics></math> be any functor that sends weak equivalences to isomorphisms. We need to show that it factors as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>F</mi></mover></mtd> <mtd></mtd> <mtd><mi>D</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>γ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>ρ</mi></msup></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mover><mi>F</mi><mo stretchy="false">˜</mo></mover></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C} && \overset{F}{\longrightarrow} && D \\ & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}} \\ && Ho(\mathcal{C}) } </annotation></semantics></math></div> <p>uniquely up to unique <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>. Now by construction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> in def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\gamma_{P,Q}</annotation></semantics></math> is the identity on the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of fibrant-cofibrant objects. It follows that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde F</annotation></semantics></math> exists at all, it must satisfy for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\to} Y</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> both fibrant and cofibrant that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \tilde F([f]) \simeq F(f) \,, </annotation></semantics></math></div> <p>(hence in particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde F(\gamma_{P,Q}(f)) = F(P Q f)</annotation></semantics></math>).</p> <p>But by def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a> that already fixes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde F</annotation></semantics></math> on all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math>, up to unique <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>. Hence it only remains to check that with this definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde F</annotation></semantics></math> there exists any <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> filling the diagram above.</p> <p>To that end, apply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to the above <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> to obtain</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟵</mo><mi>iso</mi><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mi>iso</mi><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>Q</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mrow></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟵</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>y</mi></msub><mo stretchy="false">)</mo></mrow><mi>iso</mi></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>Q</mi><mi>Y</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mi>iso</mi></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F(X) &\underoverset{iso}{F(p_X)}{\longleftarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(P Q X) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{F(Q f)}} && \downarrow^{\mathrlap{F(P Q f)}} \\ F(Y) &\underoverset{F(p_y)}{iso}{\longleftarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(P Q Y) } \,. </annotation></semantics></math></div> <p>Here now all horizontal morphisms are <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, by assumption on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>. It follows that defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>X</mi></msub><mo>≔</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>Q</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo><mo>∘</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>X</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\rho_X \coloneqq F(j_{Q X}) \circ F(p_X)^{-1}</annotation></semantics></math> makes the required natural isomorphism:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>ρ</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mi>iso</mi><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>X</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mi>iso</mi><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>Q</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mrow></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mover><mi>F</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>ρ</mi> <mi>Y</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>y</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><mi>iso</mi></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>Q</mi><mi>Y</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mi>iso</mi></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mi>Q</mi><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mover><mi>F</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \rho_X \colon & F(X) &\underoverset{iso}{F(p_X)^{-1}}{\longrightarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(P Q X) &=& \tilde F(\gamma_{P,Q}(X)) \\ & {}^{\mathllap{F(f)}}\downarrow && && \downarrow^{\mathrlap{F(P Q f)}} && \downarrow^{\tilde F(\gamma_{P,Q}(f))} \\ \rho_Y\colon& F(Y) &\underoverset{F(p_y)^{-1}}{iso}{\longrightarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(P Q Y) &=& \tilde F(\gamma_{P,Q}(X)) } \,. </annotation></semantics></math></div></div> <div class="num_remark" id="EssentialUniquenessOfLocalizationFunctorOfModelCategory"> <h6 id="remark_5">Remark</h6> <p>Due to theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a> we may suppress the choices of cofibrant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> and fibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> in def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a> and just speak of <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/localization+functor">localization functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C}) </annotation></semantics></math></div> <p>up to <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>.</p> </div> <p>In general, the localization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}[W^{-1}]</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},W)</annotation></semantics></math> (def. <a class="maruku-ref" href="#HomotopyCategoryOfACategoryWithWeakEquivalences"></a>) may invert <em>more</em> morphisms than just those in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>. However, if the category admits the structure of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>W</mi><mo>,</mo><mi>Cof</mi><mo>,</mo><mi>Fib</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},W,Cof,Fib)</annotation></semantics></math>, then its localization precisely only inverts the weak equivalences:</p> <div class="num_prop" id="MorphismIsWeakEquivalenceIfIsoInHomotopyCategoryForQuillen"> <h6 id="proposition_6">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/localization">localization</a> of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> inverts precisely the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> (def. <a class="maruku-ref" href="#ModelCategory"></a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})</annotation></semantics></math> be its <a class="existingWikiWord" href="/nlab/show/localization">localization</a> functor (def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>, theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a>). Then a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma(f)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="Simplicial+homotopy+Theory">Goerss-Jardine 96, II, prop 1.14</a>)</p> <p>While the construction of the homotopy category in def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a> combines the restriction to good (fibrant/cofibrant) objects with the passage to <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of morphisms, it is often useful to consider intermediate stages:</p> <div class="num_defn" id="FullSubcategoriesOfFibrantCofibrantObjects"> <h6 id="definition_12">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>𝒞</mi> <mrow><mi>f</mi><mi>c</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>𝒞</mi> <mi>c</mi></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>𝒞</mi> <mi>f</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>𝒞</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathcal{C}_{f c} \\ & \swarrow && \searrow \\ \mathcal{C}_c && && \mathcal{C}_f \\ & \searrow && \swarrow \\ && \mathcal{C} } </annotation></semantics></math></div> <p>for the system of <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> inclusions of:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/category+of+cofibrant+objects">category of cofibrant objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_c</annotation></semantics></math>,</p> </li> <li> <p>the category of fibrant-cofibrant objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>fc</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{fc}</annotation></semantics></math>,</p> </li> </ol> <p>all regarded a <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>), via the weak equivalences inherited from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, which we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mi>f</mi></msub><mo>,</mo><msub><mi>W</mi> <mi>f</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}_f, W_f)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mi>c</mi></msub><mo>,</mo><msub><mi>W</mi> <mi>c</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}_c, W_c)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mrow><mi>f</mi><mi>c</mi></mrow></msub><mo>,</mo><msub><mi>W</mi> <mrow><mi>f</mi><mi>c</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}_{f c}, W_{f c})</annotation></semantics></math>.</p> </div> <div class="num_remark" id="CategoriesOfFibrantObjects"> <h6 id="remark_6">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/categories+of+fibrant+objects">categories of fibrant objects</a> and <a class="existingWikiWord" href="/nlab/show/cofibration+categories">cofibration categories</a>)</strong></p> <p>Of course the subcategories in def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a> inherit more structure than just that of <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_c</annotation></semantics></math> each inherit “half” of the factorization axioms. One says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math> has the structure of a “<a class="existingWikiWord" href="/nlab/show/fibration+category">fibration category</a>” called a “Brown-<a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>”, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_c</annotation></semantics></math> has the structure of a “<a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a>”.</p> <p>We discuss properties of these categories of (co-)fibrant objects below in <em><a href="#HomotopyFiberSequences">Homotopy fiber sequences</a></em>.</p> </div> <p>The proof of theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a> immediately implies the following:</p> <div class="num_cor" id="HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects"> <h6 id="corollary_2">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, the restriction of the localization functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma\;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})</annotation></semantics></math> from def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a> (using remark <a class="maruku-ref" href="#EssentialUniquenessOfLocalizationFunctorOfModelCategory"></a>) to any of the sub-<a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a> of def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>𝒞</mi> <mrow><mi>f</mi><mi>c</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>𝒞</mi> <mi>c</mi></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>𝒞</mi> <mi>f</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>𝒞</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>γ</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathcal{C}_{f c} \\ & \swarrow && \searrow \\ \mathcal{C}_c && && \mathcal{C}_f \\ & \searrow && \swarrow \\ && \mathcal{C} \\ && \downarrow^{\mathrlap{\gamma}} \\ && Ho(\mathcal{C}) } </annotation></semantics></math></div> <p>exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math> equivalently as the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> also of these subcategories with weak equivalences, at their weak equivalences. In particular there are <a class="existingWikiWord" href="/nlab/show/equivalences+of+categories">equivalences of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>𝒞</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mo>≃</mo><msub><mi>𝒞</mi> <mi>f</mi></msub><mo stretchy="false">[</mo><msubsup><mi>W</mi> <mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">]</mo><mo>≃</mo><msub><mi>𝒞</mi> <mi>c</mi></msub><mo stretchy="false">[</mo><msubsup><mi>W</mi> <mi>c</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">]</mo><mo>≃</mo><msub><mi>𝒞</mi> <mrow><mi>f</mi><mi>c</mi></mrow></msub><mo stretchy="false">[</mo><msubsup><mi>W</mi> <mrow><mi>f</mi><mi>c</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(\mathcal{C}) \simeq \mathcal{C}[W^{-1}] \simeq \mathcal{C}_f[W_f^{-1}] \simeq \mathcal{C}_c[W_c^{-1}] \simeq \mathcal{C}_{f c}[W_{f c}^{-1}] \,. </annotation></semantics></math></div></div> <p>The following says that for computing the hom-sets in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a>, even a mixed variant of the above will do; it is sufficient that the domain is cofibrant and the codomain is fibrant:</p> <div class="num_lemma" id="HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory"> <h6 id="lemma_7">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> via mapping <a class="existingWikiWord" href="/nlab/show/cofibrant+resolutions">cofibrant resolutions</a> into <a class="existingWikiWord" href="/nlab/show/fibrant+resolutions">fibrant resolutions</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X, Y \in \mathcal{C}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> cofibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> fibrant, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P, Q</annotation></semantics></math> fibrant/cofibrant replacement functors as in def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>, then the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>P</mi><mi>X</mi><mo>,</mo><mi>Q</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mi>X</mi><mo>,</mo><mi>Q</mi><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mover><mo>⟶</mo><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><msub><mi>j</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow></mover><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex"> Hom_{Ho(\mathcal{C})}(P X,Q Y) = Hom_{\mathcal{C}}(P X, Q Y)/_{\sim} \overset{Hom_{\mathcal{C}}(j_X, p_Y)}{\longrightarrow} Hom_{\mathcal{C}}(X,Y)/_{\sim} </annotation></semantics></math></div> <p>(on <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of morphisms, well defined by prop. <a class="maruku-ref" href="#BetweenCofibFibLeftAndRightHomotopyAreEquivalentEquivalenceRelations"></a>) is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a>.</p> </div> <p>(<a href="model+category#Quillen67">Quillen 67, I.1 lemma 7</a>)</p> <div class="proof"> <h6 id="proof_12">Proof</h6> <p>We may factor the morphism in question as the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mi>X</mi><mo>,</mo><mi>Q</mi><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mover><mo>⟶</mo><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mrow><mi>P</mi><mi>X</mi></mrow></msub><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow></mover><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mover><mo>⟶</mo><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><msub><mi>j</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>id</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow></mover><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{C}}(P X, Q Y)/_{\sim} \overset{Hom_{\mathcal{C}}(id_{P X}, p_Y)/_\sim }{\longrightarrow} Hom_{\mathcal{C}}(P X, Y)/_{\sim} \overset{Hom_{\mathcal{C}}(j_X, id_Y)/_\sim}{\longrightarrow} Hom_{\mathcal{C}}(X,Y)/_{\sim} \,. </annotation></semantics></math></div> <p>This shows that it is sufficient to see that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> cofibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> fibrant, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Q</mi><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mo>→</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{C}}(id_X, p_Y)/_\sim \;\colon\; Hom_{\mathcal{C}}(X, Q Y)/_\sim \to Hom_{\mathcal{C}}(X,Y)/_\sim </annotation></semantics></math></div> <p>is an isomorphism, and dually that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><msub><mi>j</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>id</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mo>→</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{C}}(j_X, id_Y)/_\sim \;\colon\; Hom_{\mathcal{C}}(P X, Y)/_\sim \to Hom_{\mathcal{C}}(X,Y)/_\sim </annotation></semantics></math></div> <p>is an isomorphism. We discuss this for the former; the second is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>:</p> <p>First, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{\mathcal{C}}(id_X, p_Y)</annotation></semantics></math> is surjective is the <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>∅</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Q</mi><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Cof</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>Y</mi></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \emptyset &\longrightarrow& Q Y \\ {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_Y}}_{\mathrlap{\in W \cap Fib}} \\ X &\overset{f}{\longrightarrow}& Y } \,, </annotation></semantics></math></div> <p>which says that any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> comes from a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">^</mo></mover><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Q</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">\hat f \colon X \to Q Y</annotation></semantics></math> under postcomposition with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>Y</mi><mover><mo>→</mo><mrow><msub><mi>p</mi> <mi>Y</mi></msub></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">Q Y \overset{p_Y}{\to} Y</annotation></semantics></math>.</p> <p>Second, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{\mathcal{C}}(id_X, p_Y)</annotation></semantics></math> is injective is the lifting property in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>⊔</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Q</mi><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Cof</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>Y</mi></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mi>η</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X \sqcup X &\overset{(f,g)}{\longrightarrow}& Q Y \\ {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_Y}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(X) &\underset{\eta}{\longrightarrow}& Y } \,, </annotation></semantics></math></div> <p>which says that if two morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Q</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">f, g \colon X \to Q Y</annotation></semantics></math> become homotopic after postcomposition with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>Y</mi></msub><mo lspace="verythinmathspace">:</mo><mi>Q</mi><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p_Y \colon Q X \to Y</annotation></semantics></math>, then they were already homotopic before.</p> </div> <p>We record the following fact which will be used in <a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+1-1">part 1.1</a> (<a href="Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated">here</a>):</p> <div class="num_lemma" id="RepresentingHomotopyCommutativeSquaresByCommutativeSquares"> <h6 id="lemma_8">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> (def. <a class="maruku-ref" href="#ModelCategory"></a>). Then every <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> in its <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> (def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>) is, up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of squares, in the image of the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} \longrightarrow Ho(\mathcal{C})</annotation></semantics></math> of a commuting square in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (i.e.: not just commuting up to homotopy).</p> </div> <div class="proof"> <h6 id="proof_13">Proof</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>a</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>b</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>′</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>f</mi><mo>′</mo></mrow></munder></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{a}}\downarrow && \downarrow^{\mathrlap{b}} \\ A' &\underset{f'}{\longrightarrow}& B' } \;\;\;\;\; \in Ho(\mathcal{C}) </annotation></semantics></math></div> <p>be a commuting square in the homotopy category. Writing the same symbols for fibrant-cofibrant objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and for morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> representing these, then this means that in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>b</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mi>η</mi></munder></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mi>f</mi><mo>′</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mi>a</mi></munder></mtd> <mtd><mi>A</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{b}} \\ Cyl(A) &\underset{\eta}{\longrightarrow}& B' \\ {}^{\mathllap{i_0}}\uparrow && \uparrow^{\mathrlap{f'}} \\ A &\underset{a}{\longrightarrow}& A' } \,. </annotation></semantics></math></div> <p>Consider the factorization of the top square here through the <a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow></mrow></munder></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>η</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>a</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mi>f</mi><mo>′</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{i_1}}\downarrow &(po)& \downarrow^{\mathrlap{\in W}} \\ Cyl(A) &\underset{}{\longrightarrow}& Cyl(f) \\ {}^{\mathllap{i_0}}\uparrow &{}_{\mathllap{\eta}}\searrow& \downarrow^{\mathrlap{}} \\ A && B' \\ & {}_{\mathllap{a}}\searrow & \uparrow_{\mathrlap{f'}} \\ && A' } </annotation></semantics></math></div> <p>This exhibits the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \overset{i_0}{\to} Cyl(A) \to Cyl(f)</annotation></semantics></math> as an alternative representative of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">Cyl(f) \to B'</annotation></semantics></math> as an alternative representative for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>, and the commuting square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>a</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi><mo>′</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>f</mi><mo>′</mo></mrow></munder></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{}{\longrightarrow}& Cyl(f) \\ {}^{\mathllap{a}}\downarrow && \downarrow \\ A' &\underset{f'}{\longrightarrow}& B' } </annotation></semantics></math></div> <p>as an alternative representative of the given commuting square in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math>.</p> </div> <h3 id="DerivedFunctors">Derived functors</h3> <div class="num_defn" id="HomotopicalFunctor"> <h6 id="definition_13">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopical+functor">homotopical functor</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a>, def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>, then a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C}\longrightarrow \mathcal{D}</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/homotopical+functor">homotopical functor</a></strong> if it sends weak equivalences to weak equivalences.</p> </div> <div class="num_defn" id="DerivedFunctorOfAHomotopicalFunctor"> <h6 id="definition_14">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/homotopical+functor">homotopical functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C} \longrightarrow \mathcal{D}</annotation></semantics></math> (def. <a class="maruku-ref" href="#HomotopicalFunctor"></a>) between <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a> whose <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{D})</annotation></semantics></math> exist (def. <a class="maruku-ref" href="#HomotopyCategoryOfACategoryWithWeakEquivalences"></a>), then its (“<a class="existingWikiWord" href="/nlab/show/total+derived+functor">total</a>”) <em><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></em> is the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(F)</annotation></semantics></math> between these homotopy categories which is induced uniquely, up to unique isomorphism, by their universal property (def. <a class="maruku-ref" href="#HomotopyCategoryOfACategoryWithWeakEquivalences"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd><mover><mo>⟶</mo><mi>F</mi></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>γ</mi> <mi>𝒞</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>γ</mi> <mi>𝒟</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∃</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\exists \; Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>While many functors of interest between <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> are not homotopical in the sense of def. <a class="maruku-ref" href="#HomotopicalFunctor"></a>, many become homotopical after restriction to the <a class="existingWikiWord" href="/nlab/show/full+subcategories">full subcategories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">of fibrant objects</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_c</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/category+of+cofibrant+objects">of cofibrant objects</a>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>. By corollary <a class="maruku-ref" href="#HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects"></a> this is just as good for the purpose of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> </div> <p>Therefore one considers the following generalization of def. <a class="maruku-ref" href="#DerivedFunctorOfAHomotopicalFunctor"></a>:</p> <div class="num_defn" id="LeftAndRightDerivedFunctorsOnModelCategories"> <h6 id="definition_15">Definition</h6> <p><strong>(left and right <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a>)</strong></p> <p>Consider a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C} \longrightarrow \mathcal{D}</annotation></semantics></math> out of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (def. <a class="maruku-ref" href="#ModelCategory"></a>) into a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>).</p> <ol> <li> <p>If the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math> of fibrant object becomes a <a class="existingWikiWord" href="/nlab/show/homotopical+functor">homotopical functor</a> (def. <a class="maruku-ref" href="#HomotopicalFunctor"></a>), then the <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of that restriction, according to def. <a class="maruku-ref" href="#DerivedFunctorOfAHomotopicalFunctor"></a>, is called the <em><a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}F</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><msub><mi>𝒞</mi> <mi>f</mi></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>𝒞</mi></mtd> <mtd><mover><mo>⟶</mo><mi>F</mi></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mpadded width="0" lspace="-100%width"><mi>γ</mi></mpadded> <mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow></msub></mrow></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>γ</mi> <mi>𝒟</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℝ</mi><mi>F</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msub><mi>𝒞</mi> <mi>f</mi></msub><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ & \mathcal{C}_f &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ & {}^{\mathllap{\gamma}_{\mathcal{C}_f}} \downarrow && \swArrow_{\simeq} && \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ \mathbb{R} F \colon & \mathcal{C}_f[W^{-1}] &\simeq& Ho(\mathcal{C}) &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,, </annotation></semantics></math></div> <p>where we use corollary <a class="maruku-ref" href="#HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects"></a>.</p> </li> <li> <p>If the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_c</annotation></semantics></math> of cofibrant object becomes a homotopical functor (def. <a class="maruku-ref" href="#HomotopicalFunctor"></a>), then the <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of that restriction, according to def. <a class="maruku-ref" href="#DerivedFunctorOfAHomotopicalFunctor"></a>, is called the <em><a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbb{L}F</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><msub><mi>𝒞</mi> <mi>c</mi></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>𝒞</mi></mtd> <mtd><mover><mo>⟶</mo><mi>F</mi></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mpadded width="0" lspace="-100%width"><mi>γ</mi></mpadded> <mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow></msub></mrow></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>γ</mi> <mi>𝒟</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝕃</mi><mi>F</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msub><mi>𝒞</mi> <mi>c</mi></msub><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ & \mathcal{C}_c &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ & {}^{\mathllap{\gamma}_{\mathcal{C}_f}}\downarrow && \swArrow_{\simeq} && \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ \mathbb{L} F \colon & \mathcal{C}_c[W^{-1}] &\simeq& Ho(\mathcal{C}) &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,, </annotation></semantics></math></div> <p>where again we use corollary <a class="maruku-ref" href="#HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects"></a>.</p> </li> </ol> </div> <p>The key fact that makes def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a> practically relevant is the following:</p> <div class="num_prop" id="KenBrownLemma"> <h6 id="proposition_7">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Ken+Brown%27s+lemma">Ken Brown's lemma</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> with <a class="existingWikiWord" href="/nlab/show/full+subcategories">full subcategories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub><mo>,</mo><msub><mi>𝒞</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f, \mathcal{C}_c</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">of fibrant objects</a> and <a class="existingWikiWord" href="/nlab/show/cofibration+category">of cofibrant objects</a> respectively (def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>). Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>.</p> <ol> <li> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a> out of the <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒞</mi> <mi>f</mi></msub><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> F \;\colon\; \mathcal{C}_f \longrightarrow \mathcal{D} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/homotopical+functor">homotopical functor</a>, def. <a class="maruku-ref" href="#HomotopicalFunctor"></a>, already if it sends <a class="existingWikiWord" href="/nlab/show/acyclic+fibrations">acyclic fibrations</a> to <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a> out of the <a class="existingWikiWord" href="/nlab/show/category+of+cofibrant+objects">category of cofibrant objects</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒞</mi> <mi>c</mi></msub><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> F \;\colon\; \mathcal{C}_c \longrightarrow \mathcal{D} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/homotopical+functor">homotopical functor</a>, def. <a class="maruku-ref" href="#HomotopicalFunctor"></a>, already if it sends <a class="existingWikiWord" href="/nlab/show/acyclic+cofibrations">acyclic cofibrations</a> to <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>.</p> </li> </ol> </div> <p>The following proof refers to the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a>, whose full statement and proof we postpone to further below (lemma <a class="maruku-ref" href="#FactorizationLemma"></a>).</p> <div class="proof"> <h6 id="proof_14">Proof</h6> <p>We discuss the case of a functor on a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>. The other case is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> be a weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math>. Choose a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> (def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>) and consider the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></munder></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></munder><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ Path(f) &\underset{\in W \cap Fib}{\longrightarrow}& X \\ {}^{\mathllap{p_1^\ast f}}_{\mathllap{\in W}}\downarrow &(pb)& \downarrow^{\mathrlap{f}}_{\mathrlap{\in W}} \\ Path(Y) &\overset{p_1}{\underset{\in W \cap Fib}{\longrightarrow}}& Y \\ {}^{\mathllap{p_0}}_{\mathllap{\in W \cap Fib}}\downarrow \\ Y } \,, </annotation></semantics></math></div> <p>where the square is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(f)</annotation></semantics></math> on the top left is our notation for the universal <a class="existingWikiWord" href="/nlab/show/cone">cone</a> object. (Below we discuss this in more detail, it is the <em><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, def. <a class="maruku-ref" href="#MappingConeAndMappingCocone"></a>).</p> <p>Here:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math> are both acyclic fibrations, by lemma <a class="maruku-ref" href="#ComponentMapsOfCylinderAndPathSpaceInGoodSituation"></a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Path(f) \to X</annotation></semantics></math> is an acyclic fibration because it is the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi></mrow><annotation encoding="application/x-tex">p_1^\ast f</annotation></semantics></math> is a weak equivalence, because the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a> states that the composite vertical morphism factors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> through a weak equivalence, hence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a weak equivalence, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi></mrow><annotation encoding="application/x-tex">p_1^\ast f</annotation></semantics></math> is by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>).</p> </li> </ol> <p>Now apply the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to this diagram and use the assumption that it sends acyclic fibrations to weak equivalences to obtain</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></munder></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mrow><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></munder><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F(Path(f)) &\underset{\in W }{\longrightarrow}& F(X) \\ {}^{\mathllap{F(p_1^\ast f)}}_{\mathllap{}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ F(Path(Y)) &\overset{F(p_1)}{\underset{\in W }{\longrightarrow}}& F(Y) \\ {}^{\mathllap{F(p_0)}}_{\mathllap{\in W}}\downarrow \\ Y } \,. </annotation></semantics></math></div> <p>But the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a>, in addition says that the vertical composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo>∘</mo><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi></mrow><annotation encoding="application/x-tex">p_0 \circ p_1^\ast f</annotation></semantics></math> is a fibration, hence an acyclic fibration by the above. Therefore also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>∘</mo><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(p_0 \circ p_1^\ast f)</annotation></semantics></math> is a weak equivalence. Now the claim that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(f)</annotation></semantics></math> is a weak equivalence follows with applying <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) twice.</p> </div> <div class="num_cor" id="LeftAndRightDerivedFunctors"> <h6 id="corollary_3">Corollary</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}, \mathcal{D}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> and consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C}\longrightarrow \mathcal{D}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>. Then:</p> <ol> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves cofibrant objects and acyclic cofibrations between these, then its <a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a> (def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbb{L}F</annotation></semantics></math> exists, fitting into a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝒞</mi> <mi>c</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mi>F</mi></mover></mtd> <mtd><msub><mi>𝒟</mi> <mi>c</mi></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>γ</mi> <mi>𝒞</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>γ</mi> <mi>𝒟</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>𝕃</mi><mi>F</mi></mrow></mover></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}_{c} &\overset{F}{\longrightarrow}& \mathcal{D}_{c} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\overset{\mathbb{L}F}{\longrightarrow}& Ho(\mathcal{D}) } </annotation></semantics></math></div></li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves fibrant objects and acyclic fibrants between these, then its <a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a> (def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}F</annotation></semantics></math> exists, fitting into a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝒞</mi> <mi>f</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mi>F</mi></mover></mtd> <mtd><msub><mi>𝒟</mi> <mi>f</mi></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>γ</mi> <mi>𝒞</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>γ</mi> <mi>𝒟</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>ℝ</mi><mi>F</mi></mrow></munder></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}_{f} &\overset{F}{\longrightarrow}& \mathcal{D}_{f} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{R}F}{\longrightarrow}& Ho(\mathcal{D}) } \,. </annotation></semantics></math></div></li> </ol> </div> <div class="num_example" id="ComputationOfLeftRightDerivedFunctorsViaResolutions"> <h6 id="proposition_8">Proposition</h6> <p><strong>(construction of left/right <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}</annotation></semantics></math> be a functor between two <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> (def. <a class="maruku-ref" href="#ModelCategory"></a>).</p> <ol> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves fibrant objects and weak equivalences between fibrant objects, then the total <a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>F</mi><mo>≔</mo><mi>ℝ</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>𝒟</mi></msub><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}F \coloneqq \mathbb{R}(\gamma_{\mathcal{D}}\circ F)</annotation></semantics></math> (def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a>) in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝒞</mi> <mi>f</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mi>F</mi></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>γ</mi> <mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>γ</mi> <mi>𝒟</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>ℝ</mi><mi>F</mi></mrow></munder></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}_f &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}_f}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{R}F}{\longrightarrow}& Ho(\mathcal{D}) } </annotation></semantics></math></div> <p>is given, up to isomorphism, on any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi><mover><mo>⟶</mo><mrow><msub><mi>γ</mi> <mi>𝒞</mi></msub></mrow></mover><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X\in \mathcal{C} \overset{\gamma_{\mathcal{C}}}{\longrightarrow} Ho(\mathcal{C})</annotation></semantics></math> by appying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to a fibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">P X</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and then forming a cofibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(F(P X))</annotation></semantics></math> of the result:</p> </li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}F(X) \simeq Q(F(P X)) \,. </annotation></semantics></math></div> <ol> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves cofibrant objects and weak equivalences between cofibrant objects, then the total <a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>F</mi><mo>≔</mo><mi>𝕃</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>𝒟</mi></msub><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{L}F \coloneqq \mathbb{L}(\gamma_{\mathcal{D}}\circ F)</annotation></semantics></math> (def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a>) in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝒞</mi> <mi>c</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mi>F</mi></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>γ</mi> <mrow><msub><mi>𝒞</mi> <mi>c</mi></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>γ</mi> <mi>𝒟</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>𝕃</mi><mi>F</mi></mrow></munder></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}_c &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}_c}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{L}F}{\longrightarrow}& Ho(\mathcal{D}) } </annotation></semantics></math></div> <p>is given, up to isomorphism, on any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi><mover><mo>⟶</mo><mrow><msub><mi>γ</mi> <mi>𝒞</mi></msub></mrow></mover><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X\in \mathcal{C} \overset{\gamma_{\mathcal{C}}}{\longrightarrow} Ho(\mathcal{C})</annotation></semantics></math> by appying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to a cofibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Q X</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and then forming a fibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(F(Q X))</annotation></semantics></math> of the result:</p> </li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{L}F(X) \simeq P(F(Q X)) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_15">Proof</h6> <p>We discuss the first case, the second is <a class="existingWikiWord" href="/nlab/show/formal+duality">formally dual</a>. By the proof of theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>ℝ</mi><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mi>γ</mi> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>𝒞</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>γ</mi> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbb{R}F(X) & \simeq \gamma_{\mathcal{D}}(F(\gamma_{\mathcal{C}})) \\ & \simeq \gamma_{\mathcal{D}}F(Q(P(X)) ) \end{aligned} \,. </annotation></semantics></math></div> <p>But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a homotopical functor on fibrant objects, the cofibrant replacement morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(Q(P(X)))\to F(P(X))</annotation></semantics></math> is a weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>, hence becomes an isomorphism under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>𝒟</mi></msub></mrow><annotation encoding="application/x-tex">\gamma_{\mathcal{D}}</annotation></semantics></math>. Therefore</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>γ</mi> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}F(X) \simeq \gamma_{\mathcal{D}}(F(P(X))) \,. </annotation></semantics></math></div> <p>Now since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is assumed to preserve fibrant objects, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(P(X))</annotation></semantics></math> is fibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>𝒟</mi></msub></mrow><annotation encoding="application/x-tex">\gamma_{\mathcal{D}}</annotation></semantics></math> acts on it (only) by cofibrant replacement.</p> </div> <h3 id="QuillenAdjunctions">Quillen adjunctions</h3> <p>In practice it turns out to be useful to arrange for the assumptions in corollary <a class="maruku-ref" href="#LeftAndRightDerivedFunctors"></a> to be satisfied by pairs of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> (Def. <a class="maruku-ref" href="#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets"></a>). Recall that this is a pair of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> going back and forth between two categories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mo>⊥</mo><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D} </annotation></semantics></math></div> <p>such that there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> between <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> on the left and those with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> on the right <a class="maruku-eqref" href="#eq:HomIsomorphismForAdjointFunctors">(?)</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><mi>d</mi><mo>,</mo><mi>c</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mrow></mrow></munderover><msub><mi>Hom</mi> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi_{d,c} \;\colon\; Hom_{\mathcal{C}}(L(d),c) \underoverset{\simeq}{}{\longrightarrow} Hom_{\mathcal{D}}(d, R(c)) </annotation></semantics></math></div> <p>for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d\in \mathcal{D}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math>. This being <em><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a></em> (Def. <a class="maruku-ref" href="#NaturalTransformations"></a>) means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Hom</mi> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⇒</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi \colon Hom_{\mathcal{D}}(L(-),-) \Rightarrow Hom_{\mathcal{C}}(-, R(-))</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>, hence that for all morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><msub><mi>d</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>d</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">g \colon d_2 \to d_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f \colon c_1 \to c_2</annotation></semantics></math> the following is a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>ϕ</mi> <mrow><msub><mi>d</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>1</mn></msub></mrow></msub></mrow></munderover></mtd> <mtd><msub><mi>Hom</mi> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>1</mn></msub><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>g</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>g</mi><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>R</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mrow><msub><mi>d</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub></mrow><mo>≃</mo></munderover></mtd> <mtd><msub><mi>Hom</mi> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>2</mn></msub><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo><mo stretchy="false">/</mo></mrow><annotation encoding="application/x-tex"> \array{ Hom_{\mathcal{C}}(L(d_1), c_1) & \underoverset{\simeq}{\phi_{d_1,c_1}}{\longrightarrow} & Hom_{\mathcal{D}}(d_1, R(c_1)) \\ {}^{\mathllap{L(f) \circ (-)\circ g}}\downarrow && \downarrow^{\mathrlap{g\circ (-)\circ R(g)}} \\ Hom_{\mathcal{C}}(L(d_2), c_2) & \underoverset{\phi_{d_2, c_2}}{\simeq}{\longrightarrow} & Hom_{\mathcal{D}}(d_2, R(c_2)) } \,. / </annotation></semantics></math></div> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math> to indicate such an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> and call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> the <em><a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></em> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> the <em><a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a></em> of the adjoint pair.</p> <p>The archetypical example of a pair of adjoint functors is that consisting of forming <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y \times (-)</annotation></semantics></math> and forming <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">(-)^Y</annotation></semantics></math>, as in the category of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> of def. <a class="maruku-ref" href="#kTop"></a>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f \colon L(d) \to c</annotation></semantics></math> is any morphism, then the image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><mi>d</mi><mo>,</mo><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>d</mi><mo>→</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_{d,c}(f) \colon d \to R(c)</annotation></semantics></math> is called its <em><a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a></em>, and conversely. The fact that adjuncts are in bijection is also expressed by the notation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>f</mi></mover><mi>d</mi></mrow><mrow><mi>c</mi><mover><mo>⟶</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mover><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{ L(c) \overset{f}{\longrightarrow} d }{ c \overset{\tilde f}{\longrightarrow} R(d) } \,. </annotation></semantics></math></div> <p>For an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d\in \mathcal{D}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">L d</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>d</mi><mo>⟶</mo><mi>R</mi><mi>L</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">\eta_d \;\colon\; d \longrightarrow R L d</annotation></semantics></math>.</p> <p>For an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">R c</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>c</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>L</mi><mi>R</mi><mi>c</mi><mo>⟶</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\epsilon_c \;\colon\; L R c \longrightarrow c</annotation></semantics></math>.</p> <p>Adjunction units and counits turn out to encode the <a class="existingWikiWord" href="/nlab/show/adjuncts">adjuncts</a> of all other morphisms by the formulas</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mi>L</mi><mi>d</mi><mover><mo>→</mo><mi>f</mi></mover><mi>c</mi><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>d</mi><mover><mo>→</mo><mi>η</mi></mover><mi>R</mi><mi>L</mi><mi>d</mi><mover><mo>→</mo><mrow><mi>R</mi><mi>f</mi></mrow></mover><mi>R</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde{(L d\overset{f}{\to}c)} = (d\overset{\eta}{\to} R L d \overset{R f}{\to} R c)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mi>d</mi><mover><mo>→</mo><mi>g</mi></mover><mi>R</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>L</mi><mi>d</mi><mover><mo>→</mo><mrow><mi>L</mi><mi>g</mi></mrow></mover><mi>L</mi><mi>R</mi><mi>c</mi><mover><mo>→</mo><mi>ϵ</mi></mover><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde{(d\overset{g}{\to} R c)} = (L d \overset{L g}{\to} L R c \overset{\epsilon}{\to} c)</annotation></semantics></math>.</p> </li> </ul> <div class="num_defn" id="QuillenAdjunction"> <h6 id="definition_16">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}, \mathcal{D}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>. A pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> (Def. <a class="maruku-ref" href="#AdjointFunctorsInTermsOfNaturalBijectionOfHomSets"></a>) between them</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><munderover><mrow></mrow><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> (L \dashv R) \;\colon\; \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {} \mathcal{D} </annotation></semantics></math></div> <p>is called a <em><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></em>, to be denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} </annotation></semantics></math></div> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> are called left/right <em>Quillen functors</em>, respectively, if the following equivalent conditions are satisfied:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/acyclic+cofibrations">acyclic cofibrations</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/acyclic+fibrations">acyclic fibrations</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> and <a class="existingWikiWord" href="/nlab/show/acyclic+cofibrations">acyclic cofibrations</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a> and <a class="existingWikiWord" href="/nlab/show/acyclic+fibrations">acyclic fibrations</a>.</p> </li> </ol> </div> <div class="num_prop" id="ConditionsOnQuillenAdjunctionAreIndeedEquivalent"> <h6 id="proposition_9">Proposition</h6> <p>The conditions in def. <a class="maruku-ref" href="#QuillenAdjunction"></a> are indeed all equivalent.</p> </div> <p>(<a href="model+category#Quillen67">Quillen 67, I.4, theorem 3</a>)</p> <div class="proof"> <h6 id="proof_16">Proof</h6> <p>First observe that</p> <ul> <li> <p>(i) <em>A <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> preserves acyclic cofibrations precisely if its <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> preserves fibrations.</em></p> </li> <li> <p>(ii) <em>A <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> preserves cofibrations precisely if its <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> preserves acyclic fibrations.</em></p> </li> </ul> <p>We discuss statement (i), statement (ii) is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>. So let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon A \to B</annotation></semantics></math> be an acyclic cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">g \colon X \to Y</annotation></semantics></math> a fibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. Then for every <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> as on the left of the following, its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L\dashv R)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> is a commuting diagram as on the right here:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>L</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>L</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& R(X) \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{R(g)}} \\ B &\longrightarrow& R(Y) } \;\;\;\;\;\; \,, \;\;\;\;\;\; \array{ L(A) &\longrightarrow& X \\ {}^{\mathllap{L(f)}}\downarrow && \downarrow^{\mathrlap{g}} \\ L(B) &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves acyclic cofibrations, then the diagram on the right has a <a class="existingWikiWord" href="/nlab/show/lift">lift</a>, and so the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L\dashv R)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of that lift is a lift of the left diagram. This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(g)</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against all acylic cofibrations and hence is a fibration. Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> preserves fibrations, the same argument run from right to left gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves acyclic fibrations.</p> <p>Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.</p> </div> <p>The following is the analog of <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> and <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> (Def. <a class="maruku-ref" href="#AdjunctionUnitFromHomIsomorphism"></a>):</p> <div class="num_defn" id="DerivedAdjunctionUnit"> <h6 id="definition_17">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> (Def. <a class="maruku-ref" href="#ModelCategory"></a>), and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mrow><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>L</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></munderover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot_{Qu}} \mathcal{D} </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> (Def. <a class="maruku-ref" href="#QuillenAdjunction"></a>). Then</p> <ol> <li> <p>a <em>derived adjunction unit</em> at an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d \in \mathcal{D}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Q(d) \overset{\eta_{Q(d)}}{\longrightarrow} R(L(Q(d))) \overset{R( j_{L(Q(d))} )}{\longrightarrow} R(P(L(Q(d))) </annotation></semantics></math></div> <p>where</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is the ordinary <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> (Def. <a class="maruku-ref" href="#AdjunctionUnitFromHomIsomorphism"></a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>Cof</mi> <mi>𝒟</mi></msub></mrow><mrow><msub><mi>i</mi> <mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub></mrow></munderover><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>𝒟</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>𝒟</mi></msub></mrow><mrow><msub><mi>p</mi> <mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub></mrow></munderover><mi>d</mi></mrow><annotation encoding="application/x-tex">\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{Q(d)}}{\longrightarrow} Q(d) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{Q(d)}}{\longrightarrow} d</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>𝒞</mi></msub><mo>∩</mo><msub><mi>Cof</mi> <mi>𝒞</mi></msub></mrow><mrow><msub><mi>j</mi> <mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>Fib</mi> <mi>𝒞</mi></msub></mrow><mrow><msub><mi>q</mi> <mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow></munderover><mo>*</mo></mrow><annotation encoding="application/x-tex">L(Q(d)) \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_{L(Q(d))}}{\longrightarrow} P(L(Q(d))) \underoverset{\in Fib_{\mathcal{C}}}{q_{L(Q(d))}}{\longrightarrow} \ast</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/fibrant+resolution">fibrant resolution</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>);</p> </li> </ol> </li> <li> <p>a <em>derived adjunction counit</em> at an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math> is a composition of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mover><mi>L</mi><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>ϵ</mi> <mrow><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mover><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L(Q(R(P(c)))) \overset{ p_{R(P(c))} }{\longrightarrow} L R(P(c)) \overset{\epsilon_{P(c)}}{\longrightarrow} P(c) </annotation></semantics></math></div> <p>where</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> is the ordinary <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> (Def. <a class="maruku-ref" href="#AdjunctionUnitFromHomIsomorphism"></a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>𝒞</mi></msub><mo>∩</mo><msub><mi>Cof</mi> <mi>𝒞</mi></msub></mrow><mrow><msub><mi>j</mi> <mi>c</mi></msub></mrow></munderover><mi>P</mi><mi>c</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>Fib</mi> <mi>𝒞</mi></msub></mrow><mrow><msub><mi>q</mi> <mi>c</mi></msub></mrow></munderover><mo>*</mo></mrow><annotation encoding="application/x-tex">c \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_c}{\longrightarrow} P c \underoverset{\in Fib_{\mathcal{C}}}{q_c}{\longrightarrow} \ast</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/fibrant+resolution">fibrant resolution</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>Cof</mi> <mi>𝒟</mi></msub></mrow><mrow><msub><mi>i</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow></munderover><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>𝒟</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>𝒟</mi></msub></mrow><mrow><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow></munderover><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{R(P(c))}}{\longrightarrow} Q(R(P(c))) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{R(P(c))}}{\longrightarrow} R(P(c))</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>).</p> </li> </ol> </li> </ol> </div> <p>We will see that <a class="existingWikiWord" href="/nlab/show/Quillen+adjunctions">Quillen adjunctions</a> induce ordinary <a class="existingWikiWord" href="/nlab/show/adjoint+pairs">adjoint pairs</a> of <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> on <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy categories</a> (Prop. <a class="maruku-ref" href="#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories"></a>). For this we first consider the following technical observation:</p> <div class="num_lemma" id="LeftRightQuillenFunctorsPreserveCyclinderPathSpaceObjects"> <h6 id="lemma_9">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/right+Quillen+functors">right Quillen functors</a> preserve <a class="existingWikiWord" href="/nlab/show/path+space+objects">path space objects</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mover><munderover><mo>⟶</mo><mi>R</mi><mo>⊥</mo></munderover><mover><mo>⟵</mo><mi>L</mi></mover></mover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C} \stackrel{\overset{L}{\longleftarrow}}{\underoverset{R}{\bot}{\longrightarrow}} \mathcal{D}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>, def. <a class="maruku-ref" href="#QuillenAdjunction"></a>.</p> <ol> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> a fibrant object and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> (def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>), then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(Path(X))</annotation></semantics></math> is a path space object for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(X)</annotation></semantics></math>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> a cofibrant object and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(X)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> (def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>), then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(Cyl(X))</annotation></semantics></math> is a cylinder object for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(X)</annotation></semantics></math>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_17">Proof</h6> <p>Consider the second case, the first is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> <p>First Observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>⊔</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>L</mi><mi>Y</mi><mo>⊔</mo><mi>L</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">L(Y \sqcup Y) \simeq L Y \sqcup L Y</annotation></semantics></math> because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> and hence preserves <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>, hence in particular <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>.</p> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">X</mo><mo>⊔</mo><mi>X</mi><mover><mo>→</mo><mrow><mo>∈</mo><mi>Cof</mi></mrow></mover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊔</mo><mi>L</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo>∈</mo><mi>Cof</mi></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L(\X \sqcup X \overset{\in Cof}{\to} Cyl(X)) = (L(X) \sqcup L(X) \overset{\in Cof}{\to } L (Cyl(X))) </annotation></semantics></math></div> <p>is a cofibration.</p> <p>Second, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> cofibrant then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>⊔</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y \sqcup Cyl(Y)</annotation></semantics></math> is a cofibrantion, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>Y</mi><mo>⊔</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y \to Y \sqcup Y</annotation></semantics></math> is a cofibration (lemma <a class="maruku-ref" href="#ComponentMapsOfCylinderAndPathSpaceInGoodSituation"></a>). Therefore by <a class="existingWikiWord" href="/nlab/show/Ken+Brown%27s+lemma">Ken Brown's lemma</a> (prop. <a class="maruku-ref" href="#KenBrownLemma"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves the weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">Cyl(Y) \overset{\in W}{\longrightarrow} Y</annotation></semantics></math>.</p> </div> <div class="num_prop" id="QuillenAdjunctionInducesAdjunctionOnHomotopyCategories"> <h6 id="proposition_10">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/derived+adjunction">derived adjunction</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C} \underoverset{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{{}_{\phantom{Qu}}\bot_{Qu}}\mathcal{D}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>, def. <a class="maruku-ref" href="#QuillenAdjunction"></a>, also the corresponding left and right <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> (Def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a>, via cor. <a class="maruku-ref" href="#LeftAndRightDerivedFunctors"></a>) form a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mi>ℝ</mi><mi>R</mi></mrow></munder><mover><mo>⟵</mo><mrow><mi>𝕃</mi><mi>L</mi></mrow></mover></munderover><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(\mathcal{C}) \underoverset {\underset{\mathbb{R}R}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\bot} Ho(\mathcal{D}) \,. </annotation></semantics></math></div> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> and <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> of this derived adjunction are the images of the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> and <a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>) under the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> functors (Theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a>).</p> </div> <p>(<a href="model+category#Quillen67">Quillen 67, I.4 theorem 3</a>)</p> <div class="proof"> <h6 id="proof_18">Proof</h6> <p>For the first statement, by def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a> and lemma <a class="maruku-ref" href="#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory"></a> it is sufficient to see that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X, Y \in \mathcal{C}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> cofibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> fibrant, then there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></p> <div class="maruku-equation" id="eq:HomIsomorphismForDerivedQuillenAdjunctionOnHomotopyCategory"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>R</mi><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{C}}(L X , Y)/_\sim \simeq Hom_{\mathcal{C}}(X, R Y)/_\sim \,. </annotation></semantics></math></div> <p>Since by the <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjunction isomorphism</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math> such a natural bijection exists before passing to homotopy classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">(-)/_\sim</annotation></semantics></math>, it is sufficient to see that this respects homotopy classes. To that end, use from lemma <a class="maruku-ref" href="#LeftRightQuillenFunctorsPreserveCyclinderPathSpaceObjects"></a> that with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(Y)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(Cyl(Y))</annotation></semantics></math> is a cylinder object for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(Y)</annotation></semantics></math>. This implies that left homotopies</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><msub><mo>⇒</mo> <mi>L</mi></msub><mi>g</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>L</mi><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> (f \Rightarrow_L g) \;\colon\; L X \longrightarrow Y </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Cyl</mi><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>L</mi><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; Cyl(L X) = L Cyl(X) \longrightarrow Y </annotation></semantics></math></div> <p>are in bijection to left homotopies</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><msub><mo>⇒</mo> <mi>L</mi></msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>R</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex"> (\tilde f \Rightarrow_L \tilde g) \;\colon\; X \longrightarrow R Y </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo stretchy="false">˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>R</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde \eta \;\colon\; Cyl(X) \longrightarrow R X \,. </annotation></semantics></math></div> <p>This establishes the adjunction. Now regarding the (co-)units: We show this for the adjunction unit, the case of the adjunction counit is <a class="existingWikiWord" href="/nlab/show/formal+duality">formally dual</a>.</p> <p>First observe that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><msub><mi>𝒟</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">d \in \mathcal{D}_c</annotation></semantics></math>, then the defining <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> for the <a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a> from def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝒟</mi> <mi>c</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mi>L</mi></mover></mtd> <mtd><mi>𝒞</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>γ</mi> <mi>P</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>𝕃</mi><mi>L</mi></mrow></munder></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{D}_c &\overset{L}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_P}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{P,Q}}} \\ Ho(\mathcal{D}) &\underset{\mathbb{L}L}{\longrightarrow}& Ho(\mathcal{C}) } </annotation></semantics></math></div> <p>(using fibrant and <a class="existingWikiWord" href="/nlab/show/fibrant+replacement">fibrant/cofibrant replacement functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>P</mi></msub></mrow><annotation encoding="application/x-tex">\gamma_P</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mrow><mi>P</mi><mo>,</mo><mi>Q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\gamma_{P,Q}</annotation></semantics></math> from def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a> with their universal property from theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a>, corollary <a class="maruku-ref" href="#HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects"></a>) gives that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo stretchy="false">)</mo><mi>d</mi><mo>≃</mo><mi>P</mi><mi>L</mi><mi>P</mi><mi>d</mi><mo>≃</mo><mi>P</mi><mi>L</mi><mi>d</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (\mathbb{L} L ) d \simeq P L P d \simeq P L d \;\;\;\; \in Ho(\mathcal{C}) \,, </annotation></semantics></math></div> <p>where the second isomorphism holds because the left Quillen functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> sends the acyclic cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>d</mi></msub><mo lspace="verythinmathspace">:</mo><mi>d</mi><mo>→</mo><mi>P</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">j_d \colon d \to P d</annotation></semantics></math> to a weak equivalence.</p> <p>The adjunction unit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo>⊣</mo><mi>ℝ</mi><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{L}L \dashv \mathbb{R}R)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>d</mi><mo>∈</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P d \in Ho(\mathcal{C})</annotation></semantics></math> is the image of the identity under</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo stretchy="false">)</mo><mi>P</mi><mi>d</mi><mo>,</mo><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo stretchy="false">)</mo><mi>P</mi><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>Hom</mi> <mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>P</mi><mi>d</mi><mo>,</mo><mo stretchy="false">(</mo><mi>ℝ</mi><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo stretchy="false">)</mo><mi>P</mi><mi>d</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{Ho(\mathcal{C})}((\mathbb{L}L) P d, (\mathbb{L} L) P d) \overset{\simeq}{\to} Hom_{Ho(\mathcal{C})}(P d, (\mathbb{R}R)(\mathbb{L}L) P d) \,. </annotation></semantics></math></div> <p>By the above and the proof of prop. <a class="maruku-ref" href="#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories"></a>, that adjunction isomorphism is equivalently that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math> under the isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>P</mi><mi>L</mi><mi>d</mi><mo>,</mo><mi>P</mi><mi>L</mi><mi>d</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>L</mi><mi>d</mi></mrow></msub><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mi>d</mi><mo>,</mo><mi>P</mi><mi>L</mi><mi>d</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex"> Hom_{Ho(\mathcal{C})}(P L d , P L d) \overset{Hom(j_{L d}, id)}{\longrightarrow} Hom_{\mathcal{C}}(L d, P L d)/_\sim </annotation></semantics></math></div> <p>of lemma <a class="maruku-ref" href="#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory"></a>. Hence the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>) is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>d</mi><mover><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>L</mi><mi>d</mi></mrow></msub></mrow></mover><mi>P</mi><mi>L</mi><mi>d</mi><mover><mo>→</mo><mi>id</mi></mover><mi>P</mi><mi>L</mi><mi>d</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> L d \overset{j_{L d}}{\longrightarrow} P L d \overset{id}{\to} P L d \,, </annotation></semantics></math></div> <p>which indeed (by the formula for <a class="existingWikiWord" href="/nlab/show/adjuncts">adjuncts</a>, Prop. <a class="maruku-ref" href="#GeneralAdjunctsInTermsOfAdjunctionUnitCounit"></a>) is the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>η</mi></mover><mi>R</mi><mi>L</mi><mi>d</mi><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>L</mi><mi>d</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mi>P</mi><mi>L</mi><mi>d</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \overset{\eta}{\longrightarrow} R L d \overset{R (j_{L d})}{\longrightarrow} R P L d \,. </annotation></semantics></math></div></div> <p>This suggests to regard passage to <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy categories</a> and <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> as itself being a suitable <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from a category of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> to the <a class="existingWikiWord" href="/nlab/show/category+of+categories">category of categories</a>. Due to the role played by the distinction between <a class="existingWikiWord" href="/nlab/show/left+Quillen+functors">left Quillen functors</a> and <a class="existingWikiWord" href="/nlab/show/right+Quillen+functors">right Quillen functors</a>, this is usefully formulated as a <a class="existingWikiWord" href="/nlab/show/double+functor">double functor</a>:</p> <div class="num_defn" id="DoubleCategoryOfModelCategories"> <h6 id="definition_18">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/double+category">double category</a> of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>)</strong></p> <p>The (<a class="existingWikiWord" href="/nlab/show/very+large+category">very large</a>) <em><a class="existingWikiWord" href="/nlab/show/double+category">double category</a> of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ModCat</mi> <mi>dbl</mi></msub></mrow><annotation encoding="application/x-tex">ModCat_{dbl}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/double+category">double category</a> (Def. <a class="maruku-ref" href="#DoubleCategoryOfSquares"></a>) that has</p> <ol> <li> <p>as <a class="existingWikiWord" href="/nlab/show/objects">objects</a>: <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#ModelCategory"></a>);</p> </li> <li> <p>as <a class="existingWikiWord" href="/nlab/show/vertical+morphisms">vertical morphisms</a>: <a class="existingWikiWord" href="/nlab/show/left+Quillen+functors">left Quillen functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mover><mo>⟶</mo><mi>L</mi></mover><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{C} \overset{L}{\longrightarrow} \mathcal{E}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#QuillenAdjunction"></a>);</p> </li> <li> <p>as <a class="existingWikiWord" href="/nlab/show/horizontal+morphisms">horizontal morphisms</a>: <a class="existingWikiWord" href="/nlab/show/right+Quillen+functors">right Quillen functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mover><mo>⟶</mo><mi>R</mi></mover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C} \overset{R}{\longrightarrow}\mathcal{D}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#QuillenAdjunction"></a>);</p> </li> <li> <p>as <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> between the <a class="existingWikiWord" href="/nlab/show/composition">composites</a> of underlying <a class="existingWikiWord" href="/nlab/show/functors">functors</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>R</mi> <mn>1</mn></msub><mover><mo>⇒</mo><mi>ϕ</mi></mover><msub><mi>R</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>L</mi> <mn>1</mn></msub><mphantom><mi>AAAAA</mi></mphantom><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>R</mi> <mn>1</mn></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>L</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϕ</mi></mpadded></msup><mo>⇙</mo></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mi>L</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝒞</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>R</mi> <mn>2</mn></msub><mphantom><mi>AA</mi></mphantom></mrow></munder></mtd> <mtd><mi>𝒟</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> L_2\circ R_1 \overset{\phi}{\Rightarrow} R_2\circ L_1 \phantom{AAAAA} \array{ \mathcal{C} &\overset{\phantom{AA}R_1\phantom{AA}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{L_1}}\Big\downarrow &{}^{\mathllap{ \phi }}\swArrow& \Big\downarrow{}^{\mathrlap{L_2}} \\ \mathcal{C} &\underset{\phantom{AA}R_2\phantom{AA}}{\longrightarrow}& \mathcal{D} } </annotation></semantics></math></div></li> </ol> <p>and <a class="existingWikiWord" href="/nlab/show/composition">composition</a> is given by ordinary <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/functors">functors</a>, horizontally and vertically, and by <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a>-composition of <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a>.</p> </div> <p>(<a href="double+category+of+model+categories#Shulman07">Shulman 07, Example 4.6</a>)</p> <p>There is hence a <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful</a> <a class="existingWikiWord" href="/nlab/show/double+functor">double functor</a> (Remark <a class="maruku-ref" href="#StrictAndWeak2Functors"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>ModCat</mi> <mi>dbl</mi></msub><mo>⟶</mo><mi>Sq</mi><mo stretchy="false">(</mo><mi>Cat</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat) </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/double+category+of+squares">double category of squares</a> (Example <a class="maruku-ref" href="#DoubleCategoryOfSquares"></a>) in the <a class="existingWikiWord" href="/nlab/show/2-category+of+categories">2-category of categories</a> (Example <a class="maruku-ref" href="#2CategoryOfCategories"></a>), which forgets the <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> and the <a class="existingWikiWord" href="/nlab/show/Quillen+functor">Quillen functor</a>-<a class="existingWikiWord" href="/nlab/show/property">property</a>.</p> <p>The following records the 2-functoriality of sending <a class="existingWikiWord" href="/nlab/show/Quillen+adjunctions">Quillen adjunctions</a> to <a class="existingWikiWord" href="/nlab/show/adjoint+pairs">adjoint pairs</a> of <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> (Prop. <a class="maruku-ref" href="#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories"></a>):</p> <div class="num_prop" id="HomotopyDoublePseudofunctor"> <h6 id="proposition_11">Proposition</h6> <p><strong>(homotopy <a class="existingWikiWord" href="/nlab/show/double+pseudofunctor">double pseudofunctor</a> on the <a class="existingWikiWord" href="/nlab/show/double+category+of+model+categories">double category of model categories</a>)</strong></p> <p>There is a <a class="existingWikiWord" href="/nlab/show/double+pseudofunctor">double pseudofunctor</a> (Remark <a class="maruku-ref" href="#StrictAndWeak2Functors"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>ModCat</mi> <mi>dbl</mi></msub><mo>⟶</mo><mi>Sq</mi><mo stretchy="false">(</mo><mi>Cat</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho(-) \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/double+category+of+model+categories">double category of model categories</a> (Def. <a class="maruku-ref" href="#DoubleCategoryOfModelCategories"></a>) to the <a class="existingWikiWord" href="/nlab/show/double+category+of+squares">double category of squares</a> (Example <a class="maruku-ref" href="#DoubleCategoryOfSquares"></a>) in the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> (Example <a class="maruku-ref" href="#2CategoryOfCategories"></a>), which sends</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a> (Def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>);</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/left+Quillen+functor">left Quillen functor</a> (Def. <a class="maruku-ref" href="#QuillenAdjunction"></a>) to its <a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a> (Def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a>);</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/right+Quillen+functor">right Quillen functor</a> (Def. <a class="maruku-ref" href="#QuillenAdjunction"></a>) to its <a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a> (Def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a>);</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>R</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>L</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϕ</mi></mpadded></msup><mo>⇙</mo></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mi>L</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℰ</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>R</mi> <mn>2</mn></msub></mrow></munder></mtd> <mtd><mi>ℱ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C} &\overset{R_1}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{L_1}}\Big\downarrow &{}^{\mathllap{ \phi }}\swArrow& \Big\downarrow{}^{\mathrlap{L_2}} \\ \mathcal{E} &\underset{R_2}{\longrightarrow}& \mathcal{F} } </annotation></semantics></math></div> <p>to the “<a class="existingWikiWord" href="/nlab/show/derived+natural+transformation">derived natural transformation</a>”</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>ℝ</mi><msub><mi>R</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>𝕃</mi><msub><mi>L</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd><mover><mo>⇙</mo><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><mi>𝕃</mi><msub><mi>L</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>ℝ</mi><msub><mi>R</mi> <mn>2</mn></msub></mrow></munder></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Ho(\mathcal{C}) &\overset{\mathbb{R}R_1}{\longrightarrow}& Ho(\mathcal{D}) \\ {}^{\mathllap{\mathbb{L}L_1}}\Big\downarrow &\overset{Ho(\phi)}{\swArrow}& \Big\downarrow{}^{\mathrlap{\mathbb{L}L_2}} \\ Ho(\mathcal{E}) &\underset{\mathbb{R}R_2}{\longrightarrow}& Ho(\mathcal{F}) } </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a></p> <div class="maruku-equation" id="eq:DerivedNaturalTransformation"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>L</mi> <mn>2</mn></msub><mi>Q</mi><msub><mi>R</mi> <mn>1</mn></msub><mi>P</mi><mover><mo>⟵</mo><mrow></mrow></mover><msub><mi>L</mi> <mn>2</mn></msub><mi>Q</mi><msub><mi>R</mi> <mn>1</mn></msub><mi>Q</mi><mi>P</mi><mo>⟶</mo><msub><mi>L</mi> <mn>2</mn></msub><msub><mi>R</mi> <mn>1</mn></msub><mi>Q</mi><mi>P</mi><mover><mo>⟶</mo><mi>ϕ</mi></mover><msub><mi>R</mi> <mn>2</mn></msub><msub><mi>L</mi> <mn>1</mn></msub><mi>Q</mi><mi>P</mi><mo>⟶</mo><msub><mi>R</mi> <mn>2</mn></msub><mi>P</mi><mi>L</mi><mn>1</mn><mi>Q</mi><mi>P</mi><mo>⟵</mo><msub><mi>R</mi> <mn>2</mn></msub><mi>R</mi><msub><mi>L</mi> <mn>1</mn></msub><mi>Q</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Ho(\phi) \;\colon\; L_2 Q R_1 P \overset{}{\longleftarrow} L_2 Q R_1 Q P \longrightarrow L_2 R_1 Q P \overset{\phi}{\longrightarrow} R_2 L_1 Q P \longrightarrow R_2 P L1 Q P \longleftarrow R_2 R L_1 Q \,, </annotation></semantics></math></div> <p>where the unlabeled morphisms are induced by <a class="existingWikiWord" href="/nlab/show/fibrant+resolution">fibrant resolution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>→</mo><mi>P</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">c \to P c</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>c</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">Q c \to c</annotation></semantics></math>, respectively (Def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>).</p> </li> </ol> </div> <p>(<a href="double+category+of+model+categories#Shulman07">Shulman 07, Theorem 7.6</a>)</p> <div class="num_lemma" id="DerivedNaturalTransformationUpToIsos"> <h6 id="lemma_10">Lemma</h6> <p><strong>(recognizing derived natural isomorphisms)</strong></p> <p>For the <a class="existingWikiWord" href="/nlab/show/derived+natural+transformation">derived natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\phi)</annotation></semantics></math> in <a class="maruku-eqref" href="#eq:DerivedNaturalTransformation">(3)</a> to be invertible in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a>, it is sufficient that for every <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math> which is both <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant</a> and <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant</a> the following composite <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mn>2</mn></msub><mi>Q</mi><msub><mi>L</mi> <mn>1</mn></msub><mi>c</mi><mover><mo>⟶</mo><mrow><msub><mi>R</mi> <mn>2</mn></msub><msub><mi>p</mi> <mrow><msub><mi>L</mi> <mn>1</mn></msub><mi>c</mi></mrow></msub></mrow></mover><msub><mi>R</mi> <mn>2</mn></msub><msub><mi>L</mi> <mn>1</mn></msub><mi>c</mi><mover><mo>⟶</mo><mi>ϕ</mi></mover><msub><mi>L</mi> <mn>2</mn></msub><msub><mi>R</mi> <mn>1</mn></msub><mi>c</mi><mover><mo>⟶</mo><mrow><msub><mi>L</mi> <mn>2</mn></msub><msub><mi>j</mi> <mrow><msub><mi>R</mi> <mn>1</mn></msub><mi>c</mi></mrow></msub></mrow></mover><msub><mi>L</mi> <mn>2</mn></msub><mi>P</mi><msub><mi>R</mi> <mn>1</mn></msub><mi>c</mi></mrow><annotation encoding="application/x-tex"> R_2 Q L_1 c \overset{ R_2 p_{L_1 c} }{\longrightarrow} R_2 L_1 c \overset{\phi}{\longrightarrow} L_2 R_1 c \overset{ L_2 j_{R_1 c} }{\longrightarrow} L_2 P R_1 c </annotation></semantics></math></div> <p>(of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> with images of <a class="existingWikiWord" href="/nlab/show/fibrant+resolution">fibrant resolution</a>/<a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a>, Def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>) is invertible in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a>, hence that the composite is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> (by Prop. <a class="maruku-ref" href="#MorphismIsWeakEquivalenceIfIsoInHomotopyCategoryForQuillen"></a>).</p> </div> <p>(<a href="double+category+of+model+categories#Shulman07">Shulman 07, Remark 7.2</a>)</p> <div class="num_example" id="DerivedFunctorOfLeftRightQuillenFunctor"> <h6 id="example_2">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of left-right <a class="existingWikiWord" href="/nlab/show/Quillen+functor">Quillen functor</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> (Def. <a class="maruku-ref" href="#ModelCategory"></a>), and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mi>F</mi><mphantom><mi>A</mi></mphantom></mrow></mover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \overset{\phantom{A}F\phantom{A}}{\longrightarrow} \mathcal{C} </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> that is both a <a class="existingWikiWord" href="/nlab/show/left+Quillen+functor">left Quillen functor</a> as well as a <a class="existingWikiWord" href="/nlab/show/right+Quillen+functor">right Quillen functor</a> (Def. <a class="maruku-ref" href="#QuillenAdjunction"></a>). This means equivalently that there is a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> in the <a class="existingWikiWord" href="/nlab/show/double+category+of+model+categories">double category of model categories</a> (Def. <a class="maruku-ref" href="#DoubleCategoryOfModelCategories"></a>) of the form</p> <div class="maruku-equation" id="eq:DoubleMorphismExhibitingLeftRightQuillenFunctor"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>F</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>F</mi></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mi>id</mi></msup><mo>⇙</mo></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝒟</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mi>id</mi><mphantom><mi>A</mi></mphantom></mrow></munder></mtd> <mtd><mi>𝒟</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C} &\overset{\phantom{AA}F\phantom{AA}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{F}}\Big\downarrow &{}^{id}\swArrow& \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{D} &\underset{\phantom{A}id\phantom{A}}{\longrightarrow}& \mathcal{D} } </annotation></semantics></math></div> <p>It follows that the <a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbb{L}F</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}F</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> (Def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a>) are <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally isomorphic</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>𝕃</mi><mi>F</mi><mo>≃</mo><mi>ℝ</mi><mi>F</mi></mrow></mover><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(\mathcal{C}) \overset{ \mathbb{L}F \simeq \mathbb{R}F }{\longrightarrow} Ho(\mathcal{D}) \,. </annotation></semantics></math></div></div> <p>(<a href="double+category+of+model+categories#Shulman07">Shulman 07, corollary 7.8</a>)</p> <div class="proof"> <h6 id="proof_19">Proof</h6> <p>To see the <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>F</mi><mo>≃</mo><mi>ℝ</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbb{L}F \simeq \mathbb{R}F</annotation></semantics></math>: By Prop. <a class="maruku-ref" href="#HomotopyDoublePseudofunctor"></a> this is implied once the <a class="existingWikiWord" href="/nlab/show/derived+natural+transformation">derived natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>id</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(id)</annotation></semantics></math> of <a class="maruku-eqref" href="#eq:DoubleMorphismExhibitingLeftRightQuillenFunctor">(4)</a> is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>. By Prop. <a class="maruku-ref" href="#DerivedNaturalTransformationUpToIsos"></a> this is the case, in the present situation, if the composition of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>F</mi><mi>c</mi><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mrow><mi>F</mi><mi>c</mi></mrow></msub></mrow></mover><mi>F</mi><mi>c</mi><mover><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>F</mi><mi>c</mi></mrow></msub></mrow></mover><mi>P</mi><mi>F</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> Q F c \overset{ p_{F c} }{\longrightarrow} F c \overset{ j_{F c} }{\longrightarrow} P F c </annotation></semantics></math></div> <p>is a weak equivalence. But this is immediate, since the two factors are weak equivalences, by definition of <a class="existingWikiWord" href="/nlab/show/fibrant+resolution">fibrant/cofibrant resolution</a> (Def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>).</p> </div> <p>The following is the analog of <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">co-</a><a class="existingWikiWord" href="/nlab/show/reflective+subcategories">reflective subcategories</a> (Def. <a class="maruku-ref" href="#ReflectiveSubcategory"></a>) for <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>:</p> <div class="num_defn" id="QuillenReflection"> <h6 id="definition_19">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Quillen+reflection">Quillen reflection</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> (Def. <a class="maruku-ref" href="#ModelCategory"></a>), and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mrow><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot_{Qu}} \mathcal{D} </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> between them (Def. <a class="maruku-ref" href="#QuillenAdjunction"></a>). Then this may be called</p> <ol> <li> <p>a <em>Quillen reflection</em> if the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>) is componentwise a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>;</p> </li> <li> <p>a <em>Quillen co-reflection</em> if the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>) is componentwise a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>.</p> </li> </ol> </div> <p>The main class of examples of <a class="existingWikiWord" href="/nlab/show/Quillen+reflections">Quillen reflections</a> are <a class="existingWikiWord" href="/nlab/show/left+Bousfield+localizations">left Bousfield localizations</a>, discussed as Prop. <a class="maruku-ref" href="#BasicPropertiesOfLectBousfieldLocalizations"></a> below.</p> <div class="num_prop" id="QuillenReflectionViaReflectionOfHomotopyCategories"> <h6 id="proposition_12">Proposition</h6> <p><strong>(characterization of <a class="existingWikiWord" href="/nlab/show/Quillen+reflections">Quillen reflections</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mrow><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot_{Qu}} \mathcal{D} </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> (Def. <a class="maruku-ref" href="#QuillenAdjunction"></a>) and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><munderover><mrow><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>ℝ</mi><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mrow><mi>𝕃</mi><mi>L</mi></mrow></mover></munderover><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho(\mathcal{C}) \underoverset {\underset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\bot_{Qu}} Ho(\mathcal{D}) </annotation></semantics></math></div> <p>for the induced <a class="existingWikiWord" href="/nlab/show/adjoint+pair">adjoint pair</a> of <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> on the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy categories</a>, from Prop. <a class="maruku-ref" href="#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories"></a>.</p> <p>Then</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><munder><mo>⊣</mo><mi>Qu</mi></munder><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \underset{Qu}{\dashv} R)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Quillen+reflection">Quillen reflection</a> (Def. <a class="maruku-ref" href="#QuillenReflection"></a>) precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo>⊣</mo><mi>ℝ</mi><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{L}L \dashv \mathbb{R}R)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a>-inclusion (Def. <a class="maruku-ref" href="#ReflectiveSubcategory"></a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><munder><mo>⊣</mo><mi>Qu</mi></munder><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \underset{Qu}{\dashv} R)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Quillen+co-reflection">Quillen co-reflection</a>] (Def. <a class="maruku-ref" href="#QuillenReflection"></a>) precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo>⊣</mo><mi>ℝ</mi><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{L}L \dashv \mathbb{R}R)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/co-reflective+subcategory">co-reflective subcategory</a>-inclusion (Def. <a class="maruku-ref" href="#ReflectiveSubcategory"></a>);</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_20">Proof</h6> <p>By Prop. <a class="maruku-ref" href="#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories"></a> the components of the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a>/<a class="existingWikiWord" href="/nlab/show/adjunction+counit">counit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo>⊣</mo><mi>ℝ</mi><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{L}L \dashv \mathbb{R}R)</annotation></semantics></math> are precisely the images under <a class="existingWikiWord" href="/nlab/show/localization">localization</a> of the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a>/<a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">counit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><munder><mo>⊣</mo><mi>Qu</mi></munder><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \underset{Qu}{\dashv} R)</annotation></semantics></math>. Moreover, by Prop. <a class="maruku-ref" href="#MorphismIsWeakEquivalenceIfIsoInHomotopyCategoryForQuillen"></a> the localization functor of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> inverts precisely the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>. Hence the adjunction (co-)unit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo>⊣</mo><mi>ℝ</mi><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{L}L \dashv \mathbb{R}R)</annotation></semantics></math> is an isomorphism if and only if the derived (co-)unit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><munder><mo>⊣</mo><mi>Qu</mi></munder><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \underset{Qu}{\dashv} R)</annotation></semantics></math> is a weak equivalence, respectively.</p> <p>With this the statement reduces to the characterization of (co-)reflections via invertible units/counits, respectively, from Prop. <a class="maruku-ref" href="#FullyFaithfulAndInvertibleAdjoints"></a>.</p> </div> <p>The following is the analog of <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence+of+categories">adjoint equivalence of categories</a> (Def. <a class="maruku-ref" href="#AdjointEquivalenceOfCategories"></a>) for <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>:</p> <div class="num_defn" id="QuillenEquivalence"> <h6 id="definition_20">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}, \mathcal{D}</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> (Def. <a class="maruku-ref" href="#ModelCategory"></a>), a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> (def. <a class="maruku-ref" href="#QuillenAdjunction"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} </annotation></semantics></math></div> <p>is called a <em><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></em>, to be denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>≃</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>L</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></munderover><mi>𝒟</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {{}_{\phantom{Qu}}\simeq_{Qu}} \mathcal{D} \,, </annotation></semantics></math></div> <p>if the following equivalent conditions hold:</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> (via prop. <a class="maruku-ref" href="#ConditionsOnQuillenAdjunctionAreIndeedEquivalent"></a>, corollary <a class="maruku-ref" href="#LeftAndRightDerivedFunctors"></a>) is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>R</mi><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}R \colon Ho(\mathcal{C}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{D}) \,. </annotation></semantics></math></div></li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> (via prop. <a class="maruku-ref" href="#ConditionsOnQuillenAdjunctionAreIndeedEquivalent"></a>, corollary <a class="maruku-ref" href="#LeftAndRightDerivedFunctors"></a>) is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>L</mi><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{L}L \colon Ho(\mathcal{D}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{C}) \,. </annotation></semantics></math></div></li> <li> <p>For every <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d\in \mathcal{D}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>d</mi></msub></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d \overset{\eta_d}{\longrightarrow} R(L(d)) \overset{R(j_{L(d)})}{\longrightarrow} R(P(L(d))) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>;</p> <p>and for every <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>ϵ</mi></mover><mi>c</mi></mrow><annotation encoding="application/x-tex"> L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>.</p> </li> <li> <p>For every cofibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d \in \mathcal{D}</annotation></semantics></math> and every fibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>⟶</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d \longrightarrow R(c)</annotation></semantics></math> is a weak equivalence precisely if its <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">L(c) \to d</annotation></semantics></math> is:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>d</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>𝒟</mi></msub></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>𝒞</mi></msub></mrow></mover><mi>c</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{ d \overset{\in W_{\mathcal{D}}}{\longrightarrow} R(c) }{ L(d) \overset{\in W_{\mathcal{C}}}{\longrightarrow} c } \,. </annotation></semantics></math></div></li> </ol> </div> <div class="num_prop" id="ConditionsForQuillenAdjunctionAreIndeedEquivalent"> <h6 id="poposition">Poposition</h6> <p>The conditions in def. <a class="maruku-ref" href="#QuillenEquivalence"></a> are indeed all equivalent.</p> </div> <p>(<a href="model+category#Quillen67">Quillen 67, I.4, theorem 3</a>)</p> <div class="proof"> <h6 id="proof_21">Proof</h6> <p>That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">)</mo><mo>⇔</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1) \Leftrightarrow 2)</annotation></semantics></math> follows from prop. <a class="maruku-ref" href="#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories"></a> (if in an adjoint pair one is an equivalence, then so is the other).</p> <p>To see the equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⇔</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1),2) \Leftrightarrow 3)</annotation></semantics></math>, notice (<a href="adjoint+functor#FullyFaithfulAndInvertibleAdjoints">prop.</a>) that a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> precisely if both the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> and the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> are <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a>. Hence it is sufficient to see that the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a>/<a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>) indeed represent the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction (co-)unit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝕃</mi><mi>L</mi><mo>⊣</mo><mi>ℝ</mi><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{L}L \dashv \mathbb{R}R)</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a>. But this is the statement of Prop. <a class="maruku-ref" href="#QuillenAdjunctionInducesAdjunctionOnHomotopyCategories"></a>.</p> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mo stretchy="false">)</mo><mo>⇒</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">4) \Rightarrow 3)</annotation></semantics></math>:</p> <p>Consider the weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>j</mi> <mrow><mi>L</mi><mi>X</mi></mrow></msub></mrow></mover><mi>P</mi><mi>L</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">L X \overset{j_{L X}}{\longrightarrow} P L X</annotation></semantics></math>. Its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>η</mi></mover><mi>R</mi><mi>L</mi><mi>X</mi><mover><mo>⟶</mo><mrow><mi>R</mi><msub><mi>j</mi> <mrow><mi>L</mi><mi>X</mi></mrow></msub></mrow></mover><mi>R</mi><mi>P</mi><mi>L</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \overset{\eta}{\longrightarrow} R L X \overset{R j_{L X}}{\longrightarrow} R P L X </annotation></semantics></math></div> <p>by assumption 4) this is again a weak equivalence, which is the requirement for the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> in 3). Dually for <a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a>.</p> <p>To see <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn><mo stretchy="false">)</mo><mo>⇒</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">3) \Rightarrow 4)</annotation></semantics></math>:</p> <p>Consider any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>L</mi><mi>d</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f \colon L d \to c</annotation></semantics></math> a weak equivalence for cofibrant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>, firbant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>. Its <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> sits in a commuting diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>d</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>R</mi><mi>L</mi><mi>d</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>R</mi><mi>f</mi></mrow></mover></mtd> <mtd><mi>R</mi><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>=</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>R</mi><msub><mi>j</mi> <mrow><mi>L</mi><mi>d</mi></mrow></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>R</mi><msub><mi>j</mi> <mi>c</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>d</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></munder></mtd> <mtd><mi>R</mi><mi>P</mi><mi>L</mi><mi>d</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>R</mi><mi>P</mi><mi>f</mi></mrow></mover></mtd> <mtd><mi>R</mi><mi>P</mi><mi>c</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde f \colon & d &\overset{\eta}{\longrightarrow}& R L d &\overset{R f}{\longrightarrow}& R c \\ & {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{R j_{L d}}} && \downarrow^{\mathrlap{R j_c}} \\ & d &\underset{\in W}{\longrightarrow}& R P L d &\overset{R P f}{\longrightarrow}& R P c } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">P f</annotation></semantics></math> is any lift constructed as in def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>.</p> <p>This exhibits the bottom left morphism as the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>), hence a weak equivalence by assumption. But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> was a weak equivalence, so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">P f</annotation></semantics></math> (by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a>). Thereby also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>P</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">R P f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mi>j</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">R j_Y</annotation></semantics></math>, are weak equivalences by <a class="existingWikiWord" href="/nlab/show/Ken+Brown%27s+lemma">Ken Brown's lemma</a> <a class="maruku-ref" href="#KenBrownLemma"></a> and the assumed fibrancy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>. Therefore by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) also the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> is a weak equivalence.</p> </div> <div class="num_example" id="TrivialQuillenEquivalence"> <h6 id="example_3">Example</h6> <p><strong>(trivial <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> (Def. <a class="maruku-ref" href="#ModelCategory"></a>). Then the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> constitutes a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> (Def. <a class="maruku-ref" href="#QuillenEquivalence"></a>) from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> to itself:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mrow><mphantom><mrow><msub><mrow></mrow> <mi>Qu</mi></msub></mrow></mphantom><msub><mo>≃</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mi>id</mi></munder><mover><mo>⟵</mo><mi>id</mi></mover></munderover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_22">Proof</h6> <p>From prop. <a class="maruku-ref" href="#ComputationOfLeftRightDerivedFunctorsViaResolutions"></a> it is clear that in this case the <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>id</mi></mrow><annotation encoding="application/x-tex">\mathbb{L}id</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>id</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}id</annotation></semantics></math> both are themselves the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> on the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a>, hence in particular are an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>.</p> </div> <p>In certain situations the conditions on a Quillen equivalence simplify. For instance:</p> <div class="num_prop" id="InCaseTheRightAdjointCreatesWeakEquivalences"> <h6 id="proposition_13">Proposition</h6> <p><strong>(recognition of <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a>)</strong></p> <p>If in a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd><munderover><mo>⊥</mo><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></munderover></mtd> <mtd><mi>𝒟</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{\mathcal{C} &\underoverset{\underset{R}{\to}}{\overset{L}{\leftarrow}}{\bot}& \mathcal{D}}</annotation></semantics></math> (def. <a class="maruku-ref" href="#QuillenAdjunction"></a>) the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> “creates weak equivalences” (in that a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a weak equivalence precisly if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(f)</annotation></semantics></math> is) then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> (def. <a class="maruku-ref" href="#QuillenEquivalence"></a>) precisely already if for all <a class="existingWikiWord" href="/nlab/show/cofibrant+objects">cofibrant objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d \in \mathcal{D}</annotation></semantics></math> the plain <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mover><mo>⟶</mo><mi>η</mi></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d \overset{\eta}{\longrightarrow} R (L (d)) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>.</p> </div> <div class="proof"> <h6 id="proof_23">Proof</h6> <p>By prop. <a class="maruku-ref" href="#ConditionsForQuillenAdjunctionAreIndeedEquivalent"></a>, generally, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math> is a Quillen equivalence precisely if</p> <ol> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d\in \mathcal{D}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mover><mo>⟶</mo><mi>η</mi></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d \overset{\eta}{\longrightarrow} R(L(d)) \overset{R(j_{L(d)})}{\longrightarrow} R(P(L(d))) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>;</p> </li> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a> (Def. <a class="maruku-ref" href="#DerivedAdjunctionUnit"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>ϵ</mi></mover><mi>c</mi></mrow><annotation encoding="application/x-tex"> L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>.</p> </li> </ol> <p>Consider the first condition: Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> preserves the weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mrow><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">j_{L(d)}</annotation></semantics></math>, then by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) the composite in the first item is a weak equivalence precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is.</p> <p>Hence it is now sufficient to show that in this case the second condition above is automatic.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> also reflects weak equivalences, the composite in item two is a weak equivalence precisely if its image</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c) </annotation></semantics></math></div> <p>under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is.</p> <p>Moreover, assuming, by the above, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\eta_{Q(R(c))}</annotation></semantics></math> on the cofibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(R(c))</annotation></semantics></math> is a weak equivalence, then by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> this composite is a weak equivalence precisely if the further composite with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Q(R(c)) \overset{\eta_{Q(R(c))}}{\longrightarrow} R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c) \,. </annotation></semantics></math></div> <p>By the formula for <a class="existingWikiWord" href="/nlab/show/adjuncts">adjuncts</a>, this composite is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L\dashv R)</annotation></semantics></math>-adjunct of the original composite, which is just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">p_{R(c)}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>ϵ</mi></mover><mi>c</mi></mrow><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{ L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c }{ Q(R(C)) \overset{p_{R(c)}}{\longrightarrow} R(c) } \,. </annotation></semantics></math></div> <p>But <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">p_{R(c)}</annotation></semantics></math> is a weak equivalence by definition of cofibrant replacement.</p> </div> <p>The following is the analog of <a class="existingWikiWord" href="/nlab/show/adjoint+triples">adjoint triples</a>, <a class="existingWikiWord" href="/nlab/show/adjoint+quadruples">adjoint quadruples</a> (Remark <a class="maruku-ref" href="#AdjointTriples"></a>), etc. for <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>:</p> <div class="num_defn" id="QuillenAdjointTriple"> <h6 id="definition_21">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Quillen+adjoint+triple">Quillen adjoint triple</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>𝒞</mi> <mn>2</mn></msub><mo>,</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}_1, \mathcal{C}_2, \mathcal{D}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> (Def. <a class="maruku-ref" href="#ModelCategory"></a>), where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_2</annotation></semantics></math> share the same underlying <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and such that the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> constitutes a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> (Def. <a class="maruku-ref" href="#QuillenEquivalence"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>2</mn></msub><munderover><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>id</mi><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>id</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></munderover><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> \mathcal{C}_2 \underoverset {\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}} {\overset{ \phantom{AA}id\phantom{AA} }{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{C}_1 </annotation></semantics></math></div> <p>Then</p> <ol> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/Quillen+adjoint+triple">Quillen adjoint triple</a></em> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub><mrow><mtable><mtr><mtd><munderover><mo>⟶</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mi>L</mi></munderover></mtd></mtr> <mtr><mtd><munderover><mo>⟵</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mi>C</mi></munderover></mtd></mtr> <mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} </annotation></semantics></math></div> <p>is diagrams in the <a class="existingWikiWord" href="/nlab/show/double+category+of+model+categories">double category of model categories</a> (Def. <a class="maruku-ref" href="#DoubleCategoryOfModelCategories"></a>) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>𝒞</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>id</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><msub><mi>𝒞</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>L</mi></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>η</mi></mpadded></msup><mo>⇙</mo></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>𝒞</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mi>R</mi><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><mi>𝒟</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mi>C</mi><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><msub><mi>𝒞</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϵ</mi></mpadded></msup><mo>⇙</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>C</mi></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mi>id</mi></mpadded></msub></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>𝒞</mi> <mn>2</mn></msub></mtd> <mtd><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>id</mi><mphantom><mi>AA</mi></mphantom></mrow></munder></mtd> <mtd><msub><mi>𝒞</mi> <mn>2</mn></msub></mtd> <mtd><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>id</mi><mphantom><mi>AA</mi></mphantom></mrow></munder></mtd> <mtd><msub><mi>𝒞</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 \\ && {}^{\mathllap{ L }}\Big\downarrow &{}^{\mathllap{\eta}}\swArrow& \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\overset{ \phantom{A}R\phantom{A} }{\longrightarrow}& \mathcal{D} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{ id }}\Big\downarrow & {}^{\mathllap{\epsilon}}\swArrow & {}^{\mathllap{C}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 } </annotation></semantics></math></div> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit of an adjunction</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit of an adjunction</a>, thus exhibiting <a class="existingWikiWord" href="/nlab/show/Quillen+adjunctions">Quillen adjunctions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝒞</mi> <mn>1</mn></msub><munderover><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟵</mo><mi>C</mi></munder><mover><mo>⟶</mo><mi>L</mi></mover></munderover><mi>𝒟</mi></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msub><mi>𝒞</mi> <mn>2</mn></msub><munderover><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mi>C</mi></mover></munderover><mi>𝒟</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}_1 \underoverset {\underset{C}{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} \\ \\ \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} } </annotation></semantics></math></div> <p>and such that the <a class="existingWikiWord" href="/nlab/show/derived+natural+transformation">derived natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>id</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(id)</annotation></semantics></math> of the bottom right square <a class="maruku-eqref" href="#eq:DerivedNaturalTransformation">(3)</a> is invertible (a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>);</p> </li> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/Quillen+adjoint+triple">Quillen adjoint triple</a></em> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub><mrow><mtable><mtr><mtd><munderover><mo>⟵</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mi>L</mi></munderover></mtd></mtr> <mtr><mtd><munderover><mo>⟶</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mi>C</mi></munderover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longleftarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longrightarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longleftarrow} \\ } \mathcal{D} </annotation></semantics></math></div> <p>is diagram in the <a class="existingWikiWord" href="/nlab/show/double+category+of+model+categories">double category of model categories</a> (Def. <a class="maruku-ref" href="#DoubleCategoryOfModelCategories"></a>) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝒞</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>id</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><msub><mi>𝒞</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>id</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><msub><mi>𝒞</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo>⇙</mo></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>C</mi></mpadded></msup></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϵ</mi></mpadded></msup><mo>⇙</mo></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>𝒞</mi> <mn>2</mn></msub></mtd> <mtd><munder><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mi>C</mi><mphantom><mi>A</mi></mphantom></mrow></munder></mtd> <mtd><mi>𝒟</mi></mtd> <mtd><munder><mo>⟶</mo><mi>R</mi></munder></mtd> <mtd><msub><mi>𝒞</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo maxsize="1.8em" minsize="1.8em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϵ</mi></mpadded></msup><mo>⇙</mo></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>L</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>𝒞</mi> <mn>2</mn></msub></mtd> <mtd><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>id</mi><mphantom><mi>AA</mi></mphantom></mrow></munder></mtd> <mtd><msub><mi>𝒞</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}_2 &\overset{ \phantom{AA} id \phantom{AA} }{\longrightarrow}& \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}} \Big\downarrow &{}^{ \mathllap{ id } }\swArrow& \Big\downarrow{}^{ \mathrlap{ C } } & {}^{ \mathllap{\epsilon} }\swArrow & \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\underset{ \phantom{A}C\phantom{A} }{\longrightarrow}& \mathcal{D} &\underset{R}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}}\Big\downarrow &{}^{\mathllap{ \epsilon }}\swArrow& \Big\downarrow{}^{\mathrlap{L}} \\ \mathcal{C}_2 &\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 } </annotation></semantics></math></div> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit of an adjunction</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit of an adjunction</a>, thus exhibiting <a class="existingWikiWord" href="/nlab/show/Quillen+adjunctions">Quillen adjunctions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝒞</mi> <mn>2</mn></msub><munderover><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mi>C</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover><mi>𝒟</mi></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msub><mi>𝒞</mi> <mn>1</mn></msub><munderover><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟵</mo><mi>R</mi></munder><mover><mo>⟶</mo><mi>C</mi></mover></munderover><mi>𝒟</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}_2 \underoverset {\underset{C}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} \\ \\ \mathcal{C}_1 \underoverset {\underset{R}{\longleftarrow}} {\overset{C}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} } </annotation></semantics></math></div> <p>and such that the <a class="existingWikiWord" href="/nlab/show/derived+natural+transformation">derived natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>id</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(id)</annotation></semantics></math> of the top left square square (<a href="double+category+of+model+categories#DerivedNaturalTransformation">here</a>) is invertible (a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>).</p> </li> </ol> <p>If a Quillen adjoint triple of the first kind overlaps with one of the second kind</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub><mrow><mtable><mtr><mtd><munderover><mo>⟶</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mrow><msub><mi>L</mi> <mn>1</mn></msub><mphantom><mrow><mo>=</mo><msub><mi>A</mi> <mi>a</mi></msub></mrow></mphantom></mrow></munderover></mtd></mtr> <mtr><mtd><munderover><mo>⟵</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>L</mi> <mn>2</mn></msub></mrow></munderover></mtd></mtr> <mtr><mtd><munderover><mo>⟶</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mrow><msub><mi>R</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow></munderover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mrow><msub><mi>A</mi> <mi>a</mi></msub><mo>=</mo></mrow></mphantom><msub><mi>R</mi> <mn>2</mn></msub></mrow></mover></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L_1 \phantom{= A_a}}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C_1 = L_2}{\longleftarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{R_1 = C_2}{\longrightarrow} \\ \overset{\phantom{A_a = } R_2}{\longleftarrow} \\ } \mathcal{D} </annotation></semantics></math></div> <p>we speak of a <em><a class="existingWikiWord" href="/nlab/show/Quillen+adjoint+quadruple">Quillen adjoint quadruple</a></em>, and so forth.</p> </div> <div class="num_prop" id="QuillenAdjointTripleInducesAdjointTripleOfDerivedFunctors"> <h6 id="proposition_14">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Quillen+adjoint+triple">Quillen adjoint triple</a> induces <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> of <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> on <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy categories</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/Quillen+adjoint+triple">Quillen adjoint triple</a> (Def. <a class="maruku-ref" href="#QuillenAdjointTriple"></a>), the induced <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> (Def. <a class="maruku-ref" href="#DerivedFunctorOfAHomotopicalFunctor"></a>) on the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy categories</a> form an ordinary <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> (Remark <a class="maruku-ref" href="#AdjointTriples"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub><mrow><mtable><mtr><mtd><munderover><mo>⟶</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mi>L</mi></munderover></mtd></mtr> <mtr><mtd><munderover><mo>⟵</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mi>C</mi></munderover></mtd></mtr> <mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow><mi>𝒟</mi><mphantom><mi>AAAA</mi></mphantom><mover><mo>↦</mo><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mphantom><mi>AAAA</mi></mphantom><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mrow><mtable><mtr><mtd><munderover><mo>⟶</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mphantom><mi>Qu</mi></mphantom></msub></mrow><mrow><mi>𝕃</mi><mi>L</mi></mrow></munderover></mtd></mtr> <mtr><mtd><munderover><mo>⟵</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mphantom><mi>Qu</mi></mphantom></msub></mrow><mrow><mi>𝕃</mi><mi>C</mi><mo>≃</mo><mi>ℝ</mi><mi>C</mi></mrow></munderover></mtd></mtr> <mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>ℝ</mi><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D}) </annotation></semantics></math></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub><mrow><mtable><mtr><mtd><munderover><mo>⟶</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mi>L</mi></munderover></mtd></mtr> <mtr><mtd><munderover><mo>⟵</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><mi>C</mi></munderover></mtd></mtr> <mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow><mi>𝒟</mi><mphantom><mi>AAAA</mi></mphantom><mover><mo>↦</mo><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mphantom><mi>AAAA</mi></mphantom><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mrow><mtable><mtr><mtd><munderover><mo>⟶</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mphantom><mi>Qu</mi></mphantom></msub></mrow><mrow><mi>𝕃</mi><mi>L</mi></mrow></munderover></mtd></mtr> <mtr><mtd><munderover><mo>⟵</mo><mrow><msub><mrow></mrow> <mphantom><mi>Qu</mi></mphantom></msub><msub><mo>⊥</mo> <mphantom><mi>Qu</mi></mphantom></msub></mrow><mrow><mi>𝕃</mi><mi>C</mi><mo>≃</mo><mi>ℝ</mi><mi>C</mi></mrow></munderover></mtd></mtr> <mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>ℝ</mi><mi>R</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D}) </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_24">Proof</h6> <p>This follows immediately from the fact that passing to <a class="existingWikiWord" href="/nlab/show/homotopy+categories+of+model+categories">homotopy categories of model categories</a> is a <a class="existingWikiWord" href="/nlab/show/double+pseudofunctor">double pseudofunctor</a> from the <a class="existingWikiWord" href="/nlab/show/double+category+of+model+categories">double category of model categories</a> to the <a class="existingWikiWord" href="/nlab/show/double+category+of+squares">double category of squares</a> in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> (Prop. <a class="maruku-ref" href="#HomotopyDoublePseudofunctor"></a>).</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="MappingCones">Mapping cones</h3> <p>In the context of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram, such as in the definition of the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> in example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ fib(f) &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y } </annotation></semantics></math></div> <p>ought to <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commute</a> only up to a (left/right) <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> (def. <a class="maruku-ref" href="#LeftAndRightHomotopyInAModelCategory"></a>) between the outer composite morphisms. Moreover, it should satisfy its <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> up to such homotopies.</p> <p>Instead of going through the full theory of what this means, we observe that this is plausibly modeled by the following construction, and then we check (<a href="#HomotopyFibers">below</a>) that this indeed has the relevant abstract homotopy theoretic properties.</p> <div class="num_defn" id="MappingConeAndMappingCocone"> <h6 id="definition_22">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, def. <a class="maruku-ref" href="#ModelCategory"></a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> its model structure on pointed objects, prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> a morphism between cofibrant objects (hence a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><msub><mo stretchy="false">)</mo> <mi>c</mi></msub><mo>↪</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(\mathcal{C}^{\ast/})_c\hookrightarrow \mathcal{C}^{\ast/}</annotation></semantics></math>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>), its <strong>reduced <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></strong> is the object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><mo>*</mo><munder><mo>⊔</mo><mi>X</mi></munder><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munder><mo>⊔</mo><mi>X</mi></munder><mi>Y</mi></mrow><annotation encoding="application/x-tex"> Cone(f) \coloneqq \ast \underset{X}{\sqcup} Cyl(X) \underset{X}{\sqcup} Y </annotation></semantics></math></div> <p>in the colimiting diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>η</mi></mpadded></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) \\ \downarrow && & \searrow^{\mathrlap{\eta}} & \downarrow \\ {*} &\longrightarrow& &\longrightarrow& Cone(f) } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(X)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>.</p> <p>Dually, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> a morphism between fibrant objects (hence a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mo>*</mo></msup><msub><mo stretchy="false">)</mo> <mi>f</mi></msub><mo>↪</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(\mathcal{C}^{\ast})_f\hookrightarrow \mathcal{C}^{\ast/}</annotation></semantics></math>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>), its <strong><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></strong> is the object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><mo>*</mo><munder><mo>×</mo><mi>Y</mi></munder><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mi>Y</mi></munder><mi>Y</mi></mrow><annotation encoding="application/x-tex"> Path_\ast(f) \coloneqq \ast \underset{Y}{\times} Path(Y)\underset{Y}{\times} Y </annotation></semantics></math></div> <p>in the following limit diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>η</mi></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ Path_\ast(f) &\longrightarrow& &\longrightarrow& X \\ \downarrow &\searrow^{\mathrlap{\eta}}& && \downarrow^{\mathrlap{f}} \\ && Path(Y) &\underset{p_1}{\longrightarrow}& Y \\ \downarrow && \downarrow^{\mathrlap{p_0}} \\ \ast &\longrightarrow& Y } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(Y)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>.</p> </div> <div class="num_remark"> <h6 id="remark_8">Remark</h6> <p>When we write homotopies (def. <a class="maruku-ref" href="#LeftAndRightHomotopyInAModelCategory"></a>) as double arrows between morphisms, then the limit diagram in def. <a class="maruku-ref" href="#MappingConeAndMappingCocone"></a> looks just like the square in the definition of <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> in example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a>, except that it is filled by the <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> given by the component map denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>η</mi></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Path_\ast(f) &\longrightarrow& X \\ \downarrow &\swArrow_{\eta}& \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>Dually, the colimiting diagram for the mapping cone turns to look just like the square for the <a class="existingWikiWord" href="/nlab/show/cofiber">cofiber</a>, except that it is filled with a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>η</mi></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &\swArrow_{\eta}& \downarrow \\ \ast &\longrightarrow& Cone(f) } </annotation></semantics></math></div></div> <div class="num_prop" id="ConeAndMappingCylinder"> <h6 id="proposition_15">Proposition</h6> <p>The colimit appearing in the definition of the reduced <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> in def. <a class="maruku-ref" href="#MappingConeAndMappingCocone"></a> is equivalent to three consecutive <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cyl(f) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& Cone(f) } \,. </annotation></semantics></math></div> <p>The two intermediate objects appearing here are called</p> <ul> <li> <p>the plain <strong>reduced <a class="existingWikiWord" href="/nlab/show/cone">cone</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mo>*</mo><munder><mo>⊔</mo><mi>X</mi></munder><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cone(X) \coloneqq \ast \underset{X}{\sqcup} Cyl(X)</annotation></semantics></math>;</p> </li> <li> <p>the <strong>reduced <a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munder><mo>⊔</mo><mi>X</mi></munder><mi>Y</mi></mrow><annotation encoding="application/x-tex">Cyl(f) \coloneqq Cyl(X) \underset{X}{\sqcup} Y</annotation></semantics></math>.</p> </li> </ul> <p>Dually, the limit appearing in the definition of the <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> in def. <a class="maruku-ref" href="#MappingConeAndMappingCocone"></a> is equivalent to three consecutive <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Path_\ast(f) &\longrightarrow& Path(f) &\longrightarrow& X \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ Path_\ast(Y) &\longrightarrow& Path(Y) &\underset{p_1}{\longrightarrow}& Y \\ \downarrow &(pb)& \downarrow^{\mathrlap{p_0}} \\ \ast &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>The two intermediate objects appearing here are called</p> <ul> <li> <p>the <strong>based path space object</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≔</mo><mo>*</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>Y</mi></munder><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path_\ast(Y) \coloneqq \ast \underset{Y}{\prod} Path(Y)</annotation></semantics></math>;</p> </li> <li> <p>the <strong>mapping path space</strong> or <strong>mapping co-cylinder</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>X</mi><munder><mo>×</mo><mi>Y</mi></munder><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(f) \coloneqq X \underset{Y}{\times} Path(X)</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_defn" id="SuspensionAndLoopSpaceObject"> <h6 id="definition_23">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}^{\ast/}</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a>.</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, def. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to \ast</annotation></semantics></math> is called the <strong><a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced</a> <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>X</mi><mo>=</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Sigma X = Cone(X\to\ast)\,. </annotation></semantics></math></div> <p>Via prop. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a> this is equivalently the coproduct of two copies of the cone on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> over their base:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Σ</mi><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X &\stackrel{}{\longrightarrow}& \ast \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cone(X) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& \Sigma X } \,. </annotation></semantics></math></div> <p>This is also equivalently the <a class="existingWikiWord" href="/nlab/show/cofiber">cofiber</a>, example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i_0,i_1)</annotation></semantics></math>, hence (example <a class="maruku-ref" href="#WedgeSumAsCoproduct"></a>) of the <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a> inclusion:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∨</mo><mi>X</mi><mo>=</mo><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>cofib</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>Σ</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \vee X = X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \overset{cofib(i_0,i_1)}{\longrightarrow} \Sigma X \,. </annotation></semantics></math></div></li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a>, def. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\ast \to X</annotation></semantics></math> is called the <strong><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>=</mo><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mo>*</mo><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega X = Path_\ast(\ast \to X) \,. </annotation></semantics></math></div> <p>Via prop. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a> this is equivalently</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Ω</mi><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munder></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Omega X &\longrightarrow& Path_\ast(X) &\longrightarrow& \ast \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow^{} \\ Path_\ast(X) &\longrightarrow& Path(X) &\underset{p_1}{\longrightarrow}& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{p_0}} \\ \ast &\longrightarrow& X } \,. </annotation></semantics></math></div> <p>This is also equivalently the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>, example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p_0,p_1)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mover><mo>⟶</mo><mrow><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega X \overset{fib(p_0,p_1)}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X \,. </annotation></semantics></math></div></li> </ol> </div> <div class="num_prop" id="ReducedSuspensionBySmashProductWithCircle"> <h6 id="proposition_16">Proposition</h6> <p>In <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Top^{\ast/}</annotation></semantics></math>,</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a> objects (def. <a class="maruku-ref" href="#SuspensionAndLoopSpaceObject"></a>) induced from the standard <a class="existingWikiWord" href="/nlab/show/reduced+cylinder">reduced cylinder</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-)\wedge (I_+)</annotation></semantics></math> of example <a class="maruku-ref" href="#StandardReducedCyclinderInTop"></a> are isomorphic to the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> (def. <a class="maruku-ref" href="#SmashProductOfPointedObjects"></a>) with the <a class="existingWikiWord" href="/nlab/show/1-sphere">1-sphere</a>, for later purposes we choose to smash <strong>on the left</strong> and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>cofib</mi><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> cofib(X \vee X \to X \wedge (I_+)) \simeq S^1 \wedge X \,, </annotation></semantics></math></div></li> </ul> <p>Dually:</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/loop+space+objects">loop space objects</a> (def. <a class="maruku-ref" href="#SuspensionAndLoopSpaceObject"></a>) induced from the standard pointed path space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">Maps(I_+,-)_\ast</annotation></semantics></math> are isomorphic to the <a class="existingWikiWord" href="/nlab/show/pointed+mapping+space">pointed mapping space</a> (example <a class="maruku-ref" href="#PointedMappingSpace"></a>) with the <a class="existingWikiWord" href="/nlab/show/1-sphere">1-sphere</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>fib</mi><mo stretchy="false">(</mo><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo>,</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Maps</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> fib(Maps(I_+,X)_\ast \to X \times X) \simeq Maps(S^1, X)_\ast \,. </annotation></semantics></math></div></li> </ul> </div> <div class="proof"> <h6 id="proof_25">Proof</h6> <p>By immediate inspection: For instance the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo>,</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>⟶</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Maps(I_+,X)_\ast \longrightarrow X\times X</annotation></semantics></math> is clearly the subspace of the unpointed mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">X^I</annotation></semantics></math> on elements that take the endpoints of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> to the basepoint of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_example" id="MappingConesInTopologicalSpaces"> <h6 id="example_4">Example</h6> <div style="float:right;margin:0 10px 10px 0;"> <img src="http://ncatlab.org/nlab/files/mappingcone.jpg" width="560" /> </div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">Cyl(X) = X\times I</annotation></semantics></math> the standard cyclinder object, def. <a class="maruku-ref" href="#TopologicalInterval"></a>, then by example <a class="maruku-ref" href="#PushoutInTop"></a>, the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, def. <a class="maruku-ref" href="#MappingConeAndMappingCocone"></a>, of a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> is obtained by</p> <ol> <li> <p>forming the cylinder over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>attaching to one end of that cylinder the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> as specified by the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> </li> <li> <p>shrinking the other end of the cylinder to the point.</p> </li> </ol> <p>Accordingly the <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> of a topological space is the result of shrinking both ends of the cylinder on the object two the point. This is homeomoprhic to attaching two copies of the cone on the space at the base of the cone.</p> <p>(graphics taken from <a href="https://personal.us.es/fmuro/files/slides/praha.pdf">Muro 2010</a>)</p> <p>Below in example <a class="maruku-ref" href="#StandardTopologicalMappingConeIsHomotopyCofiber"></a> we find the homotopy-theoretic interpretation of this standard topological mapping cone as a model for the <em><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></em>.</p> </div> <div class="num_remark" id="UnreducedCone"> <h6 id="remark_9">Remark</h6> <p>The <em>formula</em> for the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> in prop. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a> (as opposed to that of the mapping co-cone) does not require the presence of the basepoint: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (as opposed to in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math>) we may still define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo>′</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Y</mi><munder><mo>⊔</mo><mi>X</mi></munder><mi>Cone</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Cone'(f) \coloneqq Y \underset{X}{\sqcup} Cone'(X) \,, </annotation></semantics></math></div> <p>where the prime denotes the <em>unreduced cone</em>, formed from a cylinder object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </div> <div class="num_prop" id="UnreducedMappingConeAsReducedConeOfBasedPointAdjoined"> <h6 id="proposition_17">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> a morphism in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, then its unreduced mapping cone, remark <a class="maruku-ref" href="#UnreducedCone"></a>, with respect to the standard cylinder object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \times I</annotation></semantics></math> def. <a class="maruku-ref" href="#TopologicalInterval"></a>, is isomorphic to the reduced mapping cone, def. <a class="maruku-ref" href="#MappingConeAndMappingCocone"></a>, of the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>+</mo></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>Y</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">f_+ \colon X_+ \to Y_+</annotation></semantics></math> (with a basepoint adjoined, def. <a class="maruku-ref" href="#BasePointAdjoined"></a>) with respect to the standard <a class="existingWikiWord" href="/nlab/show/reduced+cylinder">reduced cylinder</a> (example <a class="maruku-ref" href="#StandardReducedCyclinderInTop"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo>′</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Cone</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Cone'(f) \simeq Cone(f_+) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_26">Proof</h6> <p>By prop. <a class="maruku-ref" href="#LimitsAndColimitsOfPointedObjects"></a> and example <a class="maruku-ref" href="#WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>+</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cone(f_+)</annotation></semantics></math> is given by the colimit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> over the following diagram:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>⊔</mo><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>⊔</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi><mo>⊔</mo><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊔</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>+</mo></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone(f_+) } \,. </annotation></semantics></math></div> <p>We may factor the vertical maps to give</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>⊔</mo><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>⊔</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi><mo>⊔</mo><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊔</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo><mo>⊔</mo><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo>′</mo><mo stretchy="false">(</mo><mi>f</mi><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo>′</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast \sqcup \ast &\longrightarrow& &\longrightarrow& Cone'(f)_+ \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone'(f) } \,. </annotation></semantics></math></div> <p>This way the top part of the diagram (using the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> to compute the colimit in two stages) is manifestly a cocone under the result of applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">(-)_+</annotation></semantics></math> to the diagram for the unreduced cone. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">(-)_+</annotation></semantics></math> is itself given by a colimit, it preserves colimits, and hence gives the partial colimit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo>′</mo><mo stretchy="false">(</mo><mi>f</mi><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">Cone'(f)_+</annotation></semantics></math> as shown. The remaining pushout then contracts the remaining copy of the point away.</p> </div> <p>Example <a class="maruku-ref" href="#MappingConesInTopologicalSpaces"></a> makes it clear that every <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">S^n \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> that happens to be in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> can be <em>continuously</em> translated in the cylinder-direction, keeping it constant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, to the other end of the cylinder, where it shrinks away to the point. This means that every <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, def. <a class="maruku-ref" href="#HomotopyGroupsOftopologicalSpaces"></a>, in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> vanishes in the mapping cone. Hence in the mapping cone <strong>the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is removed up to homotopy</strong>. This makes it intuitively clear how <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cone(f)</annotation></semantics></math> is a homotopy-version of the <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. We now discuss this formally.</p> <div class="num_lemma" id="FactorizationLemma"> <h6 id="lemma_11">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_c</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+of+cofibrant+objects">category of cofibrant objects</a>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>. Then for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>-construction in def. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a> provides a cofibration resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, in that</p> <ol> <li> <p>the composite morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mi>f</mi></mrow></mover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \overset{i_0}{\longrightarrow} Cyl(X) \overset{(i_1)_\ast f}{\longrightarrow} Cyl(f)</annotation></semantics></math> is a cofibration;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> factors through this morphism by a weak equivalence left inverse to an acyclic cofibration</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Cof</mi></mrow><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mi>f</mi><mo>∘</mo><msub><mi>i</mi> <mn>0</mn></msub></mrow></munderover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></munder><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \underoverset{\in Cof}{(i_1)_\ast f\circ i_0}{\longrightarrow} Cyl(f) \underset{\in W}{\longrightarrow} Y \,, </annotation></semantics></math></div></li> </ol> <p>Dually:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>. Then for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a>-construction in def. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a> provides a fibration resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, in that</p> <ol> <li> <p>the composite morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi></mrow></mover><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">Path(f) \overset{p_1^\ast f}{\longrightarrow} Path(Y) \overset{p_0}{\longrightarrow} Y</annotation></semantics></math> is a fibration;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> factors through this morphism by a weak equivalence right inverse to an acyclic fibration:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></munder><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo>∘</mo><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi></mrow></munderover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \underset{\in W}{\longrightarrow} Path(f) \underoverset{\in Fib}{p_0 \circ p_1^\ast f}{\longrightarrow} Y \,, </annotation></semantics></math></div></li> </ol> </div> <div class="proof" id="ProofOfFactorizationLemma"> <h6 id="proof_27">Proof</h6> <p>We discuss the second case. The first case is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> <p>So consider the <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a>-construction from prop. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>f</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munderover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Path(f) &\overset{\in W \cap Fib}{\longrightarrow}& X \\ {}^{\mathllap{p_1^\ast f}}\downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ Path(Y) &\underoverset{\in W \cap Fib}{p_1}{\longrightarrow}& Y \\ {}^{\mathllap{\in W \cap Fib}}\downarrow^{\mathrlap{p_0}} \\ Y } \,. </annotation></semantics></math></div> <p>To see that the vertical composite is indeed a fibration, notice that, by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>, the above pullback diagram may be decomposed as a <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> of two pullback diagram as follows</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></munderover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>Y</mi><mo>×</mo><mi>Y</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mfrac linethickness="0"><mrow><msub><mi>pr</mi> <mn>2</mn></msub></mrow><mrow><mrow><mo>∈</mo><mi>Fib</mi></mrow></mrow></mfrac></mpadded></msub></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Path(f) &\underoverset{\in Fib}{(f,id)^\ast(p_1,p_0)}{\longrightarrow}& X \times Y &\stackrel{pr_1}{\to}& X \\ \downarrow && \downarrow^{\mathrlap{(f, Id)}} && \downarrow^\mathrlap{f} \\ Path(Y) &\overset{(p_1,p_0) \in Fib }{\longrightarrow}& Y \times Y &\stackrel{pr_1}{\longrightarrow}& Y \\ {}^{\mathllap{p_0}}\downarrow & \swarrow_{\mathrlap{pr_2 \atop {\in Fib}}} \\ Y } \,. </annotation></semantics></math></div> <p>Both squares are pullback squares. Since pullbacks of fibrations are fibrations by prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Path(f) \to X \times Y</annotation></semantics></math> is a fibration. Similarly, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is fibrant, also the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times Y \to Y</annotation></semantics></math> is a fibration (being the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to \ast</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">Y \to \ast</annotation></semantics></math>).</p> <p>Since the vertical composite is thereby exhibited as the composite of two fibrations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>Y</mi><mover><mo>⟶</mo><mrow><msub><mi>pr</mi> <mn>2</mn></msub><mo>∘</mo><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>pr</mi> <mn>2</mn></msub></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Path(f) \overset{(f,id)^\ast(p_1,p_0)}{\longrightarrow} X \times Y \stackrel{pr_2 \circ (f ,Id) = pr_2}{\longrightarrow} Y \,, </annotation></semantics></math></div> <p>it is itself a fibration.</p> <p>Then to see that there is a weak equivalence as claimed:</p> <p>The <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(f)</annotation></semantics></math> induces a right inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Path(f) \to X</annotation></semantics></math> fitting into this diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>id</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mo>∃</mo></munderover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>Y</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mi>i</mi></munderover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>Id</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ id_X \colon & X &\underoverset{\in W}{\exists}{\longrightarrow} & Path(f) & \overset{\in W \cap Fib}{\longrightarrow}& X \\ & {}^{\mathrlap{f}}\downarrow && \downarrow && \downarrow^{\mathrlap{f}} \\ id_Y\colon& Y &\underoverset{\in W}{i}{\longrightarrow}& Path(Y) &\stackrel{p_1}{\to}& Y \\ & & {}_{\mathllap{Id}}\searrow& \downarrow^{\mathrlap{p_0}} \\ & && Y } \,, </annotation></semantics></math></div> <p>which is a weak equivalence, as indicated, by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>).</p> <p>This establishes the claim.</p> </div> <h3 id="categories_of_fibrant_objects">Categories of fibrant objects</h3> <p><a href="#HomotopyFibers">Below</a> we discuss the homotopy-theoretic properties of the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>- and <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a>-constructions from <a href="#MappingCones">above</a>. Before we do so, we here establish a collection of general facts that hold in <a class="existingWikiWord" href="/nlab/show/categories+of+fibrant+objects">categories of fibrant objects</a> and dually in <a class="existingWikiWord" href="/nlab/show/categories+of+cofibrant+objects">categories of cofibrant objects</a>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>.</p> <p><strong>Literature</strong> (<a href="#Brown73">Brown 73, section 4</a>).</p> <div class="num_lemma" id="ReplacementOfPathObjects"> <h6 id="lemma_12">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \longrightarrow Y</annotation></semantics></math> be a morphism in a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>. Then given any choice of <a class="existingWikiWord" href="/nlab/show/path+space+objects">path space objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(Y)</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, there is a replacement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> by a path space object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{Path(X)}</annotation></semantics></math> along an acylic fibration, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{Path(X)}</annotation></semantics></math> has a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(Y)</annotation></semantics></math> which is compatible with the structure maps, in that the following diagram commutes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟵</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></munder></mtd> <mtd><mover><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover></mtd> <mtd><mover><mo>⟶</mo><mi>ϕ</mi></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>X</mi></msubsup><mo>,</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>X</mi></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>Y</mi></msubsup><mo>,</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>Y</mi></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msubsup><mover><mi>p</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn> <mi>X</mi></msubsup><mo>,</mo><msubsup><mover><mi>p</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn> <mi>X</mi></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>×</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X &\overset{f}{\longrightarrow}& Y \\ &\swarrow& \downarrow && \downarrow \\ Path(X) &\underset{\in W \cap Fib}{\longleftarrow}& \widetilde{Path(X)} &\overset{\phi}{\longrightarrow}& Path(Y) \\ &{}_{\mathllap{(p^X_0,p^X_1)}}\searrow& \downarrow^{\mathrlap{(p^Y_0,p^Y_1)}} && \downarrow^{\mathrlap{(\tilde p^X_0,\tilde p^X_1)}} \\ && X \times X &\overset{(f,f)}{\longrightarrow}& Y \times Y } \,. </annotation></semantics></math></div></div> <p>(<a href="#Brown73">Brown 73, section 2, lemma 2</a>)</p> <div class="proof"> <h6 id="proof_28">Proof</h6> <p>Consider the <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>Y</mi></msubsup><mo>,</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>Y</mi></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>X</mi></msubsup><mo>,</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>X</mi></msubsup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>×</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{f}{\longrightarrow}& Y &\longrightarrow& Path(Y) \\ \downarrow && && \downarrow^{\mathrlap{(p_0^Y, p_1^Y)}} \\ Path(X) &\overset{(p^X_0,p^X_1)}{\longrightarrow}& X \times X &\overset{(f,f)}{\longrightarrow}& Y \times Y } \,. </annotation></semantics></math></div> <p>Then consider its factorization through the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the right morphism along the bottom morphism,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>X</mi></msubsup><mo>,</mo><mi>f</mi><mo>∘</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>X</mi></msubsup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>Y</mi></msubsup><mo>,</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>Y</mi></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>X</mi></msubsup><mo>,</mo><mi>f</mi><mo>∘</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>X</mi></msubsup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>×</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& (f \circ p_0^X, f\circ p_1^X)^\ast Path(Y) &\longrightarrow& Path(Y) \\ &{}_{\mathllap{\in W}}\searrow& \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{(p_0^Y, p_1^Y)}}_{\mathrlap{\in Fib}} \\ && Path(X) &\overset{(f \circ p_0^X, f\circ p_1^X)}{\longrightarrow}& Y \times Y } \,. </annotation></semantics></math></div> <p>Finally use the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a> to factor the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>X</mi></msubsup><mo>,</mo><mi>f</mi><mo>∘</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>X</mi></msubsup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to (f \circ p_0^X, f\circ p_1^X)^\ast Path(Y)</annotation></semantics></math> through a weak equivalence followed by a fibration, the object this factors through serves as the desired path space resolution</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><mover><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>Y</mi></msubsup><mo>,</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>Y</mi></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><msubsup><mi>p</mi> <mn>0</mn> <mi>X</mi></msubsup><mo>,</mo><mi>f</mi><mo>∘</mo><msubsup><mi>p</mi> <mn>1</mn> <mi>X</mi></msubsup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Y</mi><mo>×</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{\in W}{\longrightarrow}& \widetilde{Path(X)} &\longrightarrow& Path(Y) \\ &{}_{\mathllap{\in W}}\searrow& \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{(p_0^Y, p_1^Y)}} \\ && Path(X) &\overset{(f \circ p_0^X, f\circ p_1^X)}{\longrightarrow}& Y \times Y } \,. </annotation></semantics></math></div></div> <div class="num_lemma" id="BaseChangePreservesFibrationsAndWeakEquivalences"> <h6 id="lemma_13">Lemma</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><mo>∈</mo><mi>Fib</mi></mrow></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><mo>∈</mo><mi>Fib</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A_1 &&\stackrel{f}{\longrightarrow}&& A_2 \\ & {}_{\in Fib}\searrow && \swarrow_{\in Fib} \\ && B } </annotation></semantics></math></div> <p>be a morphism over some object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>′</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">u \colon B' \to B</annotation></semantics></math> be any morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math>. Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>u</mi> <mo>*</mo></msup><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>f</mi></mrow></mover></mtd> <mtd></mtd> <mtd><msup><mi>u</mi> <mo>*</mo></msup><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><mo>∈</mo><mi>Fib</mi></mrow></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><mo>∈</mo><mi>Fib</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ u^*A_1 &&\stackrel{u^* f}{\longrightarrow}&& u^* A_2 \\ & {}_{\in Fib}\searrow && \swarrow_{\in Fib} \\ && B' } </annotation></semantics></math></div> <p>be the corresponding morphism pulled back along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>.</p> <p>Then</p> <ul> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a fibration then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>f</mi></mrow><annotation encoding="application/x-tex">u^* f</annotation></semantics></math> is a fibration;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a weak equivalence then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>f</mi></mrow><annotation encoding="application/x-tex">u^* f</annotation></semantics></math> is a weak equivalence.</p> </li> </ul> </div> <p>(<a href="#Brown73">Brown 73, section 4, lemma 1</a>)</p> <div class="num_proof"> <h6 id="proof_29">Proof</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>Fib</mi></mrow><annotation encoding="application/x-tex">f \in Fib</annotation></semantics></math> the statement follows from the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> which says that if in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>B</mi><mo>′</mo><msub><mo>×</mo> <mi>B</mi></msub><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>f</mi><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>f</mi><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi><mo>′</mo><msub><mo>×</mo> <mi>B</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mi>u</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ B' \times_B A_1 &\longrightarrow& A_1 \\ \;\;\downarrow^{\mathrlap{u^* f \in Fib}} && \;\;\downarrow^{\mathrlap{f \in Fib}} \\ B' \times_B A_2 &\longrightarrow& A_2 \\ \;\downarrow^{\mathrlap{\in Fib}} && \;\downarrow^{\mathrlap{\in Fib}} \\ B' &\stackrel{u}{\longrightarrow}& B } </annotation></semantics></math></div> <p>the bottom and the total square are pullback squares, then so is the top square. The same reasoning applies for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow><annotation encoding="application/x-tex">f \in W \cap Fib</annotation></semantics></math>.</p> <p>Now to see the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f\in W</annotation></semantics></math>:</p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}_{/B})_f</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/B}</annotation></semantics></math> (def. <a class="maruku-ref" href="#SliceCategory"></a>) on its fibrant objects, i.e. the full subcategory of the slice category on the fibrations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mi>p</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X \\ \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib}} \\ B } </annotation></semantics></math></div> <p>into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. By factorizing for every such fibration the <a class="existingWikiWord" href="/nlab/show/diagonal+morphisms">diagonal morphisms</a> into the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><munder><mo>×</mo><mi>B</mi></munder><mi>X</mi></mrow><annotation encoding="application/x-tex">X \underset{B}{\times} X</annotation></semantics></math> through a weak equivalence followed by a fibration, we obtain path space objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path_B(X)</annotation></semantics></math> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>X</mi><munder><mo>×</mo><mi>B</mi></munder><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (\Delta_X)/B \;\colon & X &\overset{\in W}{\longrightarrow}& Path_B(X) &\overset{\in Fib}{\longrightarrow}& X \underset{B}{\times} X \\ & & {}_{\mathllap{\in Fib}}\searrow & \downarrow & \swarrow_{\mathrlap{\in Fib}} \\ & && B } \,. </annotation></semantics></math></div> <p>With these, the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> (lemma <a class="maruku-ref" href="#FactorizationLemma"></a>) applies in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}_{/B})_f</annotation></semantics></math>.</p> <p>(Notice that for this we do need the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/B}</annotation></semantics></math> to the fibrations, because this ensures that the projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo>×</mo> <mi>B</mi></msub><msub><mi>X</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i \colon X_1 \times_B X_2 \to X_i</annotation></semantics></math> are still fibrations, which is used in the proof of the factorization lemma (<a href="#ProofOfFactorizationLemma">here</a>).)</p> <p>So now given any</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mi>f</mi></munderover></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X && \underoverset{\in W}{f}{\longrightarrow} && Y \\ & {}_{\mathllap{\in Fib}}\searrow && \swarrow_{\mathrlap{\in Fib}} \\ && B } </annotation></semantics></math></div> <p>apply the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}_{/B})_f</annotation></semantics></math> to factor it as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>i</mi><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{i \in W}{\longrightarrow}& Path_B(f) &\overset{\in W \cap Fib}{\longrightarrow}& Y \\ & {}_{\mathllap{\in Fib}}\searrow &\downarrow& \swarrow_{\mathrlap{\in Fib}} \\ && B } \,. </annotation></semantics></math></div> <p>By the previous discussion it is sufficient now to show that the base change of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">B'</annotation></semantics></math> is still a weak equivalence. But by the factorization lemma in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}_{/B})_f</annotation></semantics></math>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is right inverse to another acyclic fibration over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>id</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>i</mi><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ id_X \;\colon & X &\overset{i \in W}{\longrightarrow}& Path_B(f) &\overset{\in W \cap Fib}{\longrightarrow}& X \\ & & {}_{\mathllap{\in Fib}}\searrow &\downarrow& \swarrow_{\mathrlap{\in Fib}} \\ & && B } \,. </annotation></semantics></math></div> <p>(Notice that if we had applied the factorization lemma of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">(\Delta_X)/B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}_{/B})</annotation></semantics></math> then the corresponding triangle on the right here would not commute.)</p> <p>Now we may reason as before: the base change of the top morphism here is exhibited by the following pasting composite of pullbacks:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>B</mi><mo>′</mo><munder><mo>×</mo><mi>B</mi></munder><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi><mo>′</mo><munder><mo>×</mo><mi>B</mi></munder><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi><mo>′</mo><munder><mo>×</mo><mi>B</mi></munder><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi><mo>′</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ B' \underset{B}{\times} X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow \\ B' \underset{B}{\times} Path_B(f) &\longrightarrow& Path_B(f) \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in W \cap Fib}} \\ B' \underset{B}{\times}X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow \\ B' &\longrightarrow& B } \,. </annotation></semantics></math></div> <p>The acyclic fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path_B(f)</annotation></semantics></math> is preserved by this pullback, as is the identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">id_X \colon X \to Path_B(X)\to X</annotation></semantics></math>. Hence the weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to Path_B(X)</annotation></semantics></math> is preserved by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>).</p> </div> <div class="num_lemma" id="InCfPullbackAlongFibrationPreservesWeakEquivalences"> <h6 id="lemma_14">Lemma</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>, the pullback of a weak equivalence along a fibration is again a weak equivalence.</p> </div> <p>(<a href="#Brown73">Brown 73, section 4, lemma 2</a>)</p> <div class="proof"> <h6 id="proof_30">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>′</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">u \colon B' \to B</annotation></semantics></math> be a weak equivalence and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex"> p \colon E \to B</annotation></semantics></math> be a fibration. We want to show that the left vertical morphism in the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>B</mi><mo>′</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>⇒</mo><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ E \times_B B' &\longrightarrow& B' \\ \downarrow^{\mathrlap{\Rightarrow \in W} } && \;\downarrow^{\mathrlap{\in W}} \\ E &\stackrel{\in Fib}{\longrightarrow}& B } </annotation></semantics></math></div> <p>is a weak equivalence.</p> <p>First of all, using the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a> we may factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>′</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">B' \to B</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>′</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mi>Path</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>F</mi></mrow></mover><mi>B</mi></mrow><annotation encoding="application/x-tex"> B' \stackrel{\in W}{\longrightarrow} Path(u) \stackrel{\in W \cap F}{\longrightarrow} B </annotation></semantics></math></div> <p>with the first morphism a weak equivalence that is a right inverse to an acyclic fibration and the right one an acyclic fibration.</p> <p>Then the pullback diagram in question may be decomposed into two consecutive pullback diagrams</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>B</mi><mo>′</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ E \times_B B' &\to& B' \\ \downarrow && \downarrow \\ Q &\stackrel{\in Fib}{\to}& Path(u) \\ \;\;\downarrow^{\mathrlap{\in W \cap Fib}} && \;\;\downarrow^{\mathrlap{\in W \cap Fib}} \\ E &\stackrel{\in Fib}{\longrightarrow}& B } \,, </annotation></semantics></math></div> <p>where the morphisms are indicated as fibrations and acyclic fibrations using the stability of these under arbitrary pullback.</p> <p>This means that the proof reduces to proving that weak equivalences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mi>B</mi><mo>′</mo><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">u : B' \stackrel{\in W}{\to} B</annotation></semantics></math> that are right inverse to some acyclic fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>:</mo><mi>B</mi><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>F</mi></mrow></mover><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">v : B \stackrel{\in W \cap F}{\to} B'</annotation></semantics></math> map to a weak equivalence under pullback along a fibration.</p> <p>Given such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> with right inverse <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>, consider the pullback diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mfrac linethickness="0"><mrow><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mo>∈</mo><mi>W</mi></mrow></mfrac></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>E</mi> <mn>1</mn></msub><mo>≔</mo></mtd> <mtd><mi>B</mi><msub><mo>×</mo> <mrow><mi>B</mi><mo>′</mo></mrow></msub><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>p</mi><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>v</mi><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>v</mi><mo>∈</mo><mi>Fib</mi><mo>∩</mo><mi>W</mi></mrow></mover></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & E \\ & {}^{\mathllap{{(p,id)}\atop \in W}}\downarrow & \searrow^{\mathrlap{id}} \\ E_1 \coloneqq & B \times_{B'} E & \stackrel{\in W \cap Fib }{\longrightarrow} & E \\ & \downarrow^{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{p \in Fib }} \\ & &(pb)& B \\ & \downarrow && \downarrow^{\mathrlap{v \in W \cap Fib}} \\ & B &\overset{v \in Fib \cap W}{\longrightarrow}& B' } \,. </annotation></semantics></math></div> <p>Notice that the indicated universal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>×</mo><mi>Id</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p \times Id \colon E \stackrel{\in W}{\to} E_1</annotation></semantics></math> into the pullback is a weak equivalence by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>).</p> <p>The previous lemma <a class="maruku-ref" href="#BaseChangePreservesFibrationsAndWeakEquivalences"></a> says that weak equivalences between fibrations over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> are themselves preserved by base extension along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>′</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">u \colon B' \to B</annotation></semantics></math>. In total this yields the following diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>u</mi> <mo>*</mo></msup><mi>E</mi><mo>=</mo><mi>B</mi><mo>′</mo><msub><mo>×</mo> <mi>B</mi></msub><mi>E</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mfrac linethickness="0"><mrow><mrow><msup><mi>u</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>p</mi><mo>×</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mo>∈</mo><mi>W</mi></mrow></mrow></mfrac></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mfrac linethickness="0"><mrow><mrow><mi>p</mi><mo>×</mo><mi>Id</mi></mrow></mrow><mrow><mrow><mo>∈</mo><mi>W</mi></mrow></mrow></mfrac></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>u</mi> <mo>*</mo></msup><msub><mi>E</mi> <mn>1</mn></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>E</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>p</mi><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>v</mi><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mi>u</mi></mover></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>v</mi><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && u^* E = B' \times_B E &\longrightarrow & E \\ && {}^{\mathllap{ {u^*(p \times Id)} \atop {\in W} }}\downarrow && {}^{\mathllap{ {p \times Id} \atop {\in W} }}\downarrow & \searrow^{\mathrlap{id}} \\ && u^* E_1 &\longrightarrow& E_1 &\stackrel{\in W \cap Fib}{\longrightarrow}& E \\ &&\downarrow^{\mathrlap{\in Fib}}&&\downarrow^{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{p \in Fib}} \\ &&&&&& B \\ &&\downarrow&&\downarrow && \downarrow^{\mathrlap{v \in W \cap Fib}} \\ && B' &\stackrel{u}{\longrightarrow}& B &\stackrel{v \in W \cap Fib}{\longrightarrow}& B' } </annotation></semantics></math></div> <p>so that with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>×</mo><mi>Id</mi><mo>:</mo><mi>E</mi><mo>→</mo><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p \times Id : E \to E_1</annotation></semantics></math> a weak equivalence also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>p</mi><mo>×</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u^* (p \times Id)</annotation></semantics></math> is a weak equivalence, as indicated.</p> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>E</mi><mo>=</mo><mi>B</mi><mo>′</mo><msub><mo>×</mo> <mi>B</mi></msub><mi>E</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">u^* E = B' \times_B E \to E</annotation></semantics></math> is the morphism that we want to show is a weak equivalence. By <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) for that it is now sufficient to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><msub><mi>E</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">u^* E_1 \to E_1</annotation></semantics></math> is a weak equivalence.</p> <p>That finally follows now since, by assumption, the total bottom horizontal morphism is the identity. Hence so is the top horizontal morphism. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><msub><mi>E</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">u^\ast E_1 \to E_1</annotation></semantics></math> is right inverse to a weak equivalence, hence is a weak equivalence.</p> </div> <div class="num_lemma" id="UniquenessOfFibersOfQualizedMorphismsInHoC"> <h6 id="lemma_15">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}^{\ast/})_f</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a> in a <a class="existingWikiWord" href="/nlab/show/slice+model+structure">model structure on pointed objects</a> (prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>). Given any <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><msub><mo>′</mo> <mn>1</mn></msub></mtd> <mtd><munderover><mo>⟶</mo><mi>t</mi><mrow><mo>∈</mo><mi>W</mi></mrow></munderover></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msubsup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>u</mi></mover></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X'_1 &\underoverset{t}{\in W}{\longrightarrow}& X_1 &\stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}& X_2 \\ && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in Fib}} \\ && B &\overset{u}{\longrightarrow}& C } </annotation></semantics></math></div> <p>(meaning: both squares commute and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> equalizes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>) then the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><msub><mo stretchy="false">)</mo> <mi>f</mi></msub><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma \colon (\mathcal{C}^{\ast/})_f \to Ho(\mathcal{C}^{\ast/})</annotation></semantics></math> (def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>, cor <a class="maruku-ref" href="#HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects"></a>) takes the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">fib(p_1) \stackrel{\longrightarrow}{\longrightarrow} fib(p_2)</annotation></semantics></math> induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> (example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a>) to the same morphism, in the homotopy category.</p> </div> <p>(<a href="#Brown73">Brown 73, section 4, lemma 4</a>)</p> <div class="proof"> <h6 id="proof_31">Proof</h6> <p>First consider the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>: this forms the same kind of diagram but with the bottom morphism an identity. Hence it is sufficient to consider this special case.</p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}^{\ast/}_{/B})_f</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}_{/B}</annotation></semantics></math> (def. <a class="maruku-ref" href="#SliceCategory"></a>) on its fibrant objects, i.e. the full subcategory of the slice category on the fibrations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mi>p</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X \\ \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib}} \\ B } </annotation></semantics></math></div> <p>into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. By factorizing for every such fibration the <a class="existingWikiWord" href="/nlab/show/diagonal+morphisms">diagonal morphisms</a> into the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><munder><mo>×</mo><mi>B</mi></munder><mi>X</mi></mrow><annotation encoding="application/x-tex">X \underset{B}{\times} X</annotation></semantics></math> through a weak equivalence followed by a fibration, we obtain path space objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path_B(X)</annotation></semantics></math> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>X</mi><munder><mo>×</mo><mi>B</mi></munder><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (\Delta_X)/B \;\colon & X &\overset{\in W}{\longrightarrow}& Path_B(X) &\overset{\in Fib}{\longrightarrow}& X \underset{B}{\times} X \\ & & {}_{\mathllap{\in Fib}}\searrow & \downarrow & \swarrow_{\mathrlap{\in Fib}} \\ & && B } \,. </annotation></semantics></math></div> <p>With these, the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> (lemma <a class="maruku-ref" href="#FactorizationLemma"></a>) applies in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}^{\ast/}_{/B})_f</annotation></semantics></math>.</p> <p>Let then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>s</mi></mover><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><msub><mi>X</mi> <mn>2</mn></msub><msub><mo>×</mo> <mi>B</mi></msub><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X\overset{s}{\to}Path_B(X_2)\overset{(p_0,p_1)}{\to} X_2 \times_B X_2</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_2</annotation></semantics></math> in the slice over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and consider the following commuting square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><msub><mo>′</mo> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>s</mi><mi>f</mi><mi>t</mi></mrow></mover></mtd> <mtd><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>t</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub><munder><mo>×</mo><mi>B</mi></munder><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X'_1 &\overset{s f t}{\longrightarrow}& Path_B(X_2) \\ {}^{\mathllap{t}}_{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{(p_0,p_1)}}_{\mathrlap{\in Fib}} \\ X_1 &\overset{(f,g)}{\longrightarrow}& X_2\underset{B}{\times} X_2 } \,. </annotation></semantics></math></div> <p>By factoring this through the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g)^\ast(p_0,p_1)</annotation></semantics></math> and then applying the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a> and then <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) to the factoring morphisms, this may be replaced by a commuting square of the same form, where however the left morphism is an acyclic fibration</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><msub><mo>″</mo> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mi>Path</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>t</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub><munder><mo>×</mo><mi>B</mi></munder><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X''_1 &\overset{}{\longrightarrow}& Path_B(X_2) \\ {}^{\mathllap{t}}_{\mathllap{\in W\cap Fib}} \downarrow && \downarrow^{\mathrlap{(p_0,p_1)}}_{\mathrlap{\in Fib}} \\ X_1 &\overset{(f,g)}{\longrightarrow}& X_2\underset{B}{\times} X_2 } \,. </annotation></semantics></math></div> <p>This makes also the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>″</mo> <mn>1</mn></msub><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">X''_1 \to B</annotation></semantics></math> be a fibration, so that the whole diagram may now be regarded as a diagram in the category of fibrant objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}_{/B})_f</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> <p>As such, the top horizontal morphism now exhibits a <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> which under <a class="existingWikiWord" href="/nlab/show/localization">localization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>B</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>f</mi></msub><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma_B \;\colon\; (\mathcal{C}_{/B})_f \longrightarrow Ho(\mathcal{C}_{/B})</annotation></semantics></math> (def. <a class="maruku-ref" href="#FibrantCofibrantReplacementFunctorToHomotopyCategory"></a>) of the <a class="existingWikiWord" href="/nlab/show/slice+model+structure">slice model structure</a> (prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>) we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>γ</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \gamma_B(f) = \gamma_B(g) \,. </annotation></semantics></math></div> <p>The result then follows by observing that we have a commuting square of <a class="existingWikiWord" href="/nlab/show/functors">functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><msubsup><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mi>fib</mi></mover></mtd> <mtd><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>γ</mi> <mi>B</mi></msub></mrow></mpadded></msup></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>γ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><msubsup><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ (\mathcal{C}^{\ast/}_{/B})_f &\overset{fib}{\longrightarrow}& \mathcal{C}^{\ast/} \\ \downarrow^{\mathrlap{\gamma_B}} &\swArrow& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}^{\ast/}_{/B}) &\longrightarrow& Ho(\mathcal{C}^{\ast/}) } \,, </annotation></semantics></math></div> <p>because, by lemma <a class="maruku-ref" href="#BaseChangePreservesFibrationsAndWeakEquivalences"></a>, the top and right composite sends weak equivalences to isomorphisms, and hence the bottom filler exists by theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a>. This implies the claim.</p> </div> <h3 id="HomotopyFibers">Homotopy fibers</h3> <p>We now discuss the homotopy-theoretic properties of the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>- and <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a>-constructions from <a href="#MappingCones">above</a>.</p> <p><strong>Literature</strong> (<a href="#Brown73">Brown 73, section 4</a>).</p> <div class="num_remark"> <h6 id="remark_10">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a> with prop. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a> says that the <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> of a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, def. <a class="maruku-ref" href="#MappingConeAndMappingCocone"></a>, is equivalently the plain <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>, example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a>, of a fibrant resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Path</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Path_\ast(f) &\longrightarrow& Path(f) \\ \downarrow &(pb)& \downarrow^{\mathrlap{\tilde f}} \\ \ast &\longrightarrow& Y } \,. </annotation></semantics></math></div></div> <p>The following prop. <a class="maruku-ref" href="#FiberOfFibrationIsCompatibleWithWeakEquivalences"></a> says that, up to equivalence, this situation is independent of the specific fibration resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> provided by the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> (hence by the prescription for the <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a>), but only depends on it being <em>some</em> fibration resolution.</p> <div class="num_prop" id="FiberOfFibrationIsCompatibleWithWeakEquivalences"> <h6 id="proposition_18">Proposition</h6> <p>In the <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}^{\ast/})_f</annotation></semantics></math>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>, of a <a class="existingWikiWord" href="/nlab/show/slice+model+structure">model structure on pointed objects</a> (prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>) consider a morphism of <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>-diagrams, hence a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munderover></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></munderover></mtd> <mtd><msub><mi>Y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ fib(p_1) &\longrightarrow& X_1 &\underoverset{\in Fib}{p_1}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} \\ fib(p_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{p_2}{\longrightarrow}& Y_2 } \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> weak equivalences, then so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_32">Proof</h6> <p>Factor the diagram in question through the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow><mrow><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>p</mi> <mn>2</mn></msub></mrow></munderover></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></munderover></mtd> <mtd><msub><mi>Y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ fib(p_1) &\longrightarrow& X_1 \\ \downarrow^{\mathrlap{h}} && {}^{\mathllap{\in W}}\downarrow & \searrow^{\mathrlap{p_1}} & \\ fib(f^\ast p_2) &\longrightarrow& f^\ast X_2 &\underoverset{\in Fib}{f^\ast p_2}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W}} \\ fib(p_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{p_2}{\longrightarrow}& Y_2 } </annotation></semantics></math></div> <p>and observe that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>fib</mi><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mi>pt</mi> <mo>*</mo></msup><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>p</mi> <mn>2</mn></msub><mo>=</mo><msup><mi>pt</mi> <mo>*</mo></msup><msub><mi>p</mi> <mn>2</mn></msub><mo>=</mo><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">fib(f^\ast p_2) = pt^\ast f^\ast p_2 = pt^\ast p_2 = fib(p_2)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>X</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f^\ast X_2 \to X_2</annotation></semantics></math> is a weak equivalence by lemma <a class="maruku-ref" href="#InCfPullbackAlongFibrationPreservesWeakEquivalences"></a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \to f^\ast X_2</annotation></semantics></math> is a weak equivalence by assumption and by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>);</p> </li> </ol> <p>Moreover, this diagram exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>fib</mi><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h \colon fib(p_1)\to fib(f^\ast p_2) = fib(p_2)</annotation></semantics></math> as the base change, along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><msub><mi>Y</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\ast \to Y_1</annotation></semantics></math>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \to f^\ast X_2</annotation></semantics></math>. Therefore the claim now follows with lemma <a class="maruku-ref" href="#BaseChangePreservesFibrationsAndWeakEquivalences"></a>.</p> </div> <p>Hence we say:</p> <div class="num_defn" id="HomotopyFiber"> <h6 id="definition_24">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> its model category of <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a>, prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> any morphism in its <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}^{\ast/})_f</annotation></semantics></math>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>, then its <strong><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> hofib(f)\longrightarrow X </annotation></semantics></math></div> <p>is the morphism in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C}^{\ast/})</annotation></semantics></math>, def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>, which is represented by the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>, example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a>, of any fibration resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> (hence any fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> factors through a weak equivalence followed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math>).</p> <p>Dually:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> any morphism in its <a class="existingWikiWord" href="/nlab/show/category+of+cofibrant+objects">category of cofibrant objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><msub><mo stretchy="false">)</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">(\mathcal{C}^{\ast/})_c</annotation></semantics></math>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a>, then its <strong><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>⟶</mo><mi>hocofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Y \longrightarrow hocofib(f) </annotation></semantics></math></div> <p>is the morphism in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math>, def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>, which is represented by the <a class="existingWikiWord" href="/nlab/show/cofiber">cofiber</a>, example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a>, of any cofibration resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> (hence any cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> factors as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> followed by a weak equivalence).</p> </div> <div class="num_prop" id="HomotopyFiberIndependentOfChoiceOfFibrantReplacement"> <h6 id="proposition_19">Proposition</h6> <p>The homotopy fiber in def. <a class="maruku-ref" href="#HomotopyFiber"></a> is indeed well defined, in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_2</annotation></semantics></math> two fibration replacements of any morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math>, then their fibers are isomorphic in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C}^{\ast/})</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_33">Proof</h6> <p>It is sufficient to exhibit an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C}^{\ast/})</annotation></semantics></math> from the fiber of the fibration replacement given by the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a> (for any choice of <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a>) to the fiber of any other fibration resolution.</p> <p>Hence given a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon Y \longrightarrow X</annotation></semantics></math> and a factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></munder><mover><mi>X</mi><mo stretchy="false">^</mo></mover><munderover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><mrow><mo>∈</mo><mi>Fib</mi></mrow></munderover><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \underset{\in W}{\longrightarrow} \hat X \underoverset{f_1}{\in Fib}{\longrightarrow} Y </annotation></semantics></math></div> <p>consider, for any choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(Y)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> (def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>), the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mfrac linethickness="0"><mrow><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mrow><mrow><mrow><mo>∈</mo><mi>Fib</mi></mrow></mrow></mfrac></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munderover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mfrac linethickness="0"><mrow><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mrow><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mfrac></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Path(f) &\overset{\in W \cap Fib}{\longrightarrow}& X \\ {}^{\mathllap{\in W}}\downarrow &(pb)& \downarrow^{\mathrlap{\in W}} \\ Path(f_1) &\overset{\in W \cap Fib}{\longrightarrow}& \hat X \\ {}^{\mathllap{\in Fib}}\downarrow &(pb)& \downarrow^{\mathrlap{ {f_1} \atop {\in Fib}}} \\ Path(Y) &\underoverset{\in W \cap Fib}{p_1}{\longrightarrow}& Y \\ {}^{\mathllap{ {p_0} \atop \in W \cap Fib}}\downarrow \\ Y } </annotation></semantics></math></div> <p>as in the proof of lemma <a class="maruku-ref" href="#FactorizationLemma"></a>. Now by repeatedly using prop. <a class="maruku-ref" href="#FiberOfFibrationIsCompatibleWithWeakEquivalences"></a>:</p> <ol> <li> <p>the bottom square gives a weak equivalence from the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(f_1) \to Path(Y)</annotation></semantics></math> to the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_1</annotation></semantics></math>;</p> </li> <li> <p>The square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>id</mi></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Path(f_1) &\overset{id}{\longrightarrow}& Path(f_1) \\ \downarrow && \downarrow \\ Path(Y) &\underset{p_0}{\longrightarrow}& Y } </annotation></semantics></math></div> <p>gives a weak equivalence from the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(f_1) \to Path(Y)</annotation></semantics></math> to the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Path(f_1)\to Y</annotation></semantics></math>.</p> </li> <li> <p>Similarly the total vertical composite gives a weak equivalence via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mi>id</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Path(f) &\overset{\in W}{\longrightarrow}& Path(f_1) \\ \downarrow && \downarrow \\ Y &\underset{id}{\longrightarrow}& Y } </annotation></semantics></math></div></li> </ol> <p>from the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Path(f) \to Y</annotation></semantics></math> to the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Path(f_1)\to Y</annotation></semantics></math>.</p> <p>Together this is a <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> of weak equivalences of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mover><mo>⟵</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mspace width="thickmathspace"></mspace><mi>fib</mi><mo stretchy="false">(</mo><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mspace width="thickmathspace"></mspace><mi>fib</mi><mo stretchy="false">(</mo><mi>Path</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mover><mo>⟵</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mspace width="thickmathspace"></mspace><mi>fib</mi><mo stretchy="false">(</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> fib(f_1) \;\overset{\in W}{\longleftarrow}\; fib(Path(f_1)\to Path(Y)) \;\overset{\in W}{\longrightarrow}\; fib(Path(f_1)\to Y) \;\overset{\in W}{\longleftarrow}\; fib(Path(f) \to Y) </annotation></semantics></math></div> <p>between the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Path(f) \to Y</annotation></semantics></math> and the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_1</annotation></semantics></math>. This gives an isomorphism in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a>.</p> </div> <div class="num_example" id="FibersOfSerreFibrations"> <h6 id="example_5">Example</h6> <p><strong>(fibers of Serre fibrations)</strong></p> <p>In showing that <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a> are abstract fibrations in the sense of <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> theory, theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a> implies that the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> (example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a>) of a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F &\longrightarrow& X \\ && \downarrow^{\mathrlap{p}} \\ && B } </annotation></semantics></math></div> <p>over any point is actually a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> in the sense of def. <a class="maruku-ref" href="#HomotopyFiber"></a>. With prop. <a class="maruku-ref" href="#FiberOfFibrationIsCompatibleWithWeakEquivalences"></a> this implies that the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+type">weak homotopy type</a> of the fiber only depends on the Serre fibration up to weak homotopy equivalence in that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>′</mo><mo>→</mo><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p' \colon X' \to B'</annotation></semantics></math> is another Serre fibration fitting into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo>′</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>p</mi><mo>′</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{\in W_{cl}}{\longrightarrow}& X' \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{p'}} \\ B &\overset{\in W_{cl}}{\longrightarrow}& B' } </annotation></semantics></math></div> <p>then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow></mover><mi>F</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">F \overset{\in W_{cl}}{\longrightarrow} F'</annotation></semantics></math>.</p> <p>In particular this gives that the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+type">weak homotopy type</a> of the fiber of a Serre fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p \colon X \to B</annotation></semantics></math> does not change as the basepoint is moved in the same connected component. For let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>⟶</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\gamma \colon I \longrightarrow B</annotation></semantics></math> be a path between two points</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo>*</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow><mrow><msub><mi>i</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></munderover><mi>I</mi><mover><mo>⟶</mo><mi>γ</mi></mover><mi>B</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> b_{0,1} \;\colon\; \ast \underoverset{\in W_{cl}}{i_{0,1}}{\longrightarrow} I \overset{\gamma}{\longrightarrow} B \,. </annotation></semantics></math></div> <p>Then since all objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> are fibrant, and since the endpoint inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">i_{0,1}</annotation></semantics></math> are weak equivalences, lemma <a class="maruku-ref" href="#InCfPullbackAlongFibrationPreservesWeakEquivalences"></a> gives the <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> of top horizontal weak equivalences in the following diagram:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>F</mi> <mrow><msub><mi>b</mi> <mn>0</mn></msub></mrow></msub><mo>=</mo></mtd> <mtd><msubsup><mi>b</mi> <mn>0</mn> <mo>*</mo></msubsup><mi>p</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow></mover></mtd> <mtd><msup><mi>γ</mi> <mo>*</mo></msup><mi>p</mi></mtd> <mtd><mover><mo>⟵</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow></mover></mtd> <mtd><msubsup><mi>b</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>p</mi></mtd> <mtd><mo>=</mo><msub><mi>F</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mfrac linethickness="0"><mrow><mrow><msup><mi>γ</mi> <mo>*</mo></msup><mi>f</mi></mrow></mrow><mrow><mfrac linethickness="0"><mrow><mo>∈</mo></mrow><mrow><mi>Fib</mi></mrow></mfrac></mrow></mfrac></mpadded></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><munderover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow></munderover></mtd> <mtd><mi>I</mi></mtd> <mtd><munderover><mo>⟵</mo><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow></munderover></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F_{b_0} = & b_0^\ast p &\overset{\in W_{cl}}{\longrightarrow}& \gamma^{\ast}p &\overset{\in W_{cl}}{\longleftarrow}& b_1^\ast p & = F_{b_1} \\ & \downarrow &(pb)& \downarrow{\mathrlap{{\gamma^\ast f} \atop {\in \atop {Fib}}}} &\;\;(pb)& \downarrow \\ & \ast &\underoverset{i_0}{\in W_{cl}}{\longrightarrow}& I &\underoverset{i_1}{\in W_{cl}}{\longleftarrow}& \ast } </annotation></semantics></math></div> <p>and hence an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><msub><mi>b</mi> <mn>0</mn></msub></mrow></msub><mo>≃</mo><msub><mi>F</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">F_{b_0} \simeq F_{b_1}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a> (def. <a class="maruku-ref" href="#ClassicalHomotopyCategory"></a>).</p> <p>The same kind of argument applied to maps from the square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">I^2</annotation></semantics></math> gives that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>γ</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\gamma_1, \gamma_2\colon I \to B</annotation></semantics></math> are two homotopic paths with coinciding endpoints, then the isomorphisms between fibers over endpoints which they induce are equal. (But in general the isomorphism between the fibers does depend on the choice of homotopy class of paths connecting the basepoints!)</p> <p>The same kind of argument also shows that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> has the structure of a <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a> (def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>) then the restriction of the Serre fibration to one cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> may be identified in the homotopy category with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">D^n \times F</annotation></semantics></math>, and may be canonically identified so if the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is trivial. This is used when deriving the <a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre</a>-<a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> (<a href="Introduction+to+Stable+homotopy+theory+--+S#AHSSExistence">prop.</a>).</p> </div> <div class="num_example" id="StandardTopologicalMappingConeIsHomotopyCofiber"> <h6 id="example_6">Example</h6> <p>For every <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>, then the standard topological mapping cone is the <a class="existingWikiWord" href="/nlab/show/attaching+space">attaching space</a> (example <a class="maruku-ref" href="#PushoutInTop"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><msub><mo>∪</mo> <mi>f</mi></msub><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> Y \cup_f Cone(X) \;\; \in Top </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> with the standard cone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cone(X)</annotation></semantics></math> given by collapsing one end of the standard topological cyclinder <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \times I</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicalInterval"></a>) as shown in example <a class="maruku-ref" href="#MappingConesInTopologicalSpaces"></a>.</p> <p>Equipped with the canonical continuous function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>⟶</mo><mi>Y</mi><msub><mo>∪</mo> <mi>f</mi></msub><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Y \longrightarrow Y \cup_f Cone(X) </annotation></semantics></math></div> <p>this represents the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a>, def. <a class="maruku-ref" href="#HomotopyFiber"></a>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> with respect to the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}= Top_{Quillen}</annotation></semantics></math> from theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>.</p> </div> <div class="proof"> <h6 id="proof_34">Proof</h6> <p>By prop. <a class="maruku-ref" href="#TopologicalCylinderOnCWComplexIsGoodCylinderObject"></a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> then the standard topological cylinder object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X\times I</annotation></semantics></math> is indeed a cyclinder object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math>. Therefore by prop. <a class="maruku-ref" href="#ConeAndMappingCylinder"></a> and the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a>, the mapping cone construction indeed produces first a cofibrant replacement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and then the ordinary cofiber of that, hence a model for the homotopy cofiber.</p> </div> <div class="num_example"> <h6 id="example_7">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the inclusion of <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B O(n) \hookrightarrow B O(n+1)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math>. See <a href="Introduction+to+Stable+homotopy+theory+--+S#HomotopyFiberOfInclusionOfConsecutiveClassifyingSpaces">this prop.</a> at <em><a href="Introduction+to+Stable+homotopy+theory+--+S#ClassifyingSpaces">Classifying spaces and G-structure</a></em>.</p> </div> <div class="num_example" id="FiberOfFibrationWeaklyEquivalentToFiberOfItsCocylinder"> <h6 id="example_8">Example</h6> <p>Suppose a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> already happens to be a fibration between fibrant objects. The <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a> replaces it by a fibration out of the <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(f)</annotation></semantics></math>, but such that the comparison morphism is a weak equivalence:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow><mi>f</mi></munderover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></munderover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ fib(f) &\longrightarrow& X &\underoverset{\in Fib}{f}{\longrightarrow}& Y \\ \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{id}} \\ fib(\tilde f) &\longrightarrow& Path(f) &\underoverset{\in Fib}{\tilde f}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>Hence by prop. <a class="maruku-ref" href="#FiberOfFibrationIsCompatibleWithWeakEquivalences"></a> in this case the ordinary fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is weakly equivalent to the <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a>, def. <a class="maruku-ref" href="#MappingConeAndMappingCocone"></a>.</p> </div> <p>We may now state the abstract version of the statement of prop. <a class="maruku-ref" href="#SerreFibrationGivesExactSequenceOfHomotopyGroups"></a>:</p> <div class="num_prop" id="ExactSequenceOfHomotopyFiberAtOneStage"> <h6 id="proposition_20">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> any morphism of <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a>, def. <a class="maruku-ref" href="#CategoryOfPointedObjects"></a>, then the sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mo>*</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex"> [A,hofib(f)]_\ast \overset{i_\ast}{\longrightarrow} [A,X]_\ast \overset{f_\ast}{\longrightarrow} [A,Y]_{\ast} </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact</a> as a sequence of <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a>.</p> <p>(Where the sequence here is the image of the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> sequence of def. <a class="maruku-ref" href="#HomotopyFiber"></a> under the hom-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Set</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">[A,-]_\ast \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Set^{\ast/}</annotation></semantics></math> from example <a class="maruku-ref" href="#HomotopyCategoryOfPointedModelStructureIsEnrichedInPointedSets"></a>.)</p> </div> <div class="proof"> <h6 id="proof_35">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> denote fibrant-cofibrant objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> representing the given objects of the same name in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C}^{\ast/})</annotation></semantics></math>. Moreover, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> be a fibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> representing the given morphism of the same name in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C}^{\ast/})</annotation></semantics></math>.</p> <p>Then by def. <a class="maruku-ref" href="#HomotopyFiber"></a> and prop. <a class="maruku-ref" href="#HomotopyFiberIndependentOfChoiceOfFibrantReplacement"></a> there is a representative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">hofib(f) \in \mathcal{C}</annotation></semantics></math> of the homotopy fiber which fits into a pullback diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ hofib(f) &\overset{i}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y } </annotation></semantics></math></div> <p>With this the hom-sets in question are represented by genuine morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math>, modulo homotopy. From this it follows immediately that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(i_\ast)</annotation></semantics></math> includes into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ker(f_\ast)</annotation></semantics></math>. Hence it remains to show the converse: that every element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ker(f_\ast)</annotation></semantics></math> indeed comes from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(i_\ast)</annotation></semantics></math>.</p> <p>But an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ker(f_\ast)</annotation></semantics></math> is represented by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\alpha \colon A \to X</annotation></semantics></math> such that there is a left homotopy as in the following diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mover><mi>η</mi><mo stretchy="false">˜</mo></mover></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && A &\overset{\alpha}{\longrightarrow}& X \\ && {}^{\mathllap{i_0}}\downarrow &{}^{\tilde \eta}\nearrow& \downarrow^{\mathrlap{f}} \\ A &\overset{i_1}{\longrightarrow} & Cyl(A) &\overset{\eta}{\longrightarrow}& Y \\ \downarrow && && \downarrow^{\mathrlap{=}} \\ \ast && \longrightarrow && Y } \,. </annotation></semantics></math></div> <p>Now by lemma <a class="maruku-ref" href="#ComponentMapsOfCylinderAndPathSpaceInGoodSituation"></a> the square here has a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \eta</annotation></semantics></math>, as shown. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>∘</mo><mover><mi>η</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">i_1 \circ\tilde \eta</annotation></semantics></math> is left homotopic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>. But by the universal property of the fiber, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>∘</mo><mover><mi>η</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">i_1 \circ \tilde \eta</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon hofib(f) \to X</annotation></semantics></math>.</p> </div> <p>With prop. <a class="maruku-ref" href="#FiberOfFibrationIsCompatibleWithWeakEquivalences"></a> it also follows notably that the loop space construction becomes well-defined on the homotopy category:</p> <div class="num_remark" id="ConcatenatedLoopSpaceObject"> <h6 id="remark_11">Remark</h6> <p>Given an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msubsup><mi>𝒞</mi> <mi>f</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}^{\ast/}_f</annotation></semantics></math>, and picking any <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a> with induced <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega X</annotation></semantics></math>, def. <a class="maruku-ref" href="#SuspensionAndLoopSpaceObject"></a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Path</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mi>X</mi></munder><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path_2(X) = Path(X) \underset{X}{\times} Path(X)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> given by the fiber product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> with itself, via example <a class="maruku-ref" href="#ComposedPathSpaceObjects"></a>. From the pullback diagram there, the fiber inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega X \to Path(X)</annotation></semantics></math> induces a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>×</mo><mi>Ω</mi><mi>X</mi><mo>⟶</mo><mo stretchy="false">(</mo><mi>Ω</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega X \times \Omega X \longrightarrow (\Omega X)_2 \,. </annotation></semantics></math></div> <p>In the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo>=</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/} = Top^{\ast/}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> is induced, via def. <a class="maruku-ref" href="#SuspensionAndLoopSpaceObject"></a>, from the standard path space object (def. <a class="maruku-ref" href="#TopologicalPathSpace"></a>), i.e. in the case that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>=</mo><mi>fib</mi><mo stretchy="false">(</mo><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo>,</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>⟶</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Omega X = fib(Maps(I_+,X)_\ast \longrightarrow X \times X) \,, </annotation></semantics></math></div> <p>then this is the operation of concatenating two loops parameterized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I = [0,1]</annotation></semantics></math> to a single loop parameterized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,2]</annotation></semantics></math>.</p> </div> <div class="num_prop" id="LoopingAsFunctorOnHomotopyCategory"> <h6 id="proposition_21">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, def. <a class="maruku-ref" href="#ModelCategory"></a>. Then the construction of forming <a class="existingWikiWord" href="/nlab/show/loop+space+objects">loop space objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">X\mapsto \Omega X</annotation></semantics></math>, def. <a class="maruku-ref" href="#SuspensionAndLoopSpaceObject"></a> (which on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒞</mi> <mi>f</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}_f</annotation></semantics></math> depends on a choice of <a class="existingWikiWord" href="/nlab/show/path+space+objects">path space objects</a>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>) becomes unique up to isomorphism in the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a> (def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>) of the <a class="existingWikiWord" href="/nlab/show/slice+model+structure">model structure on pointed objects</a> (prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>) and extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Ho(\mathcal{C}^{\ast/}) \,. </annotation></semantics></math></div> <p>Dually, the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a> operation, def. <a class="maruku-ref" href="#SuspensionAndLoopSpaceObject"></a>, which on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> depends on a choice of <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a>, becomes a functor on the homotopy category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Sigma \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Ho(\mathcal{C}^{\ast/}) \,. </annotation></semantics></math></div> <p>Moreover, the pairing operation induced on the objects in the image of this functor via remark <a class="maruku-ref" href="#ConcatenatedLoopSpaceObject"></a> (concatenation of loops) gives the objects in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> structure, and makes this functor lift as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mo>⟶</mo><mi>Grp</mi><mo stretchy="false">(</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Grp(Ho(\mathcal{C}^{\ast/})) \,. </annotation></semantics></math></div></div> <p>(<a href="#Brown73">Brown 73, section 4, theorem 3</a>)</p> <div class="proof"> <h6 id="proof_36">Proof</h6> <p>Given an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}^{\ast/}</annotation></semantics></math> and given two choices of path space objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{Path(X)}</annotation></semantics></math>, we need to produce an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C}^{\ast/})</annotation></semantics></math> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Ω</mi><mo stretchy="false">˜</mo></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">\tilde \Omega X</annotation></semantics></math>.</p> <p>To that end, first lemma <a class="maruku-ref" href="#ReplacementOfPathObjects"></a> implies that any two choices of path space objects are connected via a third path space by a <a class="existingWikiWord" href="/nlab/show/span">span</a> of morphisms compatible with the structure maps. By <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) every morphism of path space objects compatible with the inclusion of the base object is a weak equivalence. With this, lemma <a class="maruku-ref" href="#BaseChangePreservesFibrationsAndWeakEquivalences"></a> implies that these morphisms induce weak equivalences on the corresponding loop space objects. This shows that all choices of loop space objects become isomorphic in the homotopy category.</p> <p>Moreover, all the isomorphisms produced this way are actually equal: this follows from lemma <a class="maruku-ref" href="#UniquenessOfFibersOfQualizedMorphismsInHoC"></a> applied to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>s</mi></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>⟶</mo></mover></mtd> <mtd><mover><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>id</mi></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{s}{\longrightarrow}& Path(X) &\stackrel{\longrightarrow}{\longrightarrow}& \widetilde{Path(X)} \\ && \downarrow && \downarrow \\ && X\times X &\overset{id}{\longrightarrow}& X \times X } \,. </annotation></semantics></math></div> <p>This way we obtain a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>𝒞</mi> <mi>f</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega \;\colon\; \mathcal{C}^{\ast/}_f \longrightarrow Ho(\mathcal{C}^{\ast/}) \,. </annotation></semantics></math></div> <p>By prop. <a class="maruku-ref" href="#FiberOfFibrationIsCompatibleWithWeakEquivalences"></a> (and using that Cartesian product preserves weak equivalences) this functor sends weak equivalences to isomorphisms. Therefore the functor on homotopy categories now follows with theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a>.</p> <p>It is immediate to see that the operation of loop concatenation from remark <a class="maruku-ref" href="#ConcatenatedLoopSpaceObject"></a> gives the objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>∈</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega X \in Ho(\mathcal{C}^{\ast/})</annotation></semantics></math> the structure of <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>. It is now sufficient to see that these are in fact groups:</p> <p>We claim that the inverse-assigning operation is given by the left map in the following pasting composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Ω</mi><mo>′</mo><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>swap</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ω</mi><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \Omega' X &\longrightarrow& Path'(X) &\overset{}{\longrightarrow}& X \times X \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} &(pb)& \downarrow^{\mathrlap{swap}} \\ \Omega X &\longrightarrow& Path(X) &\underset{(p_0,p_1)}{\longrightarrow}& X \times X } \,, </annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path'(X)</annotation></semantics></math>, thus defined, is the path space object obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> by “reversing the notion of source and target of a path”).</p> <p>To see that this is indeed an inverse, it is sufficient to see that the two morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mo stretchy="false">(</mo><mi>Ω</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \Omega X \stackrel{\longrightarrow}{\longrightarrow} (\Omega X)_2 </annotation></semantics></math></div> <p>induced from</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>∘</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><mi>s</mi><mo>∘</mo><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></munder><mover><mo>⟶</mo><mi>Δ</mi></mover></mover><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><mi>Path</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Path(X) \stackrel{\overset{\Delta}{\longrightarrow}}{\underset{(s\circ p_0,s \circ p_0)}{\longrightarrow}} Path(X) \times_X Path'(X) } </annotation></semantics></math></div> <p>coincide in the homotopy category. This follows with lemma <a class="maruku-ref" href="#UniquenessOfFibersOfQualizedMorphismsInHoC"></a> applied to the following commuting diagram:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>∘</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><mi>s</mi><mo>∘</mo><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></munder><mover><mo>⟶</mo><mi>Δ</mi></mover></mover></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><mi>Path</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Δ</mi><mo>∘</mo><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{i}{\longrightarrow}& Path(X) &\stackrel{\overset{\Delta}{\longrightarrow}}{\underset{(s\circ p_0,s \circ p_0)}{\longrightarrow}}& Path(X)\times_X Path'(X) \\ && {}^{\mathllap{(p_0,p_1)}}\downarrow && \downarrow^{\mathrlap{}} \\ && X\times X &\overset{\Delta \circ pr_1}{\longrightarrow}& X \times X } \,. </annotation></semantics></math></div></div> <h3 id="HomotopyPullbacks">Homotopy pullbacks</h3> <p>The concept of <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> of def. <a class="maruku-ref" href="#HomotopyFiber"></a> is a special case of the more general concept of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>.</p> <div class="num_defn" id="RightProperModelCategory"> <h6 id="definition_25">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a>)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (def. <a class="maruku-ref" href="#ModelCategory"></a>) is called</p> <ul> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper model category</a></em> if <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> along <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a> preserves <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>;</p> </li> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/left+proper+model+category">left proper model category</a></em> if <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> along <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> preserves <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>;</p> </li> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a></em> if it is both left and right proper.</p> </li> </ul> </div> <div class="num_example"> <h6 id="example_9">Example</h6> <p>By lemma <a class="maruku-ref" href="#InCfPullbackAlongFibrationPreservesWeakEquivalences"></a>, a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (def. <a class="maruku-ref" href="#ModelCategory"></a>) in which all objects are fibrant is a <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper model category</a> (def. <a class="maruku-ref" href="#RightProperModelCategory"></a>).</p> </div> <div class="num_defn" id="HomotopyPullback"> <h6 id="definition_26">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper model category</a> (def. <a class="maruku-ref" href="#RightProperModelCategory"></a>). Then a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& B \\ \downarrow && \downarrow^{\mathrlap{g}} \\ C &\underset{f}{\longrightarrow}& D } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></strong> (of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> and equivalently of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>) if the following equivalent conditions hold:</p> <ol> <li> <p>for some factorization of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover><mi>D</mi></mrow><annotation encoding="application/x-tex"> g \colon B \overset{\in W }{\longrightarrow} \hat B \overset{\in Fib}{\longrightarrow} D </annotation></semantics></math></div> <p>the universally induced morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> into the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat B</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a weak equivalence:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi><munder><mo>×</mo><mi>D</mi></munder><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& B \\ {}^{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\in W}} \\ C \underset{D}{\times} \hat B &\longrightarrow& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\longrightarrow& D } \,. </annotation></semantics></math></div></li> <li> <p>for some factorization of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mover><mi>C</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover><mi>D</mi></mrow><annotation encoding="application/x-tex"> f \colon C \overset{\in W }{\longrightarrow} \hat C \overset{\in Fib}{\longrightarrow} D </annotation></semantics></math></div> <p>the universally induced morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> into the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat D</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is a weak equivalence:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mover><mi>C</mi><mo stretchy="false">^</mo></mover><munder><mo>×</mo><mi>D</mi></munder><mi>B</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \overset{\in W}{\longrightarrow} \hat C \underset{D}{\times} B \,. </annotation></semantics></math></div></li> <li> <p>the above two conditions hold for every such factorization.</p> </li> </ol> </div> <p>(e.g. <a href="Simplicial+homotopy+Theory">Goerss-Jardine 96, II (8.14)</a>)</p> <div class="num_prop" id="ConditionsForHomotopyPullbackAreIndeedEquivalent"> <h6 id="proposition_22">Proposition</h6> <p>The conditions in def. <a class="maruku-ref" href="#HomotopyPullback"></a> are indeed equivalent.</p> </div> <div class="proof"> <h6 id="proof_37">Proof</h6> <p>First assume that the first condition holds, in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi><munder><mo>×</mo><mi>D</mi></munder><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& B \\ {}^{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\in W}} \\ C \underset{D}{\times} \hat B &\longrightarrow& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\longrightarrow& D } \,. </annotation></semantics></math></div> <p>Then let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mover><mi>C</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover><mi>D</mi></mrow><annotation encoding="application/x-tex"> f \colon C \overset{\in W }{\longrightarrow} \hat C \overset{\in Fib}{\longrightarrow} D </annotation></semantics></math></div> <p>be any factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and consider the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> diagram (using the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> for pullbacks)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mover><mi>C</mi><mo stretchy="false">^</mo></mover><munder><mo>×</mo><mi>D</mi></munder><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi><munder><mo>×</mo><mi>D</mi></munder><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><mover><mi>C</mi><mo stretchy="false">^</mo></mover><munder><mo>×</mo><mi>D</mi></munder><mover><mi>D</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mfrac linethickness="0"><mrow><mo>∈</mo></mrow><mrow><mi>Fib</mi></mrow></mfrac></mpadded></msup></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></munder></mtd> <mtd><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></munder></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{}{\longrightarrow}& \hat C \underset{D}{\times} B &\longrightarrow& B \\ {}^{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\in W}} &(pb)& \downarrow^{\mathrlap{\in W}} \\ C\underset{D}{\times} \hat B &\overset{\in W}{\longrightarrow}& \hat C \underset{D}{\times} \hat D &\overset{\in Fib}{\longrightarrow}& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in \atop Fib}} &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\underset{\in W}{\longrightarrow}& \hat C &\underset{\in Fib}{\longrightarrow}& D } \,, </annotation></semantics></math></div> <p>where the inner morphisms are fibrations and weak equivalences, as shown, by the pullback stability of fibrations (prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a>) and then since pullback along fibrations preserves weak equivalences by assumption of <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right properness</a> (def. <a class="maruku-ref" href="#RightProperModelCategory"></a>). Hence it follows by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>) that also the comparison morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>C</mi><mo stretchy="false">^</mo></mover><munder><mo>×</mo><mi>D</mi></munder><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to \hat C \underset{D}{\times} B</annotation></semantics></math> is a weak equivalence.</p> <p>In conclusion, if the homotopy pullback condition is satisfied for one factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, then it is satisfied for all factorizations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. Since the argument is symmetric in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, this proves the claim.</p> </div> <div class="num_remark" id="PullbackOfFibrationWithFibrantObjectsIsHomotopyPullback"> <h6 id="remark_12">Remark</h6> <p>In particular, an ordinary pullback square of fibrant objects, one of whose edges is a fibration, is a homotopy pullback square according to def. <a class="maruku-ref" href="#HomotopyPullback"></a>.</p> </div> <div class="num_prop" id="HomotopyPullbackPreservesWeakEquivalencesOfSpans"> <h6 id="proposition_23">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper model category</a> (def. <a class="maruku-ref" href="#RightProperModelCategory"></a>). Given a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>⟵</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover></mtd> <mtd><mi>C</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>D</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd> <mtd><munder><mo>⟵</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></munder></mtd> <mtd><mi>F</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& B &\overset{\in Fib}{\longleftarrow}& C \\ \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} \\ D &\longrightarrow& E &\underset{\in Fib}{\longleftarrow}& F } </annotation></semantics></math></div> <p>then the induced morphism on <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> is a weak equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><munder><mo>×</mo><mi>B</mi></munder><mi>C</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mi>D</mi><munder><mo>×</mo><mi>E</mi></munder><mi>F</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \underset{B}{\times} C \overset{\in W}{\longrightarrow} D \underset{E}{\times} F \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_38">Proof</h6> <p>(The reader should draw the 3-dimensional cube diagram which we describe in words now.)</p> <p>First consider the universal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>E</mi><munder><mo>×</mo><mi>F</mi></munder><mi>C</mi></mrow><annotation encoding="application/x-tex">C \to E \underset{F}{\times} C</annotation></semantics></math> and observe that it is a weak equivalence by <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right properness</a> (def. <a class="maruku-ref" href="#RightProperModelCategory"></a>) and <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>).</p> <p>Then consider the universal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><munder><mo>×</mo><mi>B</mi></munder><mi>C</mi><mo>→</mo><mi>A</mi><munder><mo>×</mo><mi>B</mi></munder><mo stretchy="false">(</mo><mi>E</mi><munder><mo>×</mo><mi>F</mi></munder><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \underset{B}{\times}C \to A \underset{B}{\times}(E \underset{F}{\times}C)</annotation></semantics></math> and observe that this is also a weak equivalence, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><munder><mo>×</mo><mi>B</mi></munder><mi>C</mi></mrow><annotation encoding="application/x-tex">A \underset{B}{\times} C</annotation></semantics></math> is the limiting cone of a homotopy pullback square by remark <a class="maruku-ref" href="#PullbackOfFibrationWithFibrantObjectsIsHomotopyPullback"></a>, and since the morphism is the comparison morphism to the pullback of the factorization constructed in the first step.</p> <p>Now by using the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>, then the commutativity of the “left” face of the cube, then the pasting law again, one finds that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><munder><mo>×</mo><mi>B</mi></munder><mo stretchy="false">(</mo><mi>E</mi><munder><mo>×</mo><mi>F</mi></munder><mi>C</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi><munder><mo>×</mo><mi>D</mi></munder><mo stretchy="false">(</mo><mi>D</mi><munder><mi>F</mi><mi>E</mi></munder><mo>×</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \underset{B}{\times} (E \underset{F}{\times} C) \simeq A \underset{D}{\times} (D \underset{E} F{\times})</annotation></semantics></math>. Again by <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right properness</a> this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><munder><mo>×</mo><mi>B</mi></munder><mo stretchy="false">(</mo><mi>E</mi><munder><mo>×</mo><mi>F</mi></munder><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>D</mi><munder><mo>×</mo><mi>E</mi></munder><mi>F</mi></mrow><annotation encoding="application/x-tex">A \underset{B}{\times} (E \underset{F}{\times} C)\to D \underset{E}{\times} F</annotation></semantics></math> is a weak equivalence.</p> <p>With this the claim follows by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a>.</p> </div> <p>Homotopy pullbacks satisfy the usual abstract properties of pullbacks:</p> <div class="num_prop" id="HomotopyPullbackOfWeakEquivalences"> <h6 id="proposition_24">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper model category</a> (def. <a class="maruku-ref" href="#RightProperModelCategory"></a>). If in a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> one edge is a weak equivalence, then the square is a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> square precisely if the opposite edge is a weak equivalence, too.</p> </div> <div class="proof"> <h6 id="proof_39">Proof</h6> <p>Consider a commuting square of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></munder></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& B \\ \downarrow && \downarrow \\ C &\underset{\in W}{\longrightarrow}& D } \,. </annotation></semantics></math></div> <p>To detect whether this is a homotopy pullback, by def. <a class="maruku-ref" href="#HomotopyPullback"></a> and prop. <a class="maruku-ref" href="#ConditionsForHomotopyPullbackAreIndeedEquivalent"></a>, we are to choose any factorization of the right vertical morphism to obtain the pasting composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi><munder><mo>×</mo><mi>D</mi></munder><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></munder></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& B \\ \downarrow && \downarrow^{\mathrlap{\in W}} \\ C \underset{D}{\times} \hat B &\overset{\in W}{\longrightarrow}& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\underset{\in W}{\longrightarrow}& D } \,. </annotation></semantics></math></div> <p>Here the morphism in the middle is a weak equivalence by <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right properness</a> (def. <a class="maruku-ref" href="#RightProperModelCategory"></a>). Hence it follows by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> that the top left comparison morphism is a weak equivalence (and so the original square is a homotopy pullback) precisely if the top morphism is a weak equivalence.</p> </div> <div class="num_prop" id="ClosurePropertiesOfHomotopyPullbacks"> <h6 id="proposition_25">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper model category</a> (def. <a class="maruku-ref" href="#RightProperModelCategory"></a>).</p> <ol> <li> <p>(<a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>) If in a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>D</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd> <mtd><munder><mo>⟶</mo><mrow></mrow></munder></mtd> <mtd><mi>F</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& B &\longrightarrow& C \\ \downarrow && \downarrow && \downarrow \\ D &\longrightarrow& E &\underset{}{\longrightarrow}& F } </annotation></semantics></math></div> <p>the square on the right is a homotoy pullback (def. <a class="maruku-ref" href="#HomotopyPullback"></a>) then the left square is, too, precisely if the total rectangle is;</p> </li> <li> <p>in the presence of <a class="existingWikiWord" href="/nlab/show/functorial+factorization">functorial factorization</a> (def. <a class="maruku-ref" href="#FunctorialFactorization"></a>) through weak equivalences followed by fibrations:</p> <p>every <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a homotopy pullback square (in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒞</mi> <mi>f</mi> <mo>□</mo></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{C}_f^{\Box}</annotation></semantics></math> of commuting squares in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math>) is itself a homotopy pullback square.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_40">Proof</h6> <p>For the first statement: choose a factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mover><mi>F</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover><mi>F</mi></mrow><annotation encoding="application/x-tex">C \overset{\in W}{\to} \hat F \overset{\in Fib}{\to} F</annotation></semantics></math>, pull it back to a factorization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover><mi>E</mi></mrow><annotation encoding="application/x-tex">B \to \hat B \overset{\in Fib}{\to} E</annotation></semantics></math> and assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">B \to \hat B</annotation></semantics></math> is a weak equivalence, i.e. that the right square is a homotopy pullback. Now use the ordinary <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> to conclude.</p> <p>For the second statement: functorially choose a factorization of the two right vertical morphisms of the squares and factor the squares through the pullbacks of the corresponding fibrations along the bottom morphisms, respectively. Now the statement that the squares are homotopy pullbacks is equivalent to their top left vertical morphisms being weak equivalences. Factor these top left morphisms functorially as cofibrations followed by acyclic fibrations. Then the statement that the squares are homotopy pullbacks is equivalent to those top left cofibrations being acyclic. Now the claim follows using that the retract of an acyclic cofibration is an acyclic cofibration (prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a>).</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="LongSequences">Long fiber sequences</h3> <p>The ordinary <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>, example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a>, of a morphism has the property that taking it <em>twice</em> is always trivial:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>≃</mo><mi>fib</mi><mo stretchy="false">(</mo><mi>fib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>fib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \ast \simeq fib(fib(f)) \longrightarrow fib(f) \longrightarrow X \overset{f}{\longrightarrow} Y \,. </annotation></semantics></math></div> <p>This is crucially different for the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>, def. <a class="maruku-ref" href="#HomotopyFiber"></a>. Here we discuss how this comes about and what the consequences are.</p> <div class="num_prop" id="HomotopyFiberOfHomotopyFiberIsLooping"> <h6 id="proposition_26">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_f</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, def. <a class="maruku-ref" href="#FullSubcategoriesOfFibrantCofibrantObjects"></a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> be a morphism in its <a class="existingWikiWord" href="/nlab/show/category+of+pointed+objects">category of pointed objects</a>, def. <a class="maruku-ref" href="#CategoryOfPointedObjects"></a>. Then the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of its <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>, def. <a class="maruku-ref" href="#HomotopyFiber"></a>, is isomorphic, in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C}^{\ast/})</annotation></semantics></math>, to the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">\Omega Y</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> (def. <a class="maruku-ref" href="#SuspensionAndLoopSpaceObject"></a>, prop. <a class="maruku-ref" href="#LoopingAsFunctorOnHomotopyCategory"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Ω</mi><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> hofib(hofib(X \overset{f}{\to}Y)) \simeq \Omega Y \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_41">Proof</h6> <p>Assume without restriction that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \;\colon\; X \longrightarrow Y</annotation></semantics></math> is already a fibration between fibrant objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (otherwise replace and rename). Then its homotopy fiber is its ordinary fiber, sitting in a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≃</mo></mtd> <mtd><mi>F</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ hofib(f) \simeq & F &\overset{i}{\longrightarrow}& X \\ & \downarrow && \downarrow^{\mathrlap{f}} \\ & \ast &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>In order to compute <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hofib(hofib(f))</annotation></semantics></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hofib(i)</annotation></semantics></math>, we need to replace the fiber inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> by a fibration. Using the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <a class="maruku-ref" href="#FactorizationLemma"></a> for this purpose yields, after a choice of <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> (def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>), a replacement of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><mi>F</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mover><mi>i</mi><mo stretchy="false">˜</mo></mover></mpadded></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F &\overset{\in W}{\longrightarrow}& F \times_X Path(X) \\ &{}_{\mathllap{i}}\searrow& \downarrow^{\mathrlap{\tilde i}}_{\mathrlap{\in Fib}} \\ && X } \,. </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hofib(i)</annotation></semantics></math> is the ordinary fiber of this map:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>F</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><mo>*</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> hofib(hofib(f)) \simeq F \times_X Path(X) \times_X \ast \;\;\;\; \in Ho(\mathcal{C}^{\ast/}) \,. </annotation></semantics></math></div> <p>Notice that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo>*</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F \times_X Path(X) \; \simeq \; \ast \times_Y Path(X) </annotation></semantics></math></div> <p>because of the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>F</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F \times_X Path(X) &\longrightarrow& Path(X) \\ \downarrow &(pb)& \downarrow \\ F &\overset{i}{\longrightarrow}& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo>*</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> hofib(hofib(f)) \;\simeq\; \ast \times_Y Path(X) \times_X \ast \,. </annotation></semantics></math></div> <p>Now we claim that there is a choice of path space objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(Y)</annotation></semantics></math> such that this model for the homotopy fiber (as an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math>) sits in a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram of the following form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>F</mi></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>Ω</mi><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \ast \times_Y Path(X) \times_X \ast &\longrightarrow& Path(X) \\ \downarrow && \downarrow\mathrlap{\in W \cap F} \\ \Omega Y &\longrightarrow& Path(Y)\times_Y X \\ \downarrow &(pb)& \downarrow \\ \ast &\longrightarrow& Y \times X } \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> and the pullback stability of acyclic fibrations, this will prove the claim.</p> <p>To see that the bottom square here is indeed a pullback, check the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a>: A morphism out of any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><munder><mo>×</mo><mrow><mi>Y</mi><mo>×</mo><mi>X</mi></mrow></munder><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\ast \underset{Y \times X}{\times} Path(Y) \times_Y X</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \colon A \to Path(Y)</annotation></semantics></math> and a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">b \colon A \to X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">p_0(a) = \ast</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_1(a) = f(b)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">b = \ast</annotation></semantics></math>. Hence it is equivalently just a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \colon A \to Path(Y)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">p_0(a) = \ast</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">p_1(a) = \ast</annotation></semantics></math>. This is the defining universal property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Y</mi><mo>≔</mo><mo>*</mo><munder><mo>×</mo><mi>Y</mi></munder><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mi>Y</mi></munder><mo>*</mo></mrow><annotation encoding="application/x-tex">\Omega Y \coloneqq \ast \underset{Y}{\times} Path(Y) \underset{Y}{\times} \ast</annotation></semantics></math>.</p> <p>Now to construct the right vertical morphism in the top square (<a href="model+category#Quillen67">Quillen 67, page 3.1</a>): Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(Y)</annotation></semantics></math> be any path space object for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> be given by a factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><mi>i</mi><mo>∘</mo><mi>f</mi><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex"> (id_X, \; i \circ f, \; id_X) \;\colon\; X \overset{\in W}{\to} Path(X) \overset{\in Fib}{\longrightarrow} X \times_Y Path(Y) \times_Y X </annotation></semantics></math></div> <p>and regarded as a path space object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by further comoposing with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>pr</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>pr</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>Fib</mi></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (pr_1,pr_3)\colon X \times_Y Path(Y) \times_Y X \overset{\in Fib}{\longrightarrow} X \times X \,. </annotation></semantics></math></div> <p>We need to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">Path(X)\to Path(Y) \times_Y X</annotation></semantics></math> is an acyclic fibration.</p> <p>It is a fibration because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X\times_Y Path(Y) \times_Y X \to Path(Y)\times_Y X</annotation></semantics></math> is a fibration, this being the pullback of the fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \overset{f}{\to} Y</annotation></semantics></math>.</p> <p>To see that it is also a weak equivalence, first observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> Path(Y)\times_Y X \overset{\in W \cap Fib}{\longrightarrow} X</annotation></semantics></math>, this being the pullback of the acyclic fibration of lemma <a class="maruku-ref" href="#ComponentMapsOfCylinderAndPathSpaceInGoodSituation"></a>. Hence we have a factorization of the identity as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mi>i</mi></munderover><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi><munder><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></munder><mi>X</mi></mrow><annotation encoding="application/x-tex"> id_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{}{\longrightarrow} Path(Y)\times_Y X \underset{\in W \cap Fib}{\longrightarrow} X </annotation></semantics></math></div> <p>and so finally the claim follows by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>).</p> </div> <div class="num_remark" id="HomotopyCommutativeSquares"> <h6 id="remark_13">Remark</h6> <p>There is a conceptual way to understand prop. <a class="maruku-ref" href="#HomotopyFiberOfHomotopyFiberIsLooping"></a> as follows: If we draw double arrows to indicate <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a>, then a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> (def. <a class="maruku-ref" href="#HomotopyFiber"></a>) is depicted by the following filled square:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ hofib(f) &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ X &\underset{f}{\longrightarrow}& Y } </annotation></semantics></math></div> <p>just like the ordinary <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> (example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a>) is given by a plain square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ fib(f) &\longrightarrow& \ast \\ \downarrow && \downarrow \\ X &\underset{f}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>One may show that just like the fiber is the <em>universal</em> solution to making such a commuting square (a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <a class="existingWikiWord" href="/nlab/show/limit">limit cone</a> def. <a class="maruku-ref" href="#Limits"></a>), so the homotopy fiber is the universal solution up to homotopy to make such a commuting square up to homotopy – a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit cone</a>.</p> <p>Now just like ordinary <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> satisfy the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> saying that attaching two pullback squares gives a pullback rectangle, the analogue is true for homotopy pullbacks. This implies that if we take the homotopy fiber of a homotopy fiber, thereby producing this double homotopy pullback square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>hofib</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ hofib(g) &\longrightarrow& hofib(f) &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow^{\mathrlap{g}} &\swArrow& \downarrow \\ \ast &\longrightarrow& X &\underset{f}{\longrightarrow}& Y } </annotation></semantics></math></div> <p>then the total outer rectangle here is itself a homotopy pullback. But the outer rectangle exhibits the homotopy fiber of the point inclusion, which, via def. <a class="maruku-ref" href="#SuspensionAndLoopSpaceObject"></a> and lemma <a class="maruku-ref" href="#FactorizationLemma"></a>, is the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Ω</mi><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Omega Y &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ \ast &\longrightarrow& Y } \,. </annotation></semantics></math></div></div> <div class="num_prop" id="LongFiberSequence"> <h6 id="proposition_27">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/long+homotopy+fiber+sequences">long homotopy fiber sequences</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a model category and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> be morphism in the pointed homotopy category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C}^{\ast/})</annotation></semantics></math> (prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>). Then:</p> <ol> <li> <p>There is a long sequence to the left in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>⟶</mo><mi>Ω</mi><mi>X</mi><mover><mo>⟶</mo><mrow><mover><mi>Ω</mi><mo>¯</mo></mover><mi>f</mi></mrow></mover><mi>Ω</mi><mi>Y</mi><mo>⟶</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \cdots \longrightarrow \Omega X \overset{\overline{\Omega} f}{\longrightarrow} \Omega Y \longrightarrow hofib(f) \longrightarrow X \overset{f}{\longrightarrow} Y \,, </annotation></semantics></math></div> <p>where each morphism is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> (def. <a class="maruku-ref" href="#HomotopyFiber"></a>) of the following one: the <strong><a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Ω</mi><mo>¯</mo></mover><mi>f</mi></mrow><annotation encoding="application/x-tex">\overline{\Omega}f</annotation></semantics></math> denotes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">\Omega f</annotation></semantics></math> followed by forming inverses with respect to the group structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(-)</annotation></semantics></math> from prop. <a class="maruku-ref" href="#LoopingAsFunctorOnHomotopyCategory"></a>.</p> <p>Moreover, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">A\in \mathcal{C}^{\ast/}</annotation></semantics></math> any object, then there is a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mi>Y</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>Ω</mi><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>Ω</mi><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>Ω</mi><mi>Y</mi><mo stretchy="false">]</mo><mo>⟶</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex"> \cdots \to [A,\Omega^2 Y]_\ast \longrightarrow [A,\Omega hofib(f)]_\ast \longrightarrow [A, \Omega X]_\ast \longrightarrow [A,\Omega Y] \longrightarrow [A,hofib(f)]_\ast \longrightarrow [A,X]_\ast \longrightarrow [A,Y]_\ast </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">[-,-]_\ast</annotation></semantics></math> denotes the pointed set valued hom-functor of example <a class="maruku-ref" href="#HomotopyCategoryOfPointedModelStructureIsEnrichedInPointedSets"></a>.</p> </li> <li> <p>Dually, there is a long sequence to the right in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi><mover><mo>⟶</mo><mrow></mrow></mover><mi>hocofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Σ</mi><mi>X</mi><mover><mo>⟶</mo><mrow><mover><mi>Σ</mi><mo>¯</mo></mover><mi>f</mi></mrow></mover><mi>Σ</mi><mi>Y</mi><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \overset{f}{\longrightarrow} Y \overset{}{\longrightarrow} hocofib(f) \longrightarrow \Sigma X \overset{\overline{\Sigma} f}{\longrightarrow} \Sigma Y \to \cdots \,, </annotation></semantics></math></div> <p>where each morphism is the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> (def. <a class="maruku-ref" href="#HomotopyFiber"></a>) of the previous one: the <strong><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber+sequence">homotopy cofiber sequence</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. Moreover, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">A\in \mathcal{C}^{\ast/}</annotation></semantics></math> any object, then there is a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>Σ</mi> <mn>2</mn></msup><mi>X</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>Σ</mi><mi>hocofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>Σ</mi><mi>Y</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>Σ</mi><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo>⟶</mo><mo stretchy="false">[</mo><mi>hocofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>Y</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex"> \cdots \to [\Sigma^2 X, A]_\ast \longrightarrow [\Sigma hocofib(f), A]_\ast \longrightarrow [\Sigma Y, A]_\ast \longrightarrow [\Sigma X, A] \longrightarrow [hocofib(f),A]_\ast \longrightarrow [Y,A]_\ast \longrightarrow [X,A]_\ast </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">[-,-]_\ast</annotation></semantics></math> denotes the pointed set valued hom-functor of example <a class="maruku-ref" href="#HomotopyCategoryOfPointedModelStructureIsEnrichedInPointedSets"></a>.</p> </li> </ol> </div> <p>(<a href="model+category#Quillen67">Quillen 67, I.3, prop. 4</a>)</p> <div class="proof"> <h6 id="proof_42">Proof</h6> <p>That there are long sequences of this form is the result of combining prop. <a class="maruku-ref" href="#HomotopyFiberOfHomotopyFiberIsLooping"></a> and prop. <a class="maruku-ref" href="#ExactSequenceOfHomotopyFiberAtOneStage"></a>.</p> <p>It only remains to see that it is indeed the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Ω</mi><mo>¯</mo></mover><mi>f</mi></mrow><annotation encoding="application/x-tex">\overline{\Omega} f</annotation></semantics></math> that appear, as indicated.</p> <p>In order to see this, it is convenient to adopt the following notation: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> a morphism, then we denote the collection of <a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a> of its homotopy fiber as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>⇝</mo><mrow><msub><mi>γ</mi> <mn>1</mn></msub></mrow></mover><mo>*</mo><mo stretchy="false">)</mo><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> hofib(f) = \left\{ (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast) \right\} </annotation></semantics></math></div> <p>indicating that these elements are pairs consisting of an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and a “path” (an element of the given path space object) from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math> to the basepoint.</p> <p>This way the canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">hofib(f) \to X</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⇝</mo><mo>*</mo><mo stretchy="false">)</mo><mo>↦</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">(x, f(x) \rightsquigarrow \ast) \mapsto x</annotation></semantics></math>. Hence in this notation the homotopy fiber of the homotopy fiber reads</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>⇝</mo><mrow><msub><mi>γ</mi> <mn>1</mn></msub></mrow></mover><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mover><mo>⇝</mo><mrow><msub><mi>γ</mi> <mn>2</mn></msub></mrow></mover><mo>*</mo><mo stretchy="false">)</mo><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> hofib(hofib(f)) = \left\{ ( (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast), x \overset{\gamma_2}{\rightsquigarrow} \ast ) \right\} \,. </annotation></semantics></math></div> <p>This identifies with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">\Omega Y</annotation></semantics></math> by forming the loops</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mn>1</mn></msub><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mover><mrow><msub><mi>γ</mi> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \gamma_1 \cdot f(\overline{\gamma_2}) \,, </annotation></semantics></math></div> <p>where the overline denotes reversal and the dot denotes concatenation.</p> <p>Then consider the next homotopy fiber</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mo>(</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>⇝</mo><mrow><msub><mi>γ</mi> <mn>1</mn></msub></mrow></mover><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mover><mo>⇝</mo><mrow><msub><mi>γ</mi> <mn>2</mn></msub></mrow></mover><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>x</mi></mtd> <mtd></mtd> <mtd><mover><mo>⇝</mo><mrow><msub><mi>γ</mi> <mn>3</mn></msub></mrow></mover></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⇝</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>γ</mi> <mn>1</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇒</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0" lspace="-100%width"><mrow></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> hofib(hofib(hofib(f))) = \left\{ \left( ( (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast), x \overset{\gamma_2}{\rightsquigarrow} \ast ), \left( \array{ x && \overset{\gamma_3}{\rightsquigarrow} && \ast \\ f(x) &&\overset{f(\gamma_3)}{\rightsquigarrow}&& \ast \\ & {}_{\mathllap{\gamma_1}}\searrow & \Rightarrow & \swarrow_{\mathllap{}} \\ && \ast } \right) \right) \right\} \,, </annotation></semantics></math></div> <p>where on the right we have a path in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hofib(f)</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>⇝</mo><mrow><msub><mi>γ</mi> <mn>1</mn></msub></mrow></mover><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, f(x)\overset{\gamma_1}{\rightsquigarrow} \ast)</annotation></semantics></math> to the basepoint element. This is a path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\gamma_3</annotation></semantics></math> together with a path-of-paths which connects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\gamma_3)</annotation></semantics></math>.</p> <p>By the above convention this is identified with the loop in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> which is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mn>2</mn></msub><mo>⋅</mo><mo stretchy="false">(</mo><msub><mover><mi>γ</mi><mo>¯</mo></mover> <mn>3</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \gamma_2 \cdot (\overline{\gamma}_3) \,. </annotation></semantics></math></div> <p>But the map to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hofib(hofib(f))</annotation></semantics></math> sends this data to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>⇝</mo><mrow><msub><mi>γ</mi> <mn>1</mn></msub></mrow></mover><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mover><mo>⇝</mo><mrow><msub><mi>γ</mi> <mn>2</mn></msub></mrow></mover><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">( (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast), x \overset{\gamma_2}{\rightsquigarrow} \ast )</annotation></semantics></math>, hence to the loop</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>γ</mi> <mn>1</mn></msub><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mover><mrow><msub><mi>γ</mi> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mover><mrow><msub><mi>γ</mi> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mn>3</mn></msub><mo>⋅</mo><mover><mrow><msub><mi>γ</mi> <mn>2</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mover><mrow><msub><mi>γ</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mover><mi>γ</mi><mo>¯</mo></mover> <mn>3</mn></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mover><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mover><mi>γ</mi><mo>¯</mo></mover> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \gamma_1 \cdot f( \overline{\gamma_2} ) & \simeq f(\gamma_3) \cdot f(\overline{\gamma_2}) \\ & = f( \gamma_3 \cdot \overline{\gamma_2} ) \\ & = f ( \overline{\gamma_2 \cdot \overline{\gamma}_3} ) \\ & = \overline{f(\gamma_2 \cdot \overline{\gamma}_3) } \end{aligned} \,, </annotation></semantics></math></div> <p>hence to the reveral of the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> of the loop in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_14">Remark</h6> <p>In (<a href="model+category#Quillen67">Quillen 67, I.3, prop. 3, prop. 4</a>) more is shown than stated in prop. <a class="maruku-ref" href="#LongFiberSequence"></a>: there the <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Y</mi><mo>→</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega Y \to hofib(f)</annotation></semantics></math> is not just shown to exist, but is described in detail via an <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">\Omega Y</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hofib(f)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math>. This takes a good bit more work. For our purposes here, however, it is sufficient to know that such a morphism exists at all, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Y</mi><mo>≃</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega Y \simeq hofib(hofib(f))</annotation></semantics></math>.</p> </div> <div class="num_example" id="LongExactSequeceOfHomotopyGroups"> <h6 id="example_10">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C} = (Top_{cg})_{Quillen}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> (<a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated</a>) from theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>, theorem <a class="maruku-ref" href="#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"></a>. Then using the standard pointed topological path space objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Maps(I_+,X)</annotation></semantics></math> from def. <a class="maruku-ref" href="#TopologicalPathSpace"></a> and example <a class="maruku-ref" href="#PointedMappingSpace"></a> as the abstract path space objects in def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, via prop. <a class="maruku-ref" href="#TopologicalPathSpaceIsGoodPathSpaceObject"></a>, this gives that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [\ast, \Omega^n X] \simeq \pi_n(X) </annotation></semantics></math></div> <p>is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>, def. <a class="maruku-ref" href="#HomotopyGroupsOftopologicalSpaces"></a>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> at its basepoint.</p> <p>Hence using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">A = \ast</annotation></semantics></math> in the first item of prop. <a class="maruku-ref" href="#LongFiberSequence"></a>, the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> this gives is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>π</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mover><msub><mi>π</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>f</mi> <mo>*</mo></msub></mrow></mover><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>hofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mover><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to \pi_3(X) \overset{f_\ast}{\longrightarrow} \pi_3(Y) \longrightarrow \pi_2(hofib(f)) \overset{}{\longrightarrow} \pi_2(X) \overset{-f_\ast}{\longrightarrow} \pi_2(Y) \longrightarrow \pi_1(hofib(f)) \overset{}{\longrightarrow} \pi_1(X) \overset{f_\ast}{\longrightarrow} \pi_1(Y) \overset{}{\longrightarrow} \ast \,. </annotation></semantics></math></div> <p>This is called the <strong><a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a></strong> induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_15">Remark</h6> <p>As we pass to <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> (in <a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+1">Part 1)</a>), the long exact sequences in example <a class="maruku-ref" href="#LongExactSequeceOfHomotopyGroups"></a> become long not just to the left, but also to the right. Given then a <a class="existingWikiWord" href="/nlab/show/tower+of+fibrations">tower of fibrations</a>, there is an induced sequence of such long exact sequences of homotopy groups, which organizes into an <em><a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a></em>. For more on this see at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+I">Interlude – Spectral sequences</a></em> (<a href="Introduction+to+Stable+homotopy+theory+--+I#UnrolledExactCoupleOfAFiltrationOnASpectrum">this remark</a>).</p> </div> <div class="num_example"> <h6 id="example_11">Example</h6> <p>Let again <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C} = (Top_{cg})_{Quillen}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> (<a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated</a>) from theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>, theorem <a class="maruku-ref" href="#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"></a>, as in example <a class="maruku-ref" href="#LongExactSequeceOfHomotopyGroups"></a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">E \in Top_{cg}^{\ast/}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon A \hookrightarrow X</annotation></semantics></math> an inclusion of pointed topological spaces, the exactness of the sequence in the second item of prop. <a class="maruku-ref" href="#LongFiberSequence"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><mo stretchy="false">[</mo><mi>hocofib</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo><mo>⟶</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \cdots \to [hocofib(i), E] \longrightarrow [X,E]_\ast \longrightarrow [A,E]_\ast \to \cdots </annotation></semantics></math></div> <p>gives that the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>⟶</mo><msup><mi>Set</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> [-,E]_\ast \;\colon\; (Top^{\ast/}_{CW})^{op} \longrightarrow Set^{\ast/} </annotation></semantics></math></div> <p>behaves like one degree in an <a href="Introduction+to+Stable+homotopy+theory+--+S#WedgeAxiom">additive</a> <a class="existingWikiWord" href="/nlab/show/reduced+cohomology+theory">reduced cohomology theory</a> (<a href="Introduction+to+Stable+homotopy+theory+--+S#ReducedGeneralizedCohomology">def.</a>). The <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a> (<a href="Introduction+to+Stable+homotopy+theory+--+S#BrownRepresentabilityTraditional">thm.</a>) implies that all additive reduced cohomology theories are degreewise representable this way (<a href="Introduction+to+Stable+homotopy+theory+--+S#AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum">prop.</a>).</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h2 id="TopologicalHomotopyTheory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Groupoids I): Topological homotopy theory</h2> <p>This section first recalls relevant concepts from actual <a class="existingWikiWord" href="/nlab/show/topology">topology</a> (“<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>”) and highlights facts that motivate the axiomatics of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> <a href="#ModelCategoryTheory">below</a>. We prove two technical lemmas (lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a> and lemma <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a>) that serve to establish the abstract homotopy theory of topological spaces <a href="#TheClassicalModelStructureOfTopologicalSpaces">further below</a>.</p> <p>Then we discuss how the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> satisfies the axioms of abstract homotopy theory (<a class="existingWikiWord" href="/nlab/show/model+category">model category</a>) theory, def. <a class="maruku-ref" href="#ModelCategory"></a>.</p> <p><strong>Literature</strong> (<a href="classical+model+structure+on+topological+spaces#Hirschhorn15">Hirschhorn 15</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>Throughout, let <em><a class="existingWikiWord" href="/nlab/show/Top">Top</a></em> denote the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between them. Its <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> are the <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a>.</p> <p>(Further <a href="#ModelStructureOnCompactlyGeneratedTopologicalSpaces">below</a> we restrict attention to the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>.)</p> <h3 id="universal_constructions">Universal constructions</h3> <p>To begin with, we recall some basics on <a class="existingWikiWord" href="/nlab/show/universal+constructions">universal constructions</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>: <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> of <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>; <a class="existingWikiWord" href="/nlab/show/exponential+objects">exponential objects</a>.</p> <p>We now discuss <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> (Def. <a class="maruku-ref" href="#Limits"></a>) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}= </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a>. The key for understanding these is the fact that there are initial and final topologies:</p> <div class="num_defn" id="InitialAndFinalTopologies"> <h6 id="definition_27">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>Top</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i = (S_i,\tau_i) \in Top\}_{i \in I}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S \in Set</annotation></semantics></math> be a bare <a class="existingWikiWord" href="/nlab/show/set">set</a>. Then</p> <ol> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>S</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mover><msub><mi>S</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{S \stackrel{f_i}{\to} S_i \}_{i \in I}</annotation></semantics></math> a set of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, the <strong><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{initial}(\{f_i\}_{i \in I})</annotation></semantics></math> is the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/minimum">minimum</a> collection of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> such that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>S</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mover><mi>S</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{S_i \stackrel{f_i}{\to} S\}_{i \in I}</annotation></semantics></math> a set of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, the <strong><a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{final}(\{f_i\}_{i \in I})</annotation></semantics></math> is the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/maximum">maximum</a> collection of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> such that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I}))</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>.</p> </li> </ol> </div> <div class="num_example" id="TopologicalSubspace"> <h6 id="example_12">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a single topological space, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>S</mi></msub><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>↪</mo><mi>U</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota_S \colon S \hookrightarrow U(X)</annotation></semantics></math> a subset of its underlying set, then the initial topology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>intial</mi></msub><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{intial}(\iota_S)</annotation></semantics></math>, def. <a class="maruku-ref" href="#InitialAndFinalTopologies"></a>, is the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, making</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>S</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>S</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> inclusion.</p> </div> <div class="num_example" id="QuotientTopology"> <h6 id="example_13">Example</h6> <p>Conversely, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>S</mi></msub><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">p_S \colon U(X) \longrightarrow S</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>, then the final topology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{final}(p_S)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is the <em><a class="existingWikiWord" href="/nlab/show/quotient+topology">quotient topology</a></em>.</p> </div> <div class="num_prop" id="DescriptionOfLimitsAndColimitsInTop"> <h6 id="proposition_28">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>⟶</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X_\bullet \colon I \longrightarrow Top</annotation></semantics></math> be an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> (a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>), with components denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X_i = (S_i, \tau_i)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S_i \in Set</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\tau_i</annotation></semantics></math> a topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math>. Then:</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/limit">limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> exists and is given by <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> topological space whose underlying set is <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> limit in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of the underlying sets in the diagram, and whose topology is the <a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, def. <a class="maruku-ref" href="#InitialAndFinalTopologies"></a>, for the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math> which are the limiting <a class="existingWikiWord" href="/nlab/show/cone">cone</a> components:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow></mrow></munder></mtd> <mtd></mtd> <mtd><msub><mi>S</mi> <mi>j</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,. </annotation></semantics></math></div> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>≃</mo><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>p</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right) </annotation></semantics></math></div></li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> exists and is the topological space whose underlying set is the colimit in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of the underlying diagram of sets, and whose topology is the <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a>, def. <a class="maruku-ref" href="#InitialAndFinalTopologies"></a> for the component maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\iota_i</annotation></semantics></math> of the colimiting <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><msub><mi>S</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mi>j</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,. </annotation></semantics></math></div> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>≃</mo><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>ι</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \underset{\longrightarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right) </annotation></semantics></math></div></li> </ol> </div> <p>(e.g. <a href="#Bourbaki71">Bourbaki 71, section I.4</a>)</p> <div class="proof"> <h6 id="proof_43">Proof</h6> <p>The required <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>p</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)</annotation></semantics></math> (def. <a class="maruku-ref" href="#Limits"></a>) is immediate: for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mi>i</mi></msub></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow></mrow></munder></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mi>j</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_j } </annotation></semantics></math></div> <p>any <a class="existingWikiWord" href="/nlab/show/cone">cone</a> over the diagram, then by construction there is a unique function of underlying sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⟶</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i</annotation></semantics></math> making the required diagrams commute, and so all that is required is that this unique function is always <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>. But this is precisely what the <a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a> ensures.</p> <p>The case of the colimit is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> </div> <div class="num_example" id="PointTopologicalSpaceAsEmptyLimit"> <h6 id="example_14">Example</h6> <p>The limit over the empty diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/point">point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> with its unique topology.</p> </div> <div class="num_example" id="DisjointUnionOfTopologicalSpacesIsCoproduct"> <h6 id="example_15">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i\}_{i \in I}</annotation></semantics></math> a set of topological spaces, their <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\sqcup} X_i \in Top</annotation></semantics></math> is their <em><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a></em>.</p> </div> <p>In particular:</p> <div class="num_example" id="DiscreteTopologicalSpaceAsCoproduct"> <h6 id="example_16">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S \in Set</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-indexed <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of the point, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mo>*</mo></mrow><annotation encoding="application/x-tex">\underset{s \in S}{\coprod}\ast </annotation></semantics></math> is the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> itself equipped with the <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a>, hence is the <a class="existingWikiWord" href="/nlab/show/discrete+topological+space">discrete topological space</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> </div> <div class="num_example" id="ProductTopologicalSpace"> <h6 id="example_17">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i\}_{i \in I}</annotation></semantics></math> a set of topological spaces, their <a class="existingWikiWord" href="/nlab/show/product">product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\prod} X_i \in Top</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the underlying sets equipped with the <em><a class="existingWikiWord" href="/nlab/show/product+topology">product topology</a></em>, also called the <em><a class="existingWikiWord" href="/nlab/show/Tychonoff+product">Tychonoff product</a></em>.</p> <p>In the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>, such as for binary product spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times Y</annotation></semantics></math>, then a <a class="existingWikiWord" href="/nlab/show/basis+for+a+topology">sub-basis</a> for the product topology is given by the <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> of the open subsets of (a basis for) each factor space.</p> </div> <div class="num_example" id="EqualizerInTop"> <h6 id="example_18">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> of two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> is the equalizer of the underlying functions of sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>eq</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>S</mi> <mi>X</mi></msub><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><msub><mi>S</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex"> eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y </annotation></semantics></math></div> <p>(hence the largets subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">S_X</annotation></semantics></math> on which both functions coincide) and equipped with the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, example <a class="maruku-ref" href="#TopologicalSubspace"></a>.</p> </div> <div class="num_example" id="CoequalizerInTop"> <h6 id="example_19">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> of two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> is the coequalizer of the underlying functions of sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><msub><mi>S</mi> <mi>Y</mi></msub><mo>⟶</mo><mi>coeq</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g) </annotation></semantics></math></div> <p>(hence the <a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a> by the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∼</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) \sim g(x)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>) and equipped with the <a class="existingWikiWord" href="/nlab/show/quotient+topology">quotient topology</a>, example <a class="maruku-ref" href="#QuotientTopology"></a>.</p> </div> <div class="num_example" id="PushoutInTop"> <h6 id="example_20">Example</h6> <p>For</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X } </annotation></semantics></math></div> <p>two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> out of the same <a class="existingWikiWord" href="/nlab/show/domain">domain</a>, then the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> under this diagram is also called the <em><a class="existingWikiWord" href="/nlab/show/pushout">pushout</a></em>, denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>g</mi> <mo>*</mo></msub><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,. </annotation></semantics></math></div> <p>(Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mo>*</mo></msub><mi>f</mi></mrow><annotation encoding="application/x-tex">g_\ast f</annotation></semantics></math> is also called the pushout of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, or the <em><a class="existingWikiWord" href="/nlab/show/base+change">cobase change</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>.)</p> <p>This is equivalently the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> of the two morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> (example <a class="maruku-ref" href="#DisjointUnionOfTopologicalSpacesIsCoproduct"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo>⟶</mo><mi>X</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \stackrel{\longrightarrow}{\longrightarrow} X \sqcup Y \longrightarrow X \sqcup_A Y \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is an inclusion, one also writes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>∪</mo> <mi>f</mi></msub><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \cup_f Y</annotation></semantics></math> and calls this the <em><a class="existingWikiWord" href="/nlab/show/attaching+space">attaching space</a></em>.</p> <div style="float:left;margin:0 10px 10px 0;"><img src="http://ncatlab.org/nlab/files/AttachingSpace.jpg" width="450" /></div> <p>By example <a class="maruku-ref" href="#CoequalizerInTop"></a> the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>/<a class="existingWikiWord" href="/nlab/show/attaching+space">attaching space</a> is the <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>Y</mi><mo>≃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo>∼</mo></mrow><annotation encoding="application/x-tex"> X \sqcup_A Y \simeq (X\sqcup Y)/\sim </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> subject to the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> which identifies a point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> if they have the same pre-image in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>(graphics from <a href="#AguilarGitlerPrieto02">Aguilar-Gitler-Prieto 02</a>)</p> <p>Notice that the defining <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of this colimit means that completing the <a class="existingWikiWord" href="/nlab/show/span">span</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& Y \\ \downarrow \\ X } </annotation></semantics></math></div> <p>to a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& Y \\ \downarrow && \downarrow \\ X &\longrightarrow& Z } </annotation></semantics></math></div> <p>is equivalent to finding a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><munder><mo>⊔</mo><mi>A</mi></munder><mi>Y</mi><mo>⟶</mo><mi>Z</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \underset{A}{\sqcup} Y \longrightarrow Z \,. </annotation></semantics></math></div></div> <div class="num_example" id="QuotientSpaceAsPushout"> <h6 id="example_21">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A\hookrightarrow X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> inclusion, example <a class="maruku-ref" href="#TopologicalSubspace"></a>, then the pushout</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\hookrightarrow& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X/A } </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> or <em><a class="existingWikiWord" href="/nlab/show/cofiber">cofiber</a></em>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X/A</annotation></semantics></math>.</p> </div> <div class="num_example" id="TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself"> <h6 id="example_22">Example</h6> <p>An important special case of example <a class="maruku-ref" href="#PushoutInTop"></a>:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>≔</mo><mo stretchy="false">{</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo><mo>≤</mo><mn>1</mn></mrow><mo stretchy="false">}</mo><mo>↪</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n \coloneqq \{ \vec x\in \mathbb{R}^n | \; {\vert \vec x \vert \leq 1}\} \hookrightarrow \mathbb{R}^n</annotation></semantics></math> for the standard topological <a class="existingWikiWord" href="/nlab/show/n-disk">n-disk</a> (equipped with its <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> as a subset of <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo>∂</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>≔</mo><mo stretchy="false">{</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">}</mo><mo>↪</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^{n-1} = \partial D^n \coloneqq \{ \vec x\in \mathbb{R}^n | \; {\vert \vec x \vert = 1}\} \hookrightarrow \mathbb{R}^n</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a>, the standard topological <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a>.</p> </li> </ul> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">S^{-1} = \emptyset</annotation></semantics></math> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mo>=</mo><mo>*</mo><mo>⊔</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">S^0 = \ast \sqcup \ast</annotation></semantics></math>.</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⟶</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> i_n \colon S^{n-1}\longrightarrow D^n </annotation></semantics></math></div> <p>be the canonical inclusion of the standard <a class="existingWikiWord" href="/nlab/show/n-sphere">(n-1)-sphere</a> as the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> of the standard <a class="existingWikiWord" href="/nlab/show/n-disk">n-disk</a> (both regarded as <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> with their <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> as subspaces of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>).</p> <div style="float:left;margin:0 10px 10px 0;"> <img src="http://ncatlab.org/nlab/files/GluingHemispheres.jpg" width="400" /></div> <p>Then the colimit in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> under the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mover><mo>⟵</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \,, </annotation></semantics></math></div> <p>i.e. the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">i_n</annotation></semantics></math> along itself, is the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,. </annotation></semantics></math></div> <p>(graphics from Ueno-Shiga-Morita 95)</p> </div> <p>Another kind of colimit that will play a role for certain technical constructions is <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a>. First recall</p> <div class="num_defn" id="PosetsWosetTosetsAndOrdinals"> <h6 id="definition_28">Definition</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/partial+order">partial order</a></em> is a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/relation">relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math> such that for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">a,b,c \in S</annotation></semantics></math></p> <p>1) (<a class="existingWikiWord" href="/nlab/show/reflexive+relation">reflexivity</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≤</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">a \leq a</annotation></semantics></math>;</p> <p>2) (<a class="existingWikiWord" href="/nlab/show/transitive+relation">transitivity</a>) if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≤</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \leq b</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>≤</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">b \leq c</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≤</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a \leq c</annotation></semantics></math>;</p> <p>3) (<a class="existingWikiWord" href="/nlab/show/antisymmetric+relation">antisymmetry</a>) if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≤</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a\leq b</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">b</mo><mo>≤</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\b \leq a</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a = b</annotation></semantics></math>.</p> <p>This we may and will equivalently think of as a <a class="existingWikiWord" href="/nlab/show/category">category</a> with <a class="existingWikiWord" href="/nlab/show/objects">objects</a> the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and a unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \to b</annotation></semantics></math> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≤</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a\leq b</annotation></semantics></math>. In particular an order-preserving function between partially ordered sets is equivalently a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> between their corresponding categories.</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/bottom+element">bottom element</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math> in a partial order is one such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo><mo>≤</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\bot \leq a</annotation></semantics></math> for all a. A <em><a class="existingWikiWord" href="/nlab/show/top+element">top element</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math> is one for wich <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≤</mo><mo>⊤</mo></mrow><annotation encoding="application/x-tex">a \leq \top</annotation></semantics></math>.</p> <p>A partial order is a <em><a class="existingWikiWord" href="/nlab/show/total+order">total order</a></em> if in addition</p> <p>4) (<a class="existingWikiWord" href="/nlab/show/total+relation">totality</a>) either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≤</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a\leq b</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>≤</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">b \leq a</annotation></semantics></math>.</p> <p>A total order is a <em><a class="existingWikiWord" href="/nlab/show/well+order">well order</a></em> if in addition</p> <p>5) (<a class="existingWikiWord" href="/nlab/show/well-founded+relation">well-foundedness</a>) every non-empty subset has a least element.</p> <p>An <em><a class="existingWikiWord" href="/nlab/show/ordinal">ordinal</a></em> is the <a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a> of a well-order.</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/successor">successor</a></em> of an ordinal is the class of the well-order with a <a class="existingWikiWord" href="/nlab/show/top+element">top element</a> freely adjoined.</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/limit+ordinal">limit ordinal</a></em> is one that is not a successor.</p> </div> <div class="num_example" id="ExamplesOfOrdinals"> <h6 id="example_23">Example</h6> <p>The finite ordinals are labeled by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, corresponding to the well-orders <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>≤</mo><mn>1</mn><mo>≤</mo><mn>2</mn><mi>⋯</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0 \leq 1 \leq 2 \cdots \leq n-1\}</annotation></semantics></math>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math> is the successor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>. The first non-empty limit ordinal is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>ℕ</mi><mo>,</mo><mo>≤</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\omega = [(\mathbb{N}, \leq)]</annotation></semantics></math>.</p> </div> <div class="num_defn" id="TransfiniteComposition"> <h6 id="definition_29">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I \subset Mor(\mathcal{C})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/class">class</a> of its morphisms.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/ordinal">ordinal</a> (regarded as a <a class="existingWikiWord" href="/nlab/show/category">category</a>), an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-indexed <em>transfinite sequence</em> of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>α</mi><mo>⟶</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> X_\bullet \;\colon\; \alpha \longrightarrow \mathcal{C} </annotation></semantics></math></div> <p>such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> takes all <a class="existingWikiWord" href="/nlab/show/successor">successor</a> morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mover><mo>→</mo><mo>≤</mo></mover><mi>β</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\beta \stackrel{\leq}{\to} \beta + 1</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> to elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>β</mi><mo>,</mo><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex"> X_{\beta,\beta + 1} \in I </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> is <em>continuous</em> in that for every nonzero <a class="existingWikiWord" href="/nlab/show/limit+ordinal">limit ordinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo><</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">\beta \lt \alpha</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> restricted to the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full-subdiagram</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>γ</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>γ</mi><mo>≤</mo><mi>β</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\gamma \;|\; \gamma \leq \beta\}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/colimit">colimiting cocone</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>γ</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>γ</mi><mo><</mo><mi>β</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\gamma \;|\; \gamma \lt \beta\}</annotation></semantics></math>.</p> </li> </ol> <p>The corresponding <strong>transfinite composition</strong> is the induced morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>⟶</mo><msub><mi>X</mi> <mi>α</mi></msub><mo>≔</mo><munder><mi>lim</mi><mo>⟶</mo></munder><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> X_0 \longrightarrow X_\alpha \coloneqq \underset{\longrightarrow}{\lim}X_\bullet </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of the diagram, schematically:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>X</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>X</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↙</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mi>α</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X_0 &\stackrel{X_{0,1}}{\to}& X_1 &\stackrel{X_{1,2}}{\to}& X_2 &\to& \cdots \\ & \searrow & \downarrow & \swarrow & \cdots \\ && X_\alpha } \,. </annotation></semantics></math></div></div> <p>We now turn to the discussion of <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>/<a class="existingWikiWord" href="/nlab/show/exponential+objects">exponential objects</a>.</p> <div class="num_defn" id="CompactOpenTopology"> <h6 id="definition_30">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact topological space</a> (in that for every point, every <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> contains a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> neighbourhood), the <strong><a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>Y</mi></msup><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> X^Y \in Top </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></p> <ul> <li> <p>whose underlying set is the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{Top}(Y,X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math>,</p> </li> <li> <p>whose <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> are <a class="existingWikiWord" href="/nlab/show/unions">unions</a> of <a class="existingWikiWord" href="/nlab/show/finitary+intersections">finitary intersections</a> of the following <a class="existingWikiWord" href="/nlab/show/topological+base">subbase</a> elements of standard open subsets:</p> <p>the standard open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mi>K</mi></msup><mo>⊂</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U^K \subset Hom_{Top}(Y,X)</annotation></semantics></math> for</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">K \hookrightarrow Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a> subset</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \hookrightarrow X</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a></p> </li> </ul> <p>is the subset of all those <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> that fit into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ K &\hookrightarrow& Y \\ \downarrow && \downarrow^{\mathrlap{f}} \\ U &\hookrightarrow& X } \,. </annotation></semantics></math></div></li> </ul> <p>Accordingly this is called the <em><a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a></em> on the set of functions.</p> <p>The construction extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Top</mi> <mi>lc</mi> <mi>op</mi></msubsup><mo>×</mo><mi>Top</mi><mo>⟶</mo><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (-)^{(-)} \;\colon\; Top_{lc}^{op} \times Top \longrightarrow Top \,. </annotation></semantics></math></div></div> <div class="num_prop" id="MappingTopologicalSpaceIsExponentialObject"> <h6 id="proposition_29">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact topological space</a> (in that for each point, each open neighbourhood contains a <a class="existingWikiWord" href="/nlab/show/compact+subset">compact</a> <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a>), the <strong>topological <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">X^Y</annotation></semantics></math> from def. <a class="maruku-ref" href="#CompactOpenTopology"></a> is an <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a>, i.e. the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">(-)^Y</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> to the product functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y \times (-)</annotation></semantics></math>: there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><mi>Z</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><msup><mi>X</mi> <mi>Y</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{Top}(Z \times Y, X) \simeq Hom_{Top}(Z, X^Y) </annotation></semantics></math></div> <p>between continuous functions out of any <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> with any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Z \in Top</annotation></semantics></math> and continuous functions from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> into the mapping space.</p> </div> <p>A proof is spelled out <a href="compact-open+topology#GivesExponentialObject">here</a> (or see e.g. <a href="#AguilarGitlerPrieto02">Aguilar-Gitler-Prieto 02, prop. 1.3.1</a>).</p> <div class="num_remark" id="UseOfHausdorffnessInCOTopology"> <h6 id="remark_16">Remark</h6> <p>In the context of prop. <a class="maruku-ref" href="#MappingTopologicalSpaceIsExponentialObject"></a> it is often assumed that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is also a <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff topological space</a>. But this is not necessary. What assuming Hausdorffness only achieves is that all alternative definitions of “locally compact” become equivalent to the one that is needed for the proposition: for every point, every open neighbourhood contains a compact neighbourhood.</p> </div> <div class="num_remark" id="ProblemWithExponentialsInTop"> <h6 id="remark_17">Remark</h6> <p>Proposition <a class="maruku-ref" href="#MappingTopologicalSpaceIsExponentialObject"></a> fails in general if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is not locally compact. Therefore the plain category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of all topological spaces is not a <a class="existingWikiWord" href="/nlab/show/Cartesian+closed+category">Cartesian closed category</a>.</p> <p>This is no problem for the construction of the homotopy theory of topological spaces as such, but it becomes a technical nuisance for various constructions that one would like to perform within that homotopy theory. For instance on general <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> is in general not <a class="existingWikiWord" href="/nlab/show/associativity">associative</a>.</p> <p>On the other hand, without changing any of the following discussion one may just pass to a more <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a> such as notably the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top_{cg} \hookrightarrow Top</annotation></semantics></math> (def. <a class="maruku-ref" href="#kTop"></a>) which is <a class="existingWikiWord" href="/nlab/show/Cartesian+closed+category">Cartesian closed</a>. This we turn to <a href="#ModelStructureOnCompactlyGeneratedTopologicalSpaces">below</a>.</p> </div> <h3 id="homotopy_2">Homotopy</h3> <p>The fundamental concept of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> is clearly that of <em><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a></em>. In the context of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> this is about <a class="existingWikiWord" href="/nlab/show/continuous+function">contiunous</a> deformations of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> parameterized by the standard interval:</p> <div class="num_defn" id="TopologicalInterval"> <h6 id="definition_31">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>≔</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> I \coloneqq [0,1] \hookrightarrow \mathbb{R} </annotation></semantics></math></div> <p>for the standard <a class="existingWikiWord" href="/nlab/show/topological+interval">topological interval</a>, a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> of the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a>.</p> <p>Equipped with the canonical inclusion of its two endpoints</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>⊔</mo><mo>*</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>I</mi><mover><mo>⟶</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><mo>*</mo></mrow><annotation encoding="application/x-tex"> \ast \sqcup \ast \stackrel{(\delta_0,\delta_1)}{\longrightarrow} I \stackrel{\exists !}{\longrightarrow} \ast </annotation></semantics></math></div> <p>this is the standard <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X \in Top</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X\times I</annotation></semantics></math>, example <a class="maruku-ref" href="#ProductTopologicalSpace"></a>, is called the standard <em><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></em> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. The endpoint inclusions of the interval make it factor the <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>I</mi><mo>⟶</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla_X \;\colon\; X \sqcup X \stackrel{((id,\delta_0),(id,\delta_1))}{\longrightarrow} X \times I \longrightarrow X \,. </annotation></semantics></math></div></div> <div class="num_defn" id="LeftHomotopy"> <h6 id="definition_32">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g\colon X \longrightarrow Y</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math>, then a <strong><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><mspace width="thinmathspace"></mspace><msub><mo>⇒</mo> <mi>L</mi></msub><mspace width="thinmathspace"></mspace><mi>g</mi></mrow><annotation encoding="application/x-tex"> \eta \colon f \,\Rightarrow_L\, g </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mi>I</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; X \times I \longrightarrow Y </annotation></semantics></math></div> <p>out of the standard <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#TopologicalInterval"></a>, such that this fits into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div style="float:right;margin:0 10px 10px 0;"> <img src="http://www.ncatlab.org/nlab/files/AHomotopy.jpg" width="400" /> </div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,. </annotation></semantics></math></div> <p>(graphics grabbed from J. Tauber <a href="http://jtauber.com/blog/2005/07/01/path_homotopy/">here</a>)</p> </div> <div class="num_example" id="PathsAsLeftHomotopyBetweenPoints"> <h6 id="example_24">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x,y \in X</annotation></semantics></math> be two of its points, regarded as functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x,y \colon \ast \longrightarrow X</annotation></semantics></math> from the point to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Then a left homotopy, def. <a class="maruku-ref" href="#LeftHomotopy"></a>, between these two functions is a commuting diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>x</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>δ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>y</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \ast \\ {}^{\mathllap{\delta_0}}\downarrow & \searrow^{\mathrlap{x}} \\ I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{\delta_1}}\uparrow & \nearrow_{\mathrlap{y}} \\ \ast } \,. </annotation></semantics></math></div> <p>This is simply a continuous path in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> whose endpoints are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>.</p> </div> <p>For instance:</p> <div class="num_example" id="StandardContractionOfStandardInterval"> <h6 id="example_25">Example</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>I</mi><mo>⟶</mo><mo>*</mo><mover><mo>⟶</mo><mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow></mover><mi>I</mi></mrow><annotation encoding="application/x-tex"> const_0 \;\colon\; I \longrightarrow \ast \overset{\delta_0}{\longrightarrow} I </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> from the standard interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I = [0,1]</annotation></semantics></math> to itself that is constant on the value 0. Then there is a left homotopy, def. <a class="maruku-ref" href="#LeftHomotopy"></a>, from the identity function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>I</mi></msub><mo>⇒</mo><msub><mi>const</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; id_I \Rightarrow const_0 </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>x</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta(x,t) \coloneqq x(1-t) \,. </annotation></semantics></math></div></div> <p>A key application of the concept of left homotopy is to the definition of <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>:</p> <div class="num_defn" id="HomotopyGroupsOftopologicalSpaces"> <h6 id="definition_33">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, then its set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(X)</annotation></semantics></math> of <em><a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a></em>, also called the <strong>0-th homotopy set</strong>, is the set of <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>-<a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> (def. <a class="maruku-ref" href="#LeftHomotopy"></a>) of points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \colon \ast \to X</annotation></semantics></math>, hence the set of path-connected components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (example <a class="maruku-ref" href="#PathsAsLeftHomotopyBetweenPoints"></a>). By <a class="existingWikiWord" href="/nlab/show/composition">composition</a> this extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><mi>Top</mi><mo>⟶</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0 \colon Top \longrightarrow Set \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \colon \ast \to X</annotation></semantics></math> any point, then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <strong><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/group">group</a></p> <ul> <li> <p>whose underlying <a class="existingWikiWord" href="/nlab/show/set">set</a> is the set of <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>-<a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mi>n</mi></msup><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">I^n \longrightarrow X</annotation></semantics></math> that take the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">I^n</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and where the left homotopies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> are constrained to be constant on the boundary;</p> </li> <li> <p>whose <a class="existingWikiWord" href="/nlab/show/group">group</a> product operation takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo lspace="verythinmathspace">:</mo><msup><mi>I</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\alpha \colon I^n \to X]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>β</mi><mo lspace="verythinmathspace">:</mo><msup><mi>I</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\beta \colon I^n \to X]</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo>⋅</mo><mi>β</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\alpha \cdot \beta]</annotation></semantics></math> with</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>⋅</mo><mi>β</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>I</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>I</mi> <mi>n</mi></msup><munder><mo>⊔</mo><mrow><msup><mi>I</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></munder><msup><mi>I</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \alpha \cdot \beta \;\colon\; I^n \stackrel{\simeq}{\longrightarrow} I^n \underset{I^{n-1}}{\sqcup} I^n \stackrel{(\alpha,\beta)}{\longrightarrow} X \,, </annotation></semantics></math></div> <p>where the first map is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> from the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cube">cube</a> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cube with one side twice the unit length (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>3</mn></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>3</mn></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, x_2, x_3, \cdots) \mapsto (2 x_1, x_2, x_3, \cdots)</annotation></semantics></math>).</p> <p>By <a class="existingWikiWord" href="/nlab/show/composition">composition</a>, this construction extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mo>•</mo><mo>≥</mo><mn>1</mn></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo>⟶</mo><msup><mi>Grp</mi> <mrow><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> \pi_{\bullet \geq 1} \;\colon\; Top^{\ast/} \longrightarrow Grp^{\mathbb{N}_{\geq 1}} </annotation></semantics></math></div> <p>from <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> to <a class="existingWikiWord" href="/nlab/show/graded+object">graded</a> <a class="existingWikiWord" href="/nlab/show/groups">groups</a>.</p> </div> <p>Notice that often one writes the value of this functor on a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>=</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_\ast = \pi_\bullet(f)</annotation></semantics></math>.</p> <div class="num_remark"> <h6 id="remark_18">Remark</h6> <p>At this point we don’t go further into the abstract reason why def. <a class="maruku-ref" href="#HomotopyGroupsOftopologicalSpaces"></a> yields group structure above degree 0, which is that <a class="existingWikiWord" href="/nlab/show/positive+dimension+spheres+are+H-cogroup+objects">positive dimension spheres are H-cogroup objects</a>. But this is important, for instance in the proof of the <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a>. See the section <em><a href="Introduction+to+Stable+homotopy+theory+--+S#BrownRepresentabilityTheorem">Brown representability theorem</a></em> in <a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+S">Part S</a>.</p> </div> <div class="num_defn" id="HomotopyEquivalence"> <h6 id="definition_34">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \;\colon\; X \longrightarrow Y</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a></strong> if there exists a continuous function the other way around, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">g \;\colon\; Y \longrightarrow X</annotation></semantics></math>, and <a class="existingWikiWord" href="/nlab/show/left+homotopies">left homotopies</a>, def. <a class="maruku-ref" href="#LeftHomotopy"></a>, from the two composites to the identity:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo>∘</mo><mi>g</mi><msub><mo>⇒</mo> <mi>L</mi></msub><msub><mi>id</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex"> \eta_1 \;\colon\; f\circ g \Rightarrow_L id_Y </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>g</mi><mo>∘</mo><mi>f</mi><msub><mo>⇒</mo> <mi>L</mi></msub><msub><mi>id</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta_2 \;\colon\; g\circ f \Rightarrow_L id_X \,. </annotation></semantics></math></div> <p>If here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\eta_2</annotation></semantics></math> is constant along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is said to exhibit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as a <strong><a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> </div> <div class="num_example" id="ProjectionFromStandardCylinderIsHomotopyEquivalence"> <h6 id="example_26">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \times I</annotation></semantics></math> its standard <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> of def. <a class="maruku-ref" href="#TopologicalInterval"></a>, then the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p \colon X \times I \longrightarrow X</annotation></semantics></math> and the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">(id, \delta_0) \colon X \longrightarrow X\times I</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a>, def. <a class="maruku-ref" href="#HomotopyEquivalence"></a>, and in fact are homotopy inverses to each other:</p> <p>The composition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>I</mi><mover><mo>⟶</mo><mi>p</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \overset{(id,\delta_0)}{\longrightarrow} X\times I \overset{p}{\longrightarrow} X </annotation></semantics></math></div> <p>is immediately the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (i.e. homotopic to the identity by a trivial homotopy), while the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi><mover><mo>⟶</mo><mi>p</mi></mover><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex"> X \times I \overset{p}{\longrightarrow} X \overset{(id, \delta_0)}{\longrightarrow} X\times I </annotation></semantics></math></div> <p>is homotopic to the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \times I</annotation></semantics></math> by a homotopy that is pointwise in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that of example <a class="maruku-ref" href="#StandardContractionOfStandardInterval"></a>.</p> </div> <div class="num_defn" id="WeakHomotopyEquivalenceOfTopologicalSpaces"> <h6 id="definition_35">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></strong> if its image under all the <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> functors of def. <a class="maruku-ref" href="#HomotopyGroupsOftopologicalSpaces"></a> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, hence if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_0(f) \;\colon\; \pi_0(X) \stackrel{\simeq}{\longrightarrow} \pi_0(X) </annotation></semantics></math></div> <p>and for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_n(f) \;\colon\; \pi_n(X,x) \stackrel{\simeq}{\longrightarrow} \pi_n(Y,f(y)) \,. </annotation></semantics></math></div></div> <div class="num_prop" id="TopologicalHomotopyEquivalencesAreWeakHomotopyEquivalences"> <h6 id="proposition_30">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, def. <a class="maruku-ref" href="#HomotopyEquivalence"></a>, is a weak homotopy equivalence, def. <a class="maruku-ref" href="#WeakHomotopyEquivalenceOfTopologicalSpaces"></a>.</p> <p>In particular a <a class="existingWikiWord" href="/nlab/show/deformation+retraction">deformation retraction</a>, def. <a class="maruku-ref" href="#HomotopyEquivalence"></a>, is a weak homotopy equivalence.</p> </div> <div class="proof"> <h6 id="proof_44">Proof</h6> <p>First observe that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X\in</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> the inclusion maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex"> X \overset{(id,\delta_0)}{\longrightarrow} X \times I </annotation></semantics></math></div> <p>into the standard <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a>, def. <a class="maruku-ref" href="#TopologicalInterval"></a>, are weak homotopy equivalences: by postcomposition with the contracting homotopy of the interval from example <a class="maruku-ref" href="#StandardContractionOfStandardInterval"></a> all homotopy groups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \times I</annotation></semantics></math> have representatives that factor through this inclusion.</p> <p>Then given a general <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, apply the homotopy groups functor to the corresponding homotopy diagrams (where for the moment we notationally suppress the choice of basepoint for readability) to get two commuting diagrams</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \pi_\bullet(X) \\ {}^{\mathllap{\pi_\bullet(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{\pi_\bullet(f)\circ \pi_\bullet(g)}} \\ \pi_\bullet(X \times I) &\stackrel{\pi_\bullet(\eta)}{\longrightarrow}& \pi_\bullet(Y) \\ {}^{\mathllap{\pi_\bullet(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{\pi_\bullet(id)}} \\ \pi_\bullet(X) } \;\;\;\;\;\;\; \,, \;\;\;\;\;\;\; \array{ \pi_\bullet(Y) \\ {}^{\mathllap{\pi_\bullet(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{\pi_\bullet(g)\circ \pi_\bullet(f)}} \\ \pi_\bullet(Y \times I) &\stackrel{\pi_\bullet(\eta)}{\longrightarrow}& \pi_\bullet(X) \\ {}^{\mathllap{\pi_\bullet(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{\pi_\bullet(id)}} \\ \pi_\bullet(Y) } \,. </annotation></semantics></math></div> <p>By the previous observation, the vertical morphisms here are isomorphisms, and hence these diagrams exhibit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(f)</annotation></semantics></math> as the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(g)</annotation></semantics></math>, hence both as isomorphisms.</p> </div> <div class="num_remark" id="NotEveryHomotopyEquivalenceIsAWeakHomotopyEquivalence"> <h6 id="remark_19">Remark</h6> <p>The converse of prop. <a class="maruku-ref" href="#TopologicalHomotopyEquivalencesAreWeakHomotopyEquivalences"></a> is not true generally: not every <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> between topological spaces is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>. (For an example with full details spelled out see for instance Fritsch, Piccinini: “Cellular Structures in Topology”, p. 289-290).</p> <p>However, as we will discuss below, it turns out that</p> <ol> <li> <p>every weak homotopy equivalence between <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> is a homotopy equivalence (<a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a>, cor. <a class="maruku-ref" href="#WhiteheadTheorem"></a>);</p> </li> <li> <p>every topological space is connected by a weak homotopy equivalence to a CW-complex (<a class="existingWikiWord" href="/nlab/show/CW+approximation">CW approximation</a>, remark <a class="maruku-ref" href="#EveryTopologicalSpaceWeaklyEquivalentToACWComplex"></a>).</p> </li> </ol> </div> <div class="num_example" id="FactoringTopologicalCodiagonalThroughCylinder"> <h6 id="example_27">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X\in Top</annotation></semantics></math>, the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X\times I \longrightarrow X</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#TopologicalInterval"></a>, is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, def. <a class="maruku-ref" href="#WeakHomotopyEquivalenceOfTopologicalSpaces"></a>. This means that the factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>↪</mo><mrow></mrow></mover><mi>X</mi><mo>×</mo><mi>I</mi><mover><mo>⟶</mo><mo>≃</mo></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> \nabla_X \;\colon\; X \sqcup X \stackrel{}{\hookrightarrow} X\times I \stackrel{\simeq}{\longrightarrow} X </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_X</annotation></semantics></math> in def. <a class="maruku-ref" href="#TopologicalInterval"></a>, which in general is far from being a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>, may be thought of as factoring it through a monomorphism after replacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, up to weak homotopy equivalence, by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X\times I</annotation></semantics></math>.</p> <p>In fact, further below (prop. <a class="maruku-ref" href="#StandardContractionOfStandardInterval"></a>) we see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \sqcup X \to X \times I</annotation></semantics></math> has better properties than the generic monomorphism has, in particular better homotopy invariant properties: it has the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against all <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>⟶</mo><mi>p</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex"> E \stackrel{p}{\longrightarrow} B</annotation></semantics></math> that are also <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>.</p> </div> <p>Of course the concept of left homotopy in def. <a class="maruku-ref" href="#LeftHomotopy"></a> is accompanied by a concept of <em><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></em>. This we turn to now.</p> <div class="num_defn" id="TopologicalPathSpace"> <h6 id="definition_36">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/path+space">path space</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, its <strong>standard topological <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></strong> is the topological <a class="existingWikiWord" href="/nlab/show/path+space">path space</a>, hence the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">X^I</annotation></semantics></math>, prop. <a class="maruku-ref" href="#MappingTopologicalSpaceIsExponentialObject"></a>, out of the standard interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> of def. <a class="maruku-ref" href="#TopologicalInterval"></a>.</p> </div> <div class="num_example"> <h6 id="example_28">Example</h6> <p>The endpoint inclusion into the standard interval, def. <a class="maruku-ref" href="#TopologicalInterval"></a>, makes the path space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">X^I</annotation></semantics></math> of def. <a class="maruku-ref" href="#TopologicalPathSpace"></a> factor the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> through the inclusion of constant paths and the endpoint evaluation of paths:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><msup><mi>X</mi> <mrow><mi>I</mi><mo>→</mo><mo>*</mo></mrow></msup></mrow></mover><msup><mi>X</mi> <mi>I</mi></msup><mover><mo>⟶</mo><mrow><msup><mi>X</mi> <mrow><mo>*</mo><mo>⊔</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow></msup></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_X \;\colon\; X \stackrel{X^{I \to \ast}}{\longrightarrow} X^I \stackrel{X^{\ast \sqcup \ast \to I}}{\longrightarrow} X \times X \,. </annotation></semantics></math></div> <p>This is the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> to example <a class="maruku-ref" href="#TopologicalInterval"></a>. As in that example, below we will see (prop. <a class="maruku-ref" href="#TopologicalPathSpaceIsGoodPathSpaceObject"></a>) that this factorization has good properties, in that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mi>I</mi><mo>→</mo><mo>*</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{I \to \ast}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mo>*</mo><mo>⊔</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow></msup></mrow><annotation encoding="application/x-tex">X^{\ast \sqcup \ast \to I}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>.</p> </li> </ol> <p>So while in general the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_X</annotation></semantics></math> is far from being an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> or even just a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, the factorization through the <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> may be thought of as replacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, up to weak homotopy equivalence, by its path space, such as to turn its diagonal into a Serre fibration after all.</p> </div> <div class="num_defn" id="RightHomotopy"> <h6 id="definition_37">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g\colon X \longrightarrow Y</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math>, then a <strong><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo>⇒</mo> <mi>R</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow_R g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><msup><mi>Y</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; X \longrightarrow Y^I </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#TopologicalPathSpace"></a>, such that this fits into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msup><mi>X</mi> <mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><msup><mi>Y</mi> <mi>I</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>Y</mi> <mrow><msub><mi>δ</mi> <mn>1</mn></msub></mrow></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Y \\ & {}^{\mathllap{f}}\nearrow & \uparrow^{\mathrlap{X^{\delta_0}}} \\ X &\stackrel{\eta}{\longrightarrow}& Y^I \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{Y^{\delta_1}}} \\ && Y } \,. </annotation></semantics></math></div></div> <h3 id="cell_complexes">Cell complexes</h3> <p>We consider topological spaces that are built consecutively by <a class="existingWikiWord" href="/nlab/show/attaching+space">attaching</a> basic cells.</p> <div class="num_defn" id="TopologicalGeneratingCofibrations"> <h6 id="definition_38">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub><mo>≔</mo><msub><mrow><mo>{</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>↪</mo><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mo>}</mo></mrow> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I_{Top} \coloneqq \left\{ S^{n-1} \stackrel{\iota_n}{\hookrightarrow} D^{n} \right\}_{n \in \mathbb{N}} \; \subset Mor(Top) </annotation></semantics></math></div> <p>for the set of canonical <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> inclusion maps of the standard <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a>, example <a class="maruku-ref" href="#TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself"></a>. This going to be called the set of standard <strong>topological <a class="existingWikiWord" href="/nlab/show/generating+cofibrations">generating cofibrations</a></strong>.</p> </div> <div class="num_defn" id="TopologicalCellComplex"> <h6 id="definition_39">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X \in Top</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cell attachment</strong> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> (“<a class="existingWikiWord" href="/nlab/show/attaching+space">attaching space</a>”, example <a class="maruku-ref" href="#PushoutInTop"></a>) of a generating cofibration, def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mi>ϕ</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><munder><mo>⊔</mo><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></munder><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>=</mo><mi>X</mi><msub><mo>∪</mo> <mi>ϕ</mi></msub><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\stackrel{\phi}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& X \underset{S^{n-1}}{\sqcup} D^n & = X \cup_\phi D^n } </annotation></semantics></math></div> <p>along some <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>.</p> <p>A continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> is called a <strong>topological <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a></strong> if it is exhibited by a (possibly infinite) sequence of cell <a class="existingWikiWord" href="/nlab/show/attaching+space">attachments</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, in that it is a <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> (def. <a class="maruku-ref" href="#TransfiniteComposition"></a>) of <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a> (example <a class="maruku-ref" href="#PushoutInTop"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mi>k</mi></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><msub><mi>ι</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{i}{\coprod} S^{n_i - 1} &\longrightarrow& X_{k} \\ {}^{\mathllap{\underset{i}{\coprod}\iota_{n_i}}}\downarrow &(po)& \downarrow \\ \underset{i}{\coprod} D^{n_i} &\longrightarrow& X_{k+1} } </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> (example <a class="maruku-ref" href="#DisjointUnionOfTopologicalSpacesIsCoproduct"></a>) of <a class="existingWikiWord" href="/nlab/show/generating+cofibrations">generating cofibrations</a> (def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>).</p> <p>A topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a></strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\emptyset \longrightarrow X</annotation></semantics></math> is a relative cell complex.</p> <p>A relative cell complex is called a <strong>finite relative cell complex</strong> if it is obtained from a <a class="existingWikiWord" href="/nlab/show/finite+number">finite number</a> of cell attachments.</p> <p>A (relative) cell complex is called a (relative) <strong><a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></strong> if the above transfinite composition is countable</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>=</mo><msub><mi>X</mi> <mn>0</mn></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi><mo>=</mo><munder><mi>lim</mi><mo>⟶</mo></munder><msub><mi>X</mi> <mo>•</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X = X_0 &\longrightarrow& X_1 &\longrightarrow& X_2 &\longrightarrow& \cdots \\ & {}_{\mathllap{f}}\searrow & \downarrow & \swarrow && \cdots \\ && Y = \underset{\longrightarrow}{\lim} X_\bullet } </annotation></semantics></math></div> <p>and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">X_k</annotation></semantics></math> is obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{k-1}</annotation></semantics></math> by attaching cells precisely only of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_20">Remark</h6> <p>Strictly speaking a relative cell complex, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>, is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math>, <em>together</em> with its cell structure, hence together with the information of the pushout diagrams and the transfinite composition of the pushout maps that exhibit it.</p> <p>In many applications, however, all that matters is that there is <em>some</em> (relative) cell decomosition, and then one tends to speak loosely and mean by a (relative) cell complex only a (relative) topological space that admits some cell decomposition.</p> </div> <p>The following lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a>, together with lemma <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a> below are the only two statements of the entire development here that involve the <a class="existingWikiWord" href="/nlab/show/concrete+particular">concrete particular</a> nature of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> (“<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>”), everything beyond that is <a class="existingWikiWord" href="/nlab/show/general+abstract">general abstract</a> homotopy theory.</p> <div class="num_lemma" id="CompactSubsetsAreSmallInCellComplexes"> <h6 id="lemma_16">Lemma</h6> <p>Assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> and the <a class="existingWikiWord" href="/nlab/show/law+of+excluded+middle">law of excluded middle</a>, every <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/topological+subspace">subspace</a> of a topological <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>, intersects the <a class="existingWikiWord" href="/nlab/show/interior">interior</a> of a <a class="existingWikiWord" href="/nlab/show/finite+number">finite number</a> of cells.</p> </div> <p>(e.g. <a href="#Hirschhorn15">Hirschhorn 15, section 3.1</a>)</p> <div class="proof"> <h6 id="proof_45">Proof</h6> <p>So let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> be a topological cell complex and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">C \hookrightarrow Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/topological+subspace">subspace</a>. Define a subset</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⊂</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> P \subset Y </annotation></semantics></math></div> <p>by <em>choosing</em> one point in the <a class="existingWikiWord" href="/nlab/show/interior">interior</a> of the intersection with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> of each cell of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> that intersects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>It is now sufficient to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> has no <a class="existingWikiWord" href="/nlab/show/accumulation+point">accumulation point</a>. Because, by the <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compactness</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, every non-finite subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> does have an accumulation point, and hence the lack of such shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> and hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> intersects the interior of finitely many cells of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> <p>To that end, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c\in C</annotation></semantics></math> be any point. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> is a 0-cell in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>c</mi></msub><mo>≔</mo><mo stretchy="false">{</mo><mi>c</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_c \coloneqq \{c\}</annotation></semantics></math>. Otherwise write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">e_c</annotation></semantics></math> for the unique cell of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> that contains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> in its <a class="existingWikiWord" href="/nlab/show/interior">interior</a>. By construction, there is exactly one point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> in the interior of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">e_c</annotation></semantics></math>. Hence there is an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><msub><mi>U</mi> <mi>c</mi></msub><mo>⊂</mo><msub><mi>e</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">c \in U_c \subset e_c</annotation></semantics></math> containing no further points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> beyond possibly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> itself, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> happens to be that single point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">e_c</annotation></semantics></math>.</p> <p>It is now sufficient to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">U_c</annotation></semantics></math> may be enlarged to an open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>U</mi><mo stretchy="false">˜</mo></mover> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\tilde U_c</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> containing no point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>, except for possibly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> itself, for that means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> is not an accumulation point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>.</p> <p>To that end, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_c</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/ordinal">ordinal</a> that labels the stage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mrow><msub><mi>α</mi> <mi>c</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">Y_{\alpha_c}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> in the <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>-presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> at which the cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">e_c</annotation></semantics></math> above appears. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> be the ordinal of the full cell complex. Then define the set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>≔</mo><mrow><mo>{</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>β</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><msub><mi>α</mi> <mi>c</mi></msub><mo>≤</mo><mi>β</mi><mo>≤</mo><mi>γ</mi><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>U</mi><munder><mo>⊂</mo><mi>open</mi></munder><msub><mi>Y</mi> <mi>β</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>∩</mo><msub><mi>Y</mi> <mi>α</mi></msub><mo>=</mo><msub><mi>U</mi> <mi>c</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>∩</mo><mi>P</mi><mo>∈</mo><mo stretchy="false">{</mo><mi>∅</mi><mo>,</mo><mo stretchy="false">{</mo><mi>c</mi><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> T \coloneqq \left\{ \; (\beta, U) \;|\; \alpha_c \leq \beta \leq \gamma \;\,,\; U \underset{open}{\subset} Y_\beta \;\,,\; U \cap Y_\alpha = U_c \;\,,\; U \cap P \in \{ \emptyset, \{c\} \} \; \right\} \,, </annotation></semantics></math></div> <p>and regard this as a <a class="existingWikiWord" href="/nlab/show/partially+ordered+set">partially ordered set</a> by declaring a partial ordering via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>β</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo><</mo><mo stretchy="false">(</mo><msub><mi>β</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>β</mi> <mn>1</mn></msub><mo><</mo><msub><mi>β</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>U</mi> <mn>2</mn></msub><mo>∩</mo><msub><mi>Y</mi> <mrow><msub><mi>β</mi> <mn>1</mn></msub></mrow></msub><mo>=</mo><msub><mi>U</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\beta_1, U_1) \lt (\beta_2, U_2) \;\;\;\; \Leftrightarrow \;\;\;\; \beta_1 \lt \beta_2 \;\,,\; U_2 \cap Y_{\beta_1} = U_1 \,. </annotation></semantics></math></div> <p>This is set up such that every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>β</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\beta, U)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> the maximum value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><mi>γ</mi></mrow><annotation encoding="application/x-tex">\beta = \gamma</annotation></semantics></math> is an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>U</mi><mo stretchy="false">˜</mo></mover> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\tilde U_c</annotation></semantics></math> that we are after.</p> <p>Observe then that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>β</mi> <mi>s</mi></msub><mo>,</mo><msub><mi>U</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\beta_s, U_s)_{s\in S}</annotation></semantics></math> a chain in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mo><</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T,\lt)</annotation></semantics></math> (a subset on which the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo><</mo></mrow><annotation encoding="application/x-tex">\lt</annotation></semantics></math> restricts to a <a class="existingWikiWord" href="/nlab/show/total+order">total order</a>), it has an upper bound in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> given by the <a class="existingWikiWord" href="/nlab/show/union">union</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mo>∪</mo> <mi>s</mi></msub><msub><mi>β</mi> <mi>s</mi></msub><mo>,</mo><msub><mo>∪</mo> <mi>s</mi></msub><msub><mi>U</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">({\cup}_s \beta_s ,\cup_s U_s)</annotation></semantics></math>. Therefore <a class="existingWikiWord" href="/nlab/show/Zorn%27s+lemma">Zorn's lemma</a> applies, saying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mo><</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T,\lt)</annotation></semantics></math> contains a <a class="existingWikiWord" href="/nlab/show/maximal+element">maximal element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>β</mi> <mi>max</mi></msub><mo>,</mo><msub><mi>U</mi> <mi>max</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\beta_{max}, U_{max})</annotation></semantics></math>.</p> <p>Hence it is now sufficient to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>max</mi></msub><mo>=</mo><mi>γ</mi></mrow><annotation encoding="application/x-tex">\beta_{max} = \gamma</annotation></semantics></math>. We argue this by showing that assuming <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>max</mi></msub><mo><</mo><mi>γ</mi></mrow><annotation encoding="application/x-tex">\beta_{\max}\lt \gamma</annotation></semantics></math> leads to a contradiction.</p> <p>So assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>max</mi></msub><mo><</mo><mi>γ</mi></mrow><annotation encoding="application/x-tex">\beta_{max}\lt \gamma</annotation></semantics></math>. Then to construct an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> that is larger than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>β</mi> <mi>max</mi></msub><mo>,</mo><msub><mi>U</mi> <mi>max</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\beta_{max},U_{max})</annotation></semantics></math>, consider for each cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> at stage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mrow><msub><mi>β</mi> <mi>max</mi></msub><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Y_{\beta_{max}+1}</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/attaching+map">attaching map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>d</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msub><mi>Y</mi> <mrow><msub><mi>β</mi> <mi>max</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">h_d \colon S^{n-1} \to Y_{\beta_{max}}</annotation></semantics></math> and the corresponding preimage open set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>h</mi> <mi>d</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>max</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">h_d^{-1}(U_{max})\subset S^{n-1}</annotation></semantics></math>. Enlarging all these preimages to open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> (such that their image back in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><msub><mi>β</mi> <mi>max</mi></msub><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{\beta_{max}+1}</annotation></semantics></math> does not contain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>), then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>β</mi> <mi>max</mi></msub><mo>,</mo><msub><mi>U</mi> <mi>max</mi></msub><mo stretchy="false">)</mo><mo><</mo><mo stretchy="false">(</mo><msub><mi>β</mi> <mi>max</mi></msub><mo>+</mo><mn>1</mn><mo>,</mo><msub><mo>∪</mo> <mi>d</mi></msub><msub><mi>U</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\beta_{max}, U_{max}) \lt (\beta_{max}+1, \cup_d U_d )</annotation></semantics></math>. This is a contradiction. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>max</mi></msub><mo>=</mo><mi>γ</mi></mrow><annotation encoding="application/x-tex">\beta_{max} = \gamma</annotation></semantics></math>, and we are done.</p> </div> <p>It is immediate and useful to generalize the concept of topological cell complexes as follows.</p> <div class="num_defn" id="TopologicalCCellComplex"> <h6 id="definition_40">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> any category and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K \subset Mor(\mathcal{C})</annotation></semantics></math> any sub-<a class="existingWikiWord" href="/nlab/show/class">class</a> of its morphisms, a <strong>relative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-cell complexes</strong> is a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> which is a <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> (def. <a class="maruku-ref" href="#TransfiniteComposition"></a>) of <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a> of <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> of morphsims in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> </div> <div class="num_defn" id="TopologicalGeneratingAcyclicCofibrations"> <h6 id="definition_41">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub><mo>≔</mo><msub><mrow><mo>{</mo><msup><mi>D</mi> <mi>n</mi></msup><mover><mo>↪</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo>}</mo></mrow> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> J_{Top} \coloneqq \left\{ D^n \stackrel{(id,\delta_0)}{\hookrightarrow} D^n \times I \right\}_{n \in \mathbb{N}} \; \subset Mor(Top) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/set">set</a> of inclusions of the topological <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a>, def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>, into their <a class="existingWikiWord" href="/nlab/show/cylinder+objects">cylinder objects</a>, def. <a class="maruku-ref" href="#TopologicalInterval"></a>, along (for definiteness) the left endpoint inclusion.</p> <p>These inclusions are similar to the standard topological <a class="existingWikiWord" href="/nlab/show/generating+cofibrations">generating cofibrations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math> of def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>, but in contrast to these they are “acyclic” (meaning: trivial on homotopy classes of maps from “cycles” given by <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>) in that they are <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> (by prop. <a class="maruku-ref" href="#TopologicalHomotopyEquivalencesAreWeakHomotopyEquivalences"></a>).</p> <p>Accordingly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math> is to be called the set of standard <strong>topological <a class="existingWikiWord" href="/nlab/show/generating+acyclic+cofibrations">generating acyclic cofibrations</a></strong>.</p> </div> <div class="num_lemma" id="CylinderOverCWComplexIsJTopRelativeCellComplex"> <h6 id="lemma_17">Lemma</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> (def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>), then its inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \overset{(id, \delta_0)}{\longrightarrow} X\times I</annotation></semantics></math> into its standard <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder</a> (def. <a class="maruku-ref" href="#TopologicalInterval"></a>) is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> (def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a>, def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>).</p> </div> <div class="proof"> <h6 id="proof_46">Proof</h6> <p>First erect a cylinder over all 0-cells</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>x</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow></munder><msup><mi>D</mi> <mn>0</mn></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>x</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow></munder><msup><mi>D</mi> <mn>1</mn></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{x \in X_0}{\coprod} D^0 &\longrightarrow& X \\ \downarrow &(po)& \downarrow \\ \underset{x\in X_0}{\coprod} D^1 &\longrightarrow& Y_1 } \,. </annotation></semantics></math></div> <p>Assume then that the cylinder over all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cells of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has been erected using attachment from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>. Then the union of any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the cylinder over its boundary is homeomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">D^{n+1}</annotation></semantics></math> and is like the cylinder over the cell “with end and interior removed”. Hence via <a class="existingWikiWord" href="/nlab/show/attaching+space">attaching</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">D^{n+1} \to D^{n+1}\times I</annotation></semantics></math> the cylinder over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> is erected.</p> </div> <div class="num_lemma" id="TopologicalGeneratingAcyclicCofibrationsAreRelativeCellComplexes"> <h6 id="lemma_18">Lemma</h6> <p>The maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">D^n \hookrightarrow D^n \times I</annotation></semantics></math> in def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a> are finite <a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a>, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>. In other words, the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a>.</p> </div> <div class="proof"> <h6 id="proof_47">Proof</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>=</mo></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ D^n & = & D^n \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow \\ D^n \times I &\simeq& D^{n+1} } </annotation></semantics></math></div> <p>such that the map on the right is the inclusion of one hemisphere into the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">D^{n+1}</annotation></semantics></math>. This inclusion is the result of <a class="existingWikiWord" href="/nlab/show/attaching+space">attaching</a> two cells:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>=</mo></msup></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mi>id</mi></mover></mtd> <mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd> <mtd><munder><mo>⟶</mo><mi>id</mi></munder></mtd> <mtd><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^{n} \\ && \downarrow^{=} \\ S^n &\overset{id}{\longrightarrow}& S^n \\ {}^{\mathllap{\iota_{n+1}}}\downarrow &(po)& \downarrow \\ D^{n+1} &\underset{id}{\longrightarrow}& D^{n+1} } \,. </annotation></semantics></math></div> <p>here the top pushout is the one from example <a class="maruku-ref" href="#TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself"></a>.</p> </div> <div class="num_lemma" id="JTopRelativeCellComplexesAreWeakHomotopyEquivalences"> <h6 id="lemma_19">Lemma</h6> <p>Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> (def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>, def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a>) is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, def. <a class="maruku-ref" href="#WeakHomotopyEquivalenceOfTopologicalSpaces"></a>.</p> </div> <div class="proof"> <h6 id="proof_48">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>=</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>β</mi><mo>≤</mo><mi>α</mi></mrow></msub><msub><mi>X</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">X \longrightarrow \hat X = \underset{\longleftarrow}{\lim}_{\beta \leq \alpha} X_\beta</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a>.</p> <p>First observe that with the elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">D^n \hookrightarrow D^n \times I</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math> being <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> (by example <a class="maruku-ref" href="#ProjectionFromStandardCylinderIsHomotopyEquivalence"></a>), each of the stages <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>β</mi></msub><mo>⟶</mo><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{\beta} \longrightarrow X_{\beta + 1}</annotation></semantics></math> in the relative cell complex is also a homotopy equivalence. We make this fully explicit:</p> <p>By definition, such a stage is a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mi>β</mi></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ {}^{\mathllap{\underset{i \in I}{\sqcup} (id, \delta_0)} }\downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} \times I &\longrightarrow& X_{\beta + 1} } \,. </annotation></semantics></math></div> <p>Then the fact that the projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub><mo lspace="verythinmathspace">:</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo>×</mo><mi>I</mi><mo>→</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">p_{n_i} \colon D^{n_i} \times I \to D^{n_i}</annotation></semantics></math> are strict left inverses to the inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(id, \delta_0)</annotation></semantics></math> gives a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mi>β</mi></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo>×</mo><mi>I</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>p</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mi>β</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ {}^{\mathllap{\underset{i \in I}{\sqcup} (id, \delta_0)} }\downarrow && \downarrow^{\mathrlap{id}} \\ \underset{i \in I}{\sqcup} D^{n_i} \times I \\ {}^{\mathllap{\underset{i \in I}{\sqcup} p_{n_i} }}\downarrow && \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta } </annotation></semantics></math></div> <p>and so the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> (<a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{\beta + 1}</annotation></semantics></math> gives a factorization of the identity morphism on the right through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{\beta + 1}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mi>β</mi></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>p</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mi>β</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ {}^{\mathllap{\underset{i \in I}{\sqcup} (id, \delta_0)} }\downarrow && \downarrow^{} \\ \underset{i \in I}{\sqcup} D^{n_i} \times I &\longrightarrow& X_{\beta + 1} \\ {}^{\mathllap{\underset{i \in I}{\sqcup} p_{n_i} }}\downarrow && \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta } </annotation></semantics></math></div> <p>which exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><msub><mi>X</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">X_{\beta + 1} \to X_\beta</annotation></semantics></math> as a strict left inverse to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>β</mi></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{\beta} \to X_{\beta + 1}</annotation></semantics></math>. Hence it is now sufficient to show that this is also a homotopy right inverse.</p> <p>To that end, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub><mo lspace="verythinmathspace">:</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo>×</mo><mi>I</mi><mo>⟶</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex"> \eta_{n_i} \colon D^{n_i}\times I \longrightarrow D^{n_i} \times I </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> that exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">p_{n_i}</annotation></semantics></math> as a homotopy right inverse to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">p_{n_i}</annotation></semantics></math> by example <a class="maruku-ref" href="#ProjectionFromStandardCylinderIsHomotopyEquivalence"></a>. For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">t \in [0,1]</annotation></semantics></math> consider the <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mi>β</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo>×</mo><mi>I</mi></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>η</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ \downarrow && \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} \times I && X_{\beta + 1} \\ {}^{\mathllap{\eta_{n_i}(-,t)}}\downarrow && \downarrow^{\mathrlap{id}} \\ \underset{i \in I}{\sqcup} D^{n_i} \times I &\longrightarrow& X_{\beta + 1} } \,. </annotation></semantics></math></div> <p>Regarded as a <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> under the <a class="existingWikiWord" href="/nlab/show/span">span</a> in the top left, the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> (<a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{\beta + 1}</annotation></semantics></math> gives a continuous function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⟶</mo><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> \eta(-,t) \;\colon\; X_{\beta + 1} \longrightarrow X_{\beta + 1} </annotation></semantics></math></div> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">t \in [0,1]</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t = 0</annotation></semantics></math> this construction reduces to the provious one in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><msub><mi>X</mi> <mi>β</mi></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\eta(-,0) \colon X_{\beta +1 } \to X_{\beta} \to X_{\beta + 1}</annotation></semantics></math> is the composite which we need to homotope to the identity; while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta(-,1)</annotation></semantics></math> is the identity. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta(-,t)</annotation></semantics></math> is clearly also continuous in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> it constitutes a continuous function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>×</mo><mi>I</mi><mo>⟶</mo><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; X_{\beta + 1}\times I \longrightarrow X_{\beta + 1} </annotation></semantics></math></div> <p>which exhibits the required left homotopy.</p> <p>So far this shows that each stage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>β</mi></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{\beta} \to X_{\beta+1}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, hence, by prop. <a class="maruku-ref" href="#TopologicalHomotopyEquivalencesAreWeakHomotopyEquivalences"></a>, a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>.</p> <p>This means that all morphisms in the following diagram (notationally suppressing basepoints and showing only the finite stages)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mo>≃</mo></mpadded></msub></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>α</mi></msub><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>α</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \pi_n(X) &\overset{\simeq}{\longrightarrow}& \pi_n(X_1) &\overset{\simeq}{\longrightarrow}& \pi_n(X_2) &\overset{\simeq}{\longrightarrow}& \pi_n(X_3) &\overset{\simeq}{\longrightarrow}& \cdots \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\simeq}} & \swarrow_{\mathrlap{\simeq}} & \cdots \\ && \underset{\longleftarrow}{\lim}_\alpha \pi_n(X_\alpha) } </annotation></semantics></math></div> <p>are isomorphisms.</p> <p>Moreover, lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a> gives that every representative and every null homotopy of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(\hat X)</annotation></semantics></math> already exists at some finite stage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">X_k</annotation></semantics></math>. This means that also the universally induced morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>α</mi></msub><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>α</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underset{\longleftarrow}{\lim}_\alpha \pi_n(X_\alpha) \overset{\simeq}{\longrightarrow} \pi_n(\hat X) </annotation></semantics></math></div> <p>is an isomorphism. Hence the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X) \overset{\simeq}{\longrightarrow} \pi_n(\hat X)</annotation></semantics></math> is an isomorphism.</p> </div> <h3 id="fibrations">Fibrations</h3> <p>Given a relative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-cell complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\iota \colon X \to Y</annotation></semantics></math>, def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a>, it is typically interesting to study the <a class="existingWikiWord" href="/nlab/show/extension">extension</a> problem along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, i.e. to ask which topological spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> are such that every <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">f\colon X \longrightarrow E</annotation></semantics></math> has an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ι</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mo>∃</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{f}{\longrightarrow}& E \\ {}^{\mathllap{\iota}}\downarrow & \nearrow_{\mathrlap{\exists \tilde f}} \\ Y } \,. </annotation></semantics></math></div> <p>If such extensions exists, it means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is sufficiently “spread out” with respect to the maps in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. More generally one considers this extension problem fiberwise, i.e. with both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> (hence also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>) equipped with a map to some base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>:</p> <div class="num_defn" id="RightLiftingProperty"> <h6 id="definition_42">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and a sub-<a class="existingWikiWord" href="/nlab/show/class">class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \subset Mor(\mathcal{C})</annotation></semantics></math> of its <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>, then a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>⟶</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p \colon E \longrightarrow B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is said to have the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> if every <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>c</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& E \\ {}^{\mathllap{c}}\downarrow && \downarrow^{\mathrlap{p}} \\ Y &\longrightarrow& B } \,, </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math>, has a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>, in that it may be completed to a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>c</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>h</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& E \\ {}^{\mathllap{c}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p}} \\ Y &\longrightarrow& B } \,. </annotation></semantics></math></div> <p>We will also say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphism">injective morphism</a></strong> if it satisfies the right lifting property against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </div> <div class="num_defn" id="SerreFibration"> <h6 id="definition_43">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>⟶</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p \colon E \longrightarrow B</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a></strong> if it is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphism">injective morphism</a>; i.e. if it has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a>, def. <a class="maruku-ref" href="#RightLiftingProperty"></a>, against all topological generating acylic cofibrations, def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>; hence if for every <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ D^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,, </annotation></semantics></math></div> <p>has a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>, in that it may be completed to a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>h</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ D^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_21">Remark</h6> <p>Def. <a class="maruku-ref" href="#SerreFibration"></a> says, in view of the definition of <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, that a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a map with the property that given a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, def. <a class="maruku-ref" href="#LeftHomotopy"></a>, between two functions into its <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a>, and given a lift of one the two functions through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, then also the homotopy between the two lifts. Therefore the condition on a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a> is also called the <em><a class="existingWikiWord" href="/nlab/show/homotopy+lifting+property">homotopy lifting property</a></em> for maps whose domain is an <a class="existingWikiWord" href="/nlab/show/n-disk">n-disk</a>.</p> <p>More generally one may ask functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> to have such <a class="existingWikiWord" href="/nlab/show/homotopy+lifting+property">homotopy lifting property</a> for functions with arbitrary domain. These are called <em><a class="existingWikiWord" href="/nlab/show/Hurewicz+fibrations">Hurewicz fibrations</a></em>.</p> </div> <div class="num_remark" id="SerreFibrationsByLiftingAgainstMapsHomeomorphicToDiskInclusions"> <h6 id="remark_22">Remark</h6> <p>The precise shape of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">D^n \times I</annotation></semantics></math> in def. <a class="maruku-ref" href="#SerreFibration"></a> turns out not to actually matter much for the nature of Serre fibrations. We will eventually find below (prop. <a class="maruku-ref" href="#LiftingPropertyInTheClassicalModelStructureOnTopologicalSpaces"></a>) that what actually matters here is only that the inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">D^n \hookrightarrow D^n \times I</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a> (lemma <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrationsAreRelativeCellComplexes"></a>) and <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> (prop. <a class="maruku-ref" href="#TopologicalHomotopyEquivalencesAreWeakHomotopyEquivalences"></a>) and that all of these may be generated from them in a suitable way.</p> <p>But for simple special cases this is readily seen directly, too. Notably we could replace the <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a> in def. <a class="maruku-ref" href="#SerreFibration"></a> with any <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> topological space. A choice important in the comparison to the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a> (<a href="#GeometricRealizationSection">below</a>) is to instead take the topological <a class="existingWikiWord" href="/nlab/show/n-simplices">n-simplices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^n</annotation></semantics></math>. Hence a Serre fibration is equivalently characterized as having lifts in all diagrams of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Δ</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Δ</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Delta^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ \Delta^n\times I &\longrightarrow& B } \,. </annotation></semantics></math></div> <p>Other deformations of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disks are useful in computations, too. For instance there is a homeomorphism from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disk to its “cylinder with interior and end removed”, formally:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo>∂</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (D^n \times \{0\})\cup (\partial D^n \times I) &\simeq& D^n \\ \downarrow && \downarrow \\ D^n \times I &\simeq& D^n\times I } </annotation></semantics></math></div> <p>and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a Serre fibration equivalently also if it admits lifts in all diagrams of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo>∂</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (D^n \times \{0\})\cup (\partial D^n \times I) &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,. </annotation></semantics></math></div></div> <p>The following is a general fact about closure of morphisms defined by lifting properties which we prove in generality below as prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a>.</p> <div class="num_prop" id="SerreFibrationHasRightLiftingAgainstJTopRelativeCellComplexes"> <h6 id="proposition_31">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a> has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against all <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> (see remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a> (def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>).</p> </div> <p>The following statement is foreshadowing the <a class="existingWikiWord" href="/nlab/show/long+exact+sequences+of+homotopy+groups">long exact sequences of homotopy groups</a> (<a href="#LongSequences">below</a>) induced by any <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>, the full version of which we come to <a href="HomotopyFiberSequences">below</a> (example <a class="maruku-ref" href="#LongExactSequeceOfHomotopyGroups"></a>) after having developed more of the abstract homotopy theory.</p> <div class="num_prop" id="SerreFibrationGivesExactSequenceOfHomotopyGroups"> <h6 id="proposition_32">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \longrightarrow Y</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">y \colon \ast \to Y</annotation></semantics></math> be any point and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>y</mi></msub><mover><mo>↪</mo><mi>ι</mi></mover><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex"> F_y \overset{\iota}{\hookrightarrow} X \overset{f}{\longrightarrow} Y </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> inclusion over that point. Then for every choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \colon \ast \to X</annotation></semantics></math> of lift of the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, the induced sequence of <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>y</mi></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>ι</mi> <mo>*</mo></msub></mrow></mover><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mover><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_{\bullet}(F_y, x) \overset{\iota_\ast}{\longrightarrow} \pi_\bullet(X, x) \overset{f_\ast}{\longrightarrow} \pi_\bullet(Y) </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact</a>, in that the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_\ast</annotation></semantics></math> is canonically identified with the <a class="existingWikiWord" href="/nlab/show/image">image</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\iota_\ast</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>im</mi><mo stretchy="false">(</mo><msub><mi>ι</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ker(f_\ast) \simeq im(\iota_\ast) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_49">Proof</h6> <p>It is clear that the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\iota_\ast</annotation></semantics></math> is in the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_\ast</annotation></semantics></math> (every sphere in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>y</mi></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">F_y\hookrightarrow X</annotation></semantics></math> becomes constant on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, hence contractible, when sent forward to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>).</p> <p>For the converse, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\alpha]\in \pi_{\bullet}(X,x)</annotation></semantics></math> be represented by some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\alpha \colon S^{n-1} \to X</annotation></semantics></math>. Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\alpha]</annotation></semantics></math> is in the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_\ast</annotation></semantics></math>. This means equivalently that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> fits into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mi>κ</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\overset{\kappa}{\longrightarrow}& Y } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> is the contracting homotopy witnessing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_\ast[\alpha] = 0</annotation></semantics></math>.</p> <p>Now since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is a lift of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, there exists a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>κ</mi><mo>⇒</mo><msub><mi>const</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; \kappa \Rightarrow const_y </annotation></semantics></math></div> <p>as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mi>κ</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>y</mi></mover></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ && {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}} \\ && D^n &\overset{\kappa}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{(id,\delta_1)}} && \downarrow^{\mathrlap{id}} \\ D^n &\overset{(id,\delta_0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ \downarrow && && \downarrow \\ \ast && \overset{y}{\longrightarrow} && Y } </annotation></semantics></math></div> <p>(for instance: regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> as embedded in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">0 \in \mathbb{R}^n</annotation></semantics></math> is identified with the basepoint on the boundary of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> and set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><mover><mi>v</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>κ</mi><mo stretchy="false">(</mo><mi>t</mi><mover><mi>v</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta(\vec v,t) \coloneqq \kappa(t \vec v)</annotation></semantics></math>).</p> <p>The <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> of the top two squares that have appeared this way is equivalent to the following commuting square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>n</mi></msub><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\longrightarrow& &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{(id,\delta_1)}}\downarrow && && \downarrow^{\mathrlap{f}} \\ S^{n-1} \times I &\overset{(\iota_n, id)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>Because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a> and by lemma <a class="maruku-ref" href="#CylinderOverCWComplexIsJTopRelativeCellComplex"></a> and prop. <a class="maruku-ref" href="#SerreFibrationHasRightLiftingAgainstJTopRelativeCellComplexes"></a>, this has a <a class="existingWikiWord" href="/nlab/show/lift">lift</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo stretchy="false">˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mi>I</mi><mo>⟶</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde \eta \;\colon\; S^{n-1} \times I \longrightarrow X \,. </annotation></semantics></math></div> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \eta</annotation></semantics></math> is a basepoint preserving <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>=</mo><mover><mi>η</mi><mo stretchy="false">˜</mo></mover><msub><mo stretchy="false">|</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\alpha = \tilde \eta|_1</annotation></semantics></math> to some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>′</mo><mo>≔</mo><mover><mi>η</mi><mo stretchy="false">˜</mo></mover><msub><mo stretchy="false">|</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\alpha' \coloneqq \tilde \eta|_0</annotation></semantics></math>. Being homotopic, they represent the same element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_{n-1}(X,x)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo>′</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\alpha'] = [\alpha] \,. </annotation></semantics></math></div> <p>But the new representative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha'</annotation></semantics></math> has the special property that its image in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is not just trivializable, but trivialized: combining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \eta</annotation></semantics></math> with the previous diagram shows that it sits in the following commuting diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>α</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mover><mi>η</mi><mo stretchy="false">˜</mo></mover></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>n</mi></msub><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>y</mi></mover></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \alpha' \colon & S^{n-1} &\overset{(id,\delta_0)}{\longrightarrow}& S^{n-1}\times I &\overset{\tilde \eta}{\longrightarrow}& X \\ & \downarrow^{\iota_n} && \downarrow^{\mathrlap{(\iota_n,id)}} && \downarrow^{\mathrlap{f}} \\ & D^n &\overset{(id,\delta_0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ & \downarrow && && \downarrow \\ & \ast && \overset{y}{\longrightarrow} && Y } \,. </annotation></semantics></math></div> <p>The commutativity of the outer square says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mi>α</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f_\ast \alpha'</annotation></semantics></math> is constant, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha'</annotation></semantics></math> is entirely contained in the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">F_y</annotation></semantics></math>. Said more abstractly, the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha'</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>y</mi></msub><mover><mo>↪</mo><mi>ι</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">F_y\overset{\iota}{\hookrightarrow} X</annotation></semantics></math>, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo>′</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\alpha'] = [\alpha]</annotation></semantics></math> is in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\iota_\ast</annotation></semantics></math>.</p> </div> <p>The following lemma <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a>, together with lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a> above, are the only two statements of the entire development here that crucially involve the <a class="existingWikiWord" href="/nlab/show/concrete+particular">concrete particular</a> nature of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> (“<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>”), everything beyond that is <a class="existingWikiWord" href="/nlab/show/general+abstract">general abstract</a> homotopy theory.</p> <div class="num_lemma" id="AcyclicSerreFibrationsAreTheJTopFibrations"> <h6 id="lemma_20">Lemma</h6> <p>The continuous functions with the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a>, def. <a class="maruku-ref" href="#RightLiftingProperty"></a> against the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub><mo>=</mo><mo stretchy="false">{</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I_{Top} = \{S^{n-1}\hookrightarrow D^n\}</annotation></semantics></math> of topological <a class="existingWikiWord" href="/nlab/show/generating+cofibrations">generating cofibrations</a>, def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>, are precisely those which are both <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>, def. <a class="maruku-ref" href="#WeakHomotopyEquivalenceOfTopologicalSpaces"></a> as well as <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a>, def. <a class="maruku-ref" href="#SerreFibration"></a>.</p> </div> <div class="proof"> <h6 id="proof_50">Proof</h6> <p>We break this up into three sub-statements:</p> <p><strong>A) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>-injective morphisms are in particular weak homotopy equivalences</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p \colon \hat X \to X</annotation></semantics></math> have the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∃</mo></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\longrightarrow & \hat X \\ {}^{\mathllap{\iota_n}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{p}} \\ D^n &\longrightarrow& X } </annotation></semantics></math></div> <p>We check that the lifts in these diagrams exhibit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(f)</annotation></semantics></math> as being an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on all <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>, def. <a class="maruku-ref" href="#HomotopyGroupsOftopologicalSpaces"></a>:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> the existence of these lifts says that every point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\hat X) \to \pi_0(X)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/surjection">surjective</a>. Let then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mo>=</mo><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo><mo>⟶</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">S^0 = \ast \coprod \ast \longrightarrow \hat X</annotation></semantics></math> be a map that hits two connected components, then the existence of the lift says that if they have the same image in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(X)</annotation></semantics></math> then they were already the same connected component in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat X</annotation></semantics></math>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\hat X)\to \pi_0(X)</annotation></semantics></math> is also <a class="existingWikiWord" href="/nlab/show/injection">injective</a> and hence is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>.</p> <p>Similarly, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">S^n \to \hat X</annotation></semantics></math> represents an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(\hat X)</annotation></semantics></math> that becomes trivial in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X)</annotation></semantics></math>, then the existence of the lift says that it already represented the trivial element itself. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(\hat X) \to \pi_n(X)</annotation></semantics></math> has trivial <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> and so is injective.</p> <p>Finally, to see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(\hat X) \to \pi_n(X)</annotation></semantics></math> is also surjective, hence bijective, observe that every elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X)</annotation></semantics></math> is equivalently represented by a commuting diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\longrightarrow& \ast &\longrightarrow& \hat X \\ \downarrow && \downarrow && \downarrow \\ D^n &\longrightarrow& X &=& X } </annotation></semantics></math></div> <p>and so here the lift gives a representative of a preimage in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_{n}(\hat X)</annotation></semantics></math>.</p> <p><strong>B) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>-injective morphisms are in particular Serre fibrations</strong></p> <p>By an immediate closure property of lifting problems (we spell this out in generality as prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a>, cor. <a class="maruku-ref" href="#SaturationOfGeneratingCofibrations"></a> below) an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphism">injective morphism</a> has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against all <a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a>, and hence, by lemma <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrationsAreRelativeCellComplexes"></a>, it is also a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-injective morphism, hence a Serre fibration.</p> <p><strong>C) Acyclic Serre fibrations are in particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphisms">injective morphisms</a></strong></p> <p>(<a href="#Hirschhorn15">Hirschhorn 15, section 6</a>).</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math> be a Serre fibration that induces isomorphisms on homotopy groups. In degree 0 this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is an isomorphism on <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a>, and this means that there is a lift in every <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>∅</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mn>0</mn></msup><mo>=</mo><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^{-1} = \emptyset &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^0 = \ast &\longrightarrow& Y } </annotation></semantics></math></div> <p>(this is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(f)</annotation></semantics></math> being surjective) and in every commuting square of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>0</mn></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mn>1</mn></msup><mo>=</mo><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^0 &\longrightarrow& X \\ {}^{\mathllap{\iota_0}}\downarrow && \downarrow^{\mathrlap{f}} \\ D^1 = \ast &\longrightarrow& Y } </annotation></semantics></math></div> <p>(this is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(f)</annotation></semantics></math> being injective). Hence we are reduced to showing that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math> every diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mi>κ</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\overset{\kappa}{\longrightarrow}& Y } </annotation></semantics></math></div> <p>has a lift.</p> <p>To that end, pick a basepoint on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n-1}</annotation></semantics></math> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> for its images in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, respectively</p> <p>Then the diagram above expresses that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mo>∈</mo><msub><mi>π</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_\ast[\alpha] = 0 \in \pi_{n-1}(Y,y)</annotation></semantics></math> and hence by assumption on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mo>∈</mo><msub><mi>π</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\alpha] = 0 \in \pi_{n-1}(X,x)</annotation></semantics></math>, which in turn mean that there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\kappa'</annotation></semantics></math> making the upper triangle of our lifting problem commute:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mi>κ</mi><mo>′</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow & \nearrow_{\mathrlap{\kappa'}} \\ D^n } \,. </annotation></semantics></math></div> <p>It is now sufficient to show that any such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\kappa'</annotation></semantics></math> may be deformed to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\rho'</annotation></semantics></math> which keeps making this upper triangle commute but also makes the remaining lower triangle commute.</p> <p>To that end, notice that by the commutativity of the original square, we already have at least this commuting square:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><munder><mo>⟶</mo><mi>κ</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f \circ \kappa'}} \\ D^n &\underset{\kappa}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>This induces the universal map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>κ</mi><mo>,</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\kappa,f \circ \kappa')</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of its <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a> in the top left, which is the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> (see <a href="Top#TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself">this</a> example):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><munder><mo>⟶</mo><mi>κ</mi></munder></mtd> <mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mrow><mo stretchy="false">(</mo><mi>κ</mi><mo>,</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow^{\mathrlap{f \circ \kappa'}} \\ D^n &\underset{\kappa}{\longrightarrow}& S^n \\ && & \searrow^{(\kappa,f \circ \kappa')} \\ && && Y } \,. </annotation></semantics></math></div> <p>This universal morphism represents an element of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th homotopy group:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>κ</mi><mo>,</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [(\kappa,f \circ \kappa')] \in \pi_n(Y,y) \,. </annotation></semantics></math></div> <p>By assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a weak homotopy equivalence, there is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ρ</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\rho] \in \pi_{n}(X,x)</annotation></semantics></math> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">[</mo><mi>ρ</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>κ</mi><mo>,</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> f_\ast [\rho] = [(\kappa,f \circ \kappa')] </annotation></semantics></math></div> <p>hence on representatives there is a lift up to homotopy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ρ</mi></mpadded></msup><msub><mo>↗</mo> <mpadded width="0"><mo>⇓</mo></mpadded></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>κ</mi><mo>,</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X \\ &{}^{\mathllap{\rho}}\nearrow_{\mathrlap{\Downarrow}} & \downarrow^{\mathrlap{f}} \\ S^n &\underset{(\kappa,f\circ \kappa')}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>Morever, we may always find <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ρ</mi><mo>′</mo><mo>,</mo><mi>κ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rho', \kappa')</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\rho' \colon D^n \to X</annotation></semantics></math>. (“Paste <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\kappa'</annotation></semantics></math> to the reverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>.”)</p> <p>Consider then the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>ρ</mi><mo>′</mo><mo>,</mo><mi>κ</mi><mo stretchy="false">)</mo></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex"> S^n \overset{(f\circ \rho', \kappa)}{\longrightarrow} Y </annotation></semantics></math></div> <p>and observe that this represents the trivial class:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>ρ</mi><mo>′</mo><mo>,</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>ρ</mi><mo>′</mo><mo>,</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo>,</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>f</mi> <mo>*</mo></msub><munder><munder><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>ρ</mi><mo>′</mo><mo>,</mo><mi>κ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mo stretchy="false">[</mo><mi>ρ</mi><mo stretchy="false">]</mo></mrow></munder><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo>,</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>κ</mi><mo>,</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>κ</mi><mo>′</mo><mo>,</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} [(f \circ \rho', \kappa)] & = [(f\circ \rho', f\circ \kappa')] + [(f\circ \kappa', \kappa)] \\ & = f_\ast \underset{= [\rho]}{\underbrace{[(\rho',\kappa')]}} + [(f\circ \kappa', \kappa)] \\ & = [(\kappa,f \circ \kappa')] + [(f\circ \kappa', \kappa)] \\ & = 0 \end{aligned} \,. </annotation></semantics></math></div> <p>This means equivalently that there is a homotopy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo>∘</mo><mi>ρ</mi><mo>′</mo><mo>⇒</mo><mi>κ</mi></mrow><annotation encoding="application/x-tex"> \phi \; \colon \; f\circ \rho' \Rightarrow \kappa </annotation></semantics></math></div> <p>fixing the boundary of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disk.</p> <p>Hence if we denote homotopy by double arrows, then we have now achieved the following situation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mi>ρ</mi><mo>′</mo></mrow></msup><msub><mo>↗</mo> <mrow><msup><mo>⇓</mo> <mi>ϕ</mi></msup></mrow></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow & {}^{\rho'}\nearrow_{\Downarrow^{\phi}} & \downarrow^{\mathrlap{f}} \\ D^n &\longrightarrow& Y } </annotation></semantics></math></div> <p>and it now suffices to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> may be lifted to a homotopy of just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\rho'</annotation></semantics></math>, fixing the boundary, for then the resulting homotopic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">\rho''</annotation></semantics></math> is the desired lift.</p> <p>To that end, notice that the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\phi \colon D^n \times I \to Y</annotation></semantics></math> fixes the boundary of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disk means equivalently that it extends to a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><munder><mo>⊔</mo><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>α</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex"> S^{n-1} \underset{S^{n-1}\times I}{\sqcup} D^n \times I \overset{(f\circ \alpha,\phi)}{\longrightarrow} Y </annotation></semantics></math></div> <p>out of the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> that identifies in the cylinder over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> all points lying over the boundary. Hence we are reduced to finding a lift in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>ρ</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><munder><mo>⊔</mo><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>α</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ D^n &\overset{\rho'}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ S^{n-1}\underset{S^{n-1}\times I}{\sqcup} D^n \times I &\overset{(f\circ \alpha,\phi)}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>But inspection of the left map reveals that it is homeomorphic again to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">D^n \to D^n \times I</annotation></semantics></math>, and hence the lift does indeed exist.</p> </div> <h3 id="the_classical_model_structure_on_topological_spaces">The classical model structure on topological spaces</h3> <div class="num_defn" id="ClassesOfMorhismsInTopQuillen"> <h6 id="definition_44">Definition</h6> <p>Say that a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, hence a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, is</p> <ul> <li> <p>a <strong>classical weak equivalence</strong> if it is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, def. <a class="maruku-ref" href="#WeakHomotopyEquivalenceOfTopologicalSpaces"></a>;</p> </li> <li> <p>a <strong>classical fibration</strong> if it is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a>;</p> </li> <li> <p>a <strong>classical cofibration</strong> if it is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> (rem. <a class="maruku-ref" href="#RetractsOfMorphisms"></a>) of a <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a>, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>.</p> </li> </ul> <p>and hence</p> <ul> <li> <p>a <strong>acyclic classical cofibration</strong> if it is a classical cofibration as well as a classical weak equivalence;</p> </li> <li> <p>a <strong>acyclic classical fibration</strong> if it is a classical fibration as well as a classical weak equivalence.</p> </li> </ul> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>cl</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>Fib</mi> <mi>cl</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>Cof</mi> <mi>cl</mi></msub><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> W_{cl},\;Fib_{cl},\;Cof_{cl} \subset Mor(Top) </annotation></semantics></math></div> <p>for the classes of these morphisms, respectively.</p> </div> <p>We first prove now that the classes of morphisms in def. <a class="maruku-ref" href="#ClassesOfMorhismsInTopQuillen"></a> satisfy the conditions for a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure, def. <a class="maruku-ref" href="#ModelCategory"></a> (after some lemmas, this is theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a> below). Then we discuss the resulting <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a> (<a href="#TheClassicalHomotopyCategory">below</a>) and then a few variant model structures whose proof follows immediately along the line of the proof of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math>:</p> <ul> <li> <p><a href="#ModelstructureOnPointedTopologicalSpaces">The model structure on pointed topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>Quillen</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top^{\ast/}_{Quillen}</annotation></semantics></math>;</p> </li> <li> <p><a href="#ModelStructureOnCompactlyGeneratedTopologicalSpaces">The model structure on compactly generated topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top^{\ast/}_{cg})_{Quillen}</annotation></semantics></math>;</p> </li> <li> <p><a href="#ModelStructureOnTopEnrichedFunctors">The model structure on topologically enriched functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[\mathcal{C}, (Top_{cg})_{Quillen}]_{proj}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mo>*</mo></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[\mathcal{C},(Top^{\ast}_{cg})_{Quillen}]_{proj}</annotation></semantics></math>.</p> </li> </ul> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_prop" id="QuillenWeakEquivalencesSatisfyTwoOutOfThree"> <h6 id="proposition_33">Proposition</h6> <p>The classical weak equivalences, def. <a class="maruku-ref" href="#ClassesOfMorhismsInTopQuillen"></a>, satify <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>).</p> </div> <div class="proof"> <h6 id="proof_51">Proof</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> (of <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>) satisfy 2-out-of-3, this property is directly inherited via the very definition of <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, def. <a class="maruku-ref" href="#WeakHomotopyEquivalenceOfTopologicalSpaces"></a>.</p> </div> <div class="num_lemma" id="FactorizationInTopQuillen"> <h6 id="lemma_21">Lemma</h6> <p>Every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \longrightarrow Y</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> factors as a classical cofibration followed by an acyclic classical fibration, def. <a class="maruku-ref" href="#ClassesOfMorhismsInTopQuillen"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>cl</mi></msub></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \stackrel{\in Cof_{cl}}{\longrightarrow} \hat X \stackrel{\in W_{cl} \cap Fib_{cl}}{\longrightarrow} Y \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_52">Proof</h6> <p>By lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a> the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub><mo>=</mo><mo stretchy="false">{</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I_{Top} = \{S^{n-1}\hookrightarrow D^n\}</annotation></semantics></math> of topological <a class="existingWikiWord" href="/nlab/show/generating+cofibrations">generating cofibrations</a>, def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>, has small domains, in the sense of def. <a class="maruku-ref" href="#ClassOfMorphismsWithSmallDomains"></a> (the <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> are <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a>). Hence by the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>, prop. <a class="maruku-ref" href="#SmallObjectArgument"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> factors as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a>, def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a>, hence just a plain relative cell complex, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>, followed by an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphisms">injective morphisms</a>, def. <a class="maruku-ref" href="#RightLiftingProperty"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>I</mi> <mi>Top</mi></msub><mi>Inj</mi></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \stackrel{\in Cof_{cl}}{\longrightarrow} \hat X \stackrel{\in I_{Top} Inj}{\longrightarrow} Y \,. </annotation></semantics></math></div> <p>By lemma <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a> the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\hat X \to Y</annotation></semantics></math> is both a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> as well as a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>.</p> </div> <div class="num_lemma" id="ContinuousFunctionsFactorAsQuillenAcyclicCofibrationFollowedBySerreFibration"> <h6 id="lemma_22">Lemma</h6> <p>Every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \longrightarrow Y</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> factors as an acyclic classical cofibration followed by a fibration, def. <a class="maruku-ref" href="#ClassesOfMorhismsInTopQuillen"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>Fib</mi> <mi>cl</mi></msub></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \stackrel{\in W_{cl} \cap Cof_{cl}}{\longrightarrow} \hat X \stackrel{\in Fib_{cl}}{\longrightarrow} Y \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_53">Proof</h6> <p>By lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a> the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub><mo>=</mo><mo stretchy="false">{</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">J_{Top} = \{D^n \hookrightarrow D^n\times I\}</annotation></semantics></math> of topological <a class="existingWikiWord" href="/nlab/show/generating+acyclic+cofibrations">generating acyclic cofibrations</a>, def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>, has small domains, in the sense of def. <a class="maruku-ref" href="#ClassOfMorphismsWithSmallDomains"></a> (the <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a> are <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a>). Hence by the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>, prop. <a class="maruku-ref" href="#SmallObjectArgument"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> factors as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a>, def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a>, followed by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>top</mi></msub></mrow><annotation encoding="application/x-tex">J_{top}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphisms">injective morphisms</a>, def. <a class="maruku-ref" href="#RightLiftingProperty"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>J</mi> <mi>Top</mi></msub><mi>Cell</mi></mrow></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>J</mi> <mi>Top</mi></msub><mi>Inj</mi></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \stackrel{\in J_{Top} Cell}{\longrightarrow} \hat X \stackrel{\in J_{Top} Inj}{\longrightarrow} Y \,. </annotation></semantics></math></div> <p>By definition this makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\hat X \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, hence a fibration.</p> <p>By lemma <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrationsAreRelativeCellComplexes"></a> a relative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-cell complex is in particular a relative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>-cell complex. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">X \to \hat X</annotation></semantics></math> is a classical cofibration. By lemma <a class="maruku-ref" href="#JTopRelativeCellComplexesAreWeakHomotopyEquivalences"></a> it is also a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, hence a clasical weak equivalence.</p> </div> <div class="num_lemma" id="LiftingPropertyInTheClassicalModelStructureOnTopologicalSpaces"> <h6 id="lemma_23">Lemma</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> with the left morphism a classical cofibration and the right morphism a fibration, def. <a class="maruku-ref" href="#ClassesOfMorhismsInTopQuillen"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mfrac linethickness="0"><mrow><mrow><mi>g</mi><mo>∈</mo></mrow></mrow><mrow><mrow><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow></mrow></mfrac></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mfrac linethickness="0"><mrow><mrow><mi>f</mi><mo>∈</mo></mrow></mrow><mrow><msub><mi>Fib</mi> <mi>cl</mi></msub></mrow></mfrac></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ &\longrightarrow& \\ {}^{\mathllap{{g \in} \atop {Cof_{cl}}}}\downarrow && \downarrow^{\mathrlap{{f \in }\atop Fib_{cl}}} \\ &\longrightarrow& } </annotation></semantics></math></div> <p>admits a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> as soon as one of the two is also a classical weak equivalence.</p> </div> <div class="proof"> <h6 id="proof_54">Proof</h6> <p><strong>A)</strong> If the fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is also a weak equivalence, then lemma <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a> says that it has the right lifting property against the generating cofibrations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>, and cor. <a class="maruku-ref" href="#SaturationOfGeneratingCofibrations"></a> implies the claim.</p> <p><strong>B)</strong> If the cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> on the left is also a weak equivalence, consider any factorization into a relative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-cell complex, def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>, def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a>, followed by a fibration,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>J</mi> <mi>Top</mi></msub><mi>Cell</mi></mrow></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>Fib</mi> <mi>cl</mi></msub></mrow></mover><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> g \;\colon\; \stackrel{\in J_{Top} Cell}{\longrightarrow} \stackrel{\in Fib_{cl}}{\longrightarrow} \,, </annotation></semantics></math></div> <p>as in the proof of lemma <a class="maruku-ref" href="#ContinuousFunctionsFactorAsQuillenAcyclicCofibrationFollowedBySerreFibration"></a>. By lemma <a class="maruku-ref" href="#JTopRelativeCellComplexesAreWeakHomotopyEquivalences"></a> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>J</mi> <mi>Top</mi></msub><mi>Cell</mi></mrow></mover></mrow><annotation encoding="application/x-tex">\overset{\in J_{Top} Cell}{\longrightarrow}</annotation></semantics></math> is a weak homotopy equivalence, and so by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> (prop. <a class="maruku-ref" href="#QuillenWeakEquivalencesSatisfyTwoOutOfThree"></a>) the factorizing fibration is actually an acyclic fibration. By case A), this acyclic fibration has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against the cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> itself, and so the <a class="existingWikiWord" href="/nlab/show/retract+argument">retract argument</a>, lemma <a class="maruku-ref" href="#RetractArgument"></a> gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a relative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-cell complex. With this, finally cor. <a class="maruku-ref" href="#SaturationOfGeneratingCofibrations"></a> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>.</p> </div> <p>Finally:</p> <div class="num_prop" id="LiftingExhausted"> <h6 id="proposition_34">Proposition</h6> <p>The systems <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Cof</mi> <mi>cl</mi></msub><mo>,</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>cl</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Cof_{cl} , W_{cl} \cap Fib_{cl})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Cof</mi> <mi>cl</mi></msub><mo>,</mo><msub><mi>Fib</mi> <mi>cl</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(W_{cl} \cap Cof_{cl}, Fib_{cl})</annotation></semantics></math> from def. <a class="maruku-ref" href="#ClassesOfMorhismsInTopQuillen"></a> are <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a>.</p> </div> <div class="proof"> <h6 id="proof_55">Proof</h6> <p>Since we have already seen the factorization property (lemma <a class="maruku-ref" href="#FactorizationInTopQuillen"></a>, lemma <a class="maruku-ref" href="#ContinuousFunctionsFactorAsQuillenAcyclicCofibrationFollowedBySerreFibration"></a>) and the lifting properties (lemma <a class="maruku-ref" href="#LiftingPropertyInTheClassicalModelStructureOnTopologicalSpaces"></a>), it only remains to see that the given left/right classes exhaust the class of morphisms with the given lifting property.</p> <p>For the classical fibrations this is by definition, for the the classical acyclic fibrations this is by lemma <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a>.</p> <p>The remaining statement for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow><annotation encoding="application/x-tex">Cof_{cl}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow><annotation encoding="application/x-tex">W_{cl}\cap Cof_{cl}</annotation></semantics></math> follows from a general argument (<a href="cofibrantly+generated+model+category#RetractsOfCellComplexesExchaustLLPOfRLP">here</a>) for <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+categories">cofibrantly generated model categories</a> (def. <a class="maruku-ref" href="#CofibrantlyGeneratedModelCategory"></a>), which we spell out:</p> <p>So let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> be in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>Top</mi></msub><mi>Inj</mi><mo stretchy="false">)</mo><mi>Proj</mi></mrow><annotation encoding="application/x-tex">(I_{Top} Inj) Proj</annotation></semantics></math>, we need to show that then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a retract (remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>) of a <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a>. To that end, apply the <a class="existingWikiWord" href="/nlab/show/small+object">small object</a> argument as in lemma <a class="maruku-ref" href="#FactorizationInTopQuillen"></a> to factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>I</mi> <mi>Top</mi></msub><mi>Cell</mi></mrow></mover><mover><mi>Y</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>I</mi> <mi>Top</mi></msub><mi>Inj</mi></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon \; X \overset{I_{Top} Cell}{\longrightarrow} \hat Y \overset{\in I_{Top} Inj}{\longrightarrow} Y \,. </annotation></semantics></math></div> <p>It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Y</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\hat Y \to Y</annotation></semantics></math>, and hence by the <a class="existingWikiWord" href="/nlab/show/retract+argument">retract argument</a> (lemma <a class="maruku-ref" href="#RetractArgument"></a>) it is a retract of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mrow><mi>I</mi><mi>Cell</mi></mrow></mover><mover><mi>Y</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">X \overset{I Cell}{\to} \hat Y</annotation></semantics></math>. This proves the claim for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow><annotation encoding="application/x-tex">Cof_{cl}</annotation></semantics></math>.</p> <p>The analogous argument for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow><annotation encoding="application/x-tex">W_{cl} \cap Cof_{cl}</annotation></semantics></math>, using the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>, shows that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mo stretchy="false">(</mo><msub><mi>J</mi> <mi>Top</mi></msub><mi>Inj</mi><mo stretchy="false">)</mo><mi>Proj</mi></mrow><annotation encoding="application/x-tex">f \in (J_{Top} Inj) Proj</annotation></semantics></math> is a retract of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-cell complex. By lemma <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrationsAreRelativeCellComplexes"></a> and lemma <a class="maruku-ref" href="#JTopRelativeCellComplexesAreWeakHomotopyEquivalences"></a> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math>-cell complex is both an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math>-cell complex and a weak homotopy equivalence. Retracts of the former are cofibrations by definition, and retracts of the latter are still weak homotopy equivalences by lemma <a class="maruku-ref" href="#RetractPreservesIsomorphism"></a>. Hence such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is an acyclic cofibration.</p> </div> <p>In conclusion, prop. <a class="maruku-ref" href="#QuillenWeakEquivalencesSatisfyTwoOutOfThree"></a> and prop. <a class="maruku-ref" href="#LiftingExhausted"></a> say that:</p> <div class="num_theorem" id="TopQuillenModelStructure"> <h6 id="theorem_2">Theorem</h6> <p>The classes of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mor</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mor(Top)</annotation></semantics></math> of def. <a class="maruku-ref" href="#ClassesOfMorhismsInTopQuillen"></a>,</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>cl</mi></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">W_{cl} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Fib</mi> <mi>cl</mi></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">Fib_{cl} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cof</mi> <mi>cl</mi></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">Cof_{cl} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> of <a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a></p> </li> </ul> <p>define a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure (def. <a class="maruku-ref" href="#ModelCategory"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math>, the <strong><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></strong> or <strong>Serre-Quillen model structure</strong> .</p> </div> <p>In particular</p> <ol> <li> <p>every object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> is fibrant;</p> </li> <li> <p>the cofibrant objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> of <a class="existingWikiWord" href="/nlab/show/cell+complexes">cell complexes</a>.</p> </li> </ol> <p>Hence in particular the following classical statement is an immediate corollary:</p> <div class="num_cor" id="WhiteheadTheorem"> <h6 id="corollary_4">Corollary</h6> <p><strong>(Whitehead theorem)</strong></p> <p>Every <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> (def. <a class="maruku-ref" href="#WeakHomotopyEquivalenceOfTopologicalSpaces"></a>) between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> that are <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, in particular to a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> (def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>), is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> (def. <a class="maruku-ref" href="#HomotopyEquivalence"></a>).</p> </div> <div class="proof"> <h6 id="proof_56">Proof</h6> <p>This is the “Whitehead theorem in model categories”, lemma <a class="maruku-ref" href="#WhiteheadTheoremInModelCategories"></a>, specialized to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> via theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>.</p> </div> <p>In proving theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a> we have in fact shown a bit more that stated. Looking back, all the structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> is entirely induced by the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>) of generating cofibrations and the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>) of generating acyclic cofibrations (whence the terminology). This situation is usefully summarized by the concept of <em><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></em> (Def. <a class="maruku-ref" href="#CofibrantlyGeneratedModelCategory"></a>).</p> <p>This phenomenon will keep recurring and will keep being useful as we construct further model categories, such as the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+pointed+topological+spaces">classical model structure on pointed topological spaces</a> (def. <a class="maruku-ref" href="#ClassicalModelStructureOnPointedTopologicalSpaces"></a>), the <a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+enriched+functors">projective model structure on topological functors</a> (thm. <a class="maruku-ref" href="#ProjectiveModelStructureOnTopologicalFunctors"></a>), and finally various <a class="existingWikiWord" href="/nlab/show/model+structures+on+spectra">model structures on spectra</a> which we turn to in the <a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+1">section on stable homotopy theory</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="TheClassicalHomotopyCategory">The classical homotopy category</h3> <p>With the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> in hand, we now have good control over the <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a>:</p> <div class="num_defn" id="ClassicalHomotopyCategory"> <h6 id="definition_45">Definition</h6> <p>The <strong>Serre-Quillen <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a></strong> is the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a>, def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>, of the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> from theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>: we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(Top) \coloneqq Ho(Top_{Quillen}) \,. </annotation></semantics></math></div></div> <div class="num_remark" id="EveryTopologicalSpaceWeaklyEquivalentToACWComplex"> <h6 id="remark_23">Remark</h6> <p>From just theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>, the definition <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a> (def. <a class="maruku-ref" href="#ClassicalHomotopyCategory"></a>) gives that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mrow><mi>Retract</mi><mo stretchy="false">(</mo><mi>Cell</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex"> Ho(Top_{Quillen}) \simeq (Top_{Retract(Cell)})/_\sim </annotation></semantics></math></div> <p>is the category whose objects are <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> of <a class="existingWikiWord" href="/nlab/show/cell+complexes">cell complexes</a> (def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>) and whose morphisms are <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a>. But in fact more is true:</p> <p>Theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a> in itself implies that every topological space is weakly equivalent to a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>. But by the existence of <a class="existingWikiWord" href="/nlab/show/CW+approximations">CW approximations</a>, this cell complex may even be taken to be a <a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a>.</p> <p>(Better yet, there is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> to the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a> which implies a <em>functorial</em> <a class="existingWikiWord" href="/nlab/show/CW+approximation">CW approximation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">|</mo></mrow><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">{\vert Sing X\vert} \overset{\in W_{cl}}{\longrightarrow} X</annotation></semantics></math> given by forming the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.)</p> <p>Hence the Serre-Quillen <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a> is also equivalently the category of just the <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> whith <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between them</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mrow><mi>Retract</mi><mo stretchy="false">(</mo><mi>Cell</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>CW</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} Ho(Top_{Quillen}) & \simeq (Top_{Retract(Cell)})/_\sim \\ & \simeq (Top_{CW})/_{\sim} \end{aligned} \,. </annotation></semantics></math></div> <p>It follows that the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the homotopy category (theorem <a class="maruku-ref" href="#UniversalPropertyOfHomotopyCategoryOfAModelCategory"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>Top</mi><mo stretchy="false">[</mo><msubsup><mi>W</mi> <mi>cl</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Ho(Top_{Quillen}) \simeq Top[W_{cl}^{-1}] </annotation></semantics></math></div> <p>implies that there is a bijection, up to <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>, between</p> <ol> <li> <p>functors out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>CW</mi></msub></mrow><annotation encoding="application/x-tex">Top_{CW}</annotation></semantics></math> which agree on homotopy-equivalent maps;</p> </li> <li> <p>functors out of all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> which send weak homotopy equivalences to isomorphisms.</p> </li> </ol> <p>This statement in particular serves to show that two different axiomatizations of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theories are equivalent to each other. See at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+S">Introduction to Stable homotopy theory – S</a></em> the section <em><a href="Introduction+to+Stable+homotopy+theory+--+S#GeneralizedHomologyAndCohomologyFunctors">generalized cohomology functors</a></em> (<a href="Introduction+to+Stable+homotopy+theory+--+S#HomotopyTheoreticVersionOfCohomologyFunctorDefIsEquivalent">this prop.</a>)</p> <p><strong>Beware</strong> that, by remark <a class="maruku-ref" href="#NotEveryHomotopyEquivalenceIsAWeakHomotopyEquivalence"></a>, what is <strong>not</strong> equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top_{Quillen})</annotation></semantics></math> is the category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hTop</mi><mo>≔</mo><mi>Top</mi><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex"> hTop \coloneqq Top/_\sim </annotation></semantics></math></div> <p>obtained from <em>all</em> topological spaces with morphisms the homotopy classes of continuous functions. This category is “too large”, the correct homotopy category is just the genuine <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mrow><mi>Retract</mi><mo stretchy="false">(</mo><mi>Cell</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mo>≃</mo><mi>Top</mi><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mo>=</mo><mo>↪</mo><mi>hTop</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(Top_{Quillen}) \simeq (Top_{Retract(Cell)})/_\sim \simeq Top/_\sim = \hookrightarrow hTop \,. </annotation></semantics></math></div> <p>Beware also the ambiguity of terminology: “classical homotopy category” some literature refers to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hTop</mi></mrow><annotation encoding="application/x-tex">hTop</annotation></semantics></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top_{Quillen})</annotation></semantics></math>. However, here we never have any use for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hTop</mi></mrow><annotation encoding="application/x-tex">hTop</annotation></semantics></math> and will not mention it again.</p> </div> <div class="num_prop" id="TopologicalCylinderOnCWComplexIsGoodCylinderObject"> <h6 id="proposition_35">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>. Then the standard topological cylinder of def. <a class="maruku-ref" href="#TopologicalInterval"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>I</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} X\times I \longrightarrow X </annotation></semantics></math></div> <p>(obtained by forming the <a class="existingWikiWord" href="/nlab/show/product+space">product space</a> with the standard <a class="existingWikiWord" href="/nlab/show/topological+interval">topological interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I = [0,1]</annotation></semantics></math>) is indeed a <em><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></em> in the abstract sense of def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>.</p> </div> <div class="proof"> <h6 id="proof_57">Proof</h6> <p>We describe the proof informally. It is immediate how to turn this into a formal proof, but the notation becomes tedious. (One place where it is spelled out completely is <a href="cylinder+object#Ottina14">Ottina 14, prop. 2.9</a>.)</p> <p>So let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>→</mo><mi>⋯</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_0 \to X_1 \to X_2\to \cdots \to X</annotation></semantics></math> be a presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as a CW-complex. Proceed by induction on the cell dimension.</p> <p>First observe that the cylinder <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X_0 \times I</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> is a cell complex: First <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> itself is a disjoint union of points. Adding a second copy for every point (i.e. <a class="existingWikiWord" href="/nlab/show/attaching+space">attaching</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>D</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">S^{-1}\to D^0</annotation></semantics></math>) yields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>⊔</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0 \sqcup X_0</annotation></semantics></math>, then attaching an inteval between any two corresponding points (along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mo>→</mo><msup><mi>D</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^0 \to D^1</annotation></semantics></math>) yields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X_0 \times I</annotation></semantics></math>.</p> <p>So assume that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> it has been shown that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X_n \times I</annotation></semantics></math> has the structure of a CW-complex of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>. Then for each cell of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n+1}</annotation></semantics></math>, attach it <em>twice</em> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X_n \times I</annotation></semantics></math>, once at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X_n \times \{0\}</annotation></semantics></math>, and once at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>×</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X_n \times \{1\}</annotation></semantics></math>.</p> <p>The result is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n+1}</annotation></semantics></math> with a <em>hollow cylinder</em> erected over each of its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-cells. Now fill these hollow cylinders (along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n+1} \to D^{n+1}</annotation></semantics></math>) to obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X_{n+1}\times I</annotation></semantics></math>.</p> <p>This completes the induction, hence the proof of the CW-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X\times I</annotation></semantics></math>.</p> <p>The construction also manifestly exhibits the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover></mrow><annotation encoding="application/x-tex">X \sqcup X \overset{(i_0,i_1)}{\longrightarrow}</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a>.</p> <p>Finally, it is clear (prop. <a class="maruku-ref" href="#TopologicalHomotopyEquivalencesAreWeakHomotopyEquivalences"></a>) that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times I \to X</annotation></semantics></math> is a weak homotopy equivalence.</p> </div> <p>Conversely:</p> <div class="num_prop" id="TopologicalPathSpaceIsGoodPathSpaceObject"> <h6 id="proposition_36">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. Then the standard topological <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> (def. <a class="maruku-ref" href="#TopologicalPathSpace"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><msup><mi>X</mi> <mi>I</mi></msup><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow></msup><mo>,</mo><msup><mi>X</mi> <mrow><msub><mi>δ</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \longrightarrow X^I \overset{(X^{\delta_0}, X^{\delta_1})}{\longrightarrow} X \times X </annotation></semantics></math></div> <p>(obtained by forming the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a>, def. <a class="maruku-ref" href="#CompactOpenTopology"></a>, with the standard <a class="existingWikiWord" href="/nlab/show/topological+interval">topological interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I = [0,1]</annotation></semantics></math>) is indeed a <em><a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></em> in the abstract sense of def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>.</p> </div> <div class="proof"> <h6 id="proof_58">Proof</h6> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msup><mi>X</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">const \colon X\to X^I</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> it is sufficient, by prop. <a class="maruku-ref" href="#TopologicalHomotopyEquivalencesAreWeakHomotopyEquivalences"></a>, to exhibit a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>. Let the homotopy inverse be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow></msup><mo lspace="verythinmathspace">:</mo><msup><mi>X</mi> <mi>I</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X^{\delta_0} \colon X^I \to X</annotation></semantics></math>. Then the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>const</mi></mover><msup><mi>X</mi> <mi>I</mi></msup><mover><mo>⟶</mo><mrow><msup><mi>X</mi> <mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow></msup></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \overset{const}{\longrightarrow} X^I \overset{X^{\delta_0}}{\longrightarrow} X </annotation></semantics></math></div> <p>is already equal to the identity. The other we round, the rescaling of paths provides the required homotopy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>I</mi><mo>×</mo><msup><mi>X</mi> <mi>I</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>γ</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mi>X</mi> <mi>I</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ I \times X^I &\overset{(t,\gamma)\mapsto \gamma(t\cdot(-))}{\longrightarrow}& X^I } \,. </annotation></semantics></math></div> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>I</mi></msup><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X^I \to X\times X</annotation></semantics></math> is a fibration, we need to show that every commuting square of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>X</mi> <mi>I</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ D^n &\longrightarrow& X^I \\ {}^{\mathllap{i_0}}\downarrow && \downarrow^{} \\ D^n \times I &\longrightarrow& X \times X } </annotation></semantics></math></div> <p>has a lift.</p> <p>Now first use the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>I</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊣</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">(I \times (-))\dashv (-)^I</annotation></semantics></math> from prop. <a class="maruku-ref" href="#MappingTopologicalSpaceIsExponentialObject"></a> to rewrite this equivalently as the following commuting square:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>⊔</mo><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊔</mo><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ D^n \sqcup D^n &\overset{(i_0, i_0)}{\longrightarrow}& (D^n \times I) \sqcup (D^n \times I) \\ {}^{\mathllap{(i_0, i_1)}}\downarrow && \downarrow \\ D^n \times I &\longrightarrow& X } \,. </annotation></semantics></math></div> <p>This square is equivalently (example <a class="maruku-ref" href="#PushoutInTop"></a>) a morphism out of the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><munder><mo>⊔</mo><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>⊔</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow></munder><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊔</mo><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⟶</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> D^n \times I \underset{D^n \sqcup D^n }{\sqcup} \left((D^n \times I) \sqcup (D^n \times I)\right) \longrightarrow X \,. </annotation></semantics></math></div> <p>By the same reasoning, a lift in the original diagram is now equivalently a lifting in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><munder><mo>⊔</mo><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>⊔</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow></munder><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊔</mo><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ D^n \times I \underset{D^n \sqcup D^n }{\sqcup} \left((D^n \times I) \sqcup (D^n \times I)\right) &\longrightarrow& X \\ \downarrow && \downarrow \\ (D^n \times I)\times I &\longrightarrow& \ast } \,. </annotation></semantics></math></div> <p>Inspection of the component maps shows that the left vertical morphism here is the inclusion into the square times <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> of three of its faces times <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math>. This is homeomorphic to the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>D</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">D^{n+1} \to D^{n+1} \times I</annotation></semantics></math> (as in remark <a class="maruku-ref" href="#SerreFibrationsByLiftingAgainstMapsHomeomorphicToDiskInclusions"></a>). Therefore a lift in this square exsists, and hence a lift in the original square exists.</p> </div> <h3 id="ModelstructureOnPointedTopologicalSpaces">Model structure on pointed spaces</h3> <p>A <em><a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> is of course an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/point">point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \colon \ast \to X</annotation></semantics></math>, and a morphism of pointed objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x) \longrightarrow (Y,y)</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \longrightarrow Y</annotation></semantics></math> that takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>. Trivial as this is in itself, it is good to record some basic facts, which we do here.</p> <p>Passing to pointed objects is also the first step in linearizing classical homotopy theory to <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>. In particular, every category of pointed objects has a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a>, hence has <a class="existingWikiWord" href="/nlab/show/zero+morphisms">zero morphisms</a>. And crucially, if the original category had <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a>, then its pointed objects canonically inherit a non-cartesian <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>: the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a>. These ingredients will be key below in the <a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+1">section on stable homotopy theory</a>.</p> <div class="num_defn" id="SliceCategory"> <h6 id="definition_46">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/object">object</a>.</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/X}</annotation></semantics></math> is the category whose</p> <ul> <li> <p>objects are morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{A \\ \downarrow \\ X}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>morphisms are <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting triangles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow></mrow></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ A && \longrightarrow && B \\ & {}_{}\searrow && \swarrow \\ && X}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> </ul> <p>Dually, the <em><a class="existingWikiWord" href="/nlab/show/coslice+category">coslice category</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{X/}</annotation></semantics></math> is the category whose</p> <ul> <li> <p>objects are morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{X \\ \downarrow \\ A}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>morphisms are <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting triangles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ && X \\ & \swarrow && \searrow \\ A && \longrightarrow && B }</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> </ul> <p>There are the canonical <a class="existingWikiWord" href="/nlab/show/forgetful+functors">forgetful functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo>,</mo><msup><mi>𝒞</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo></mrow></msup><mo>⟶</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> U \;\colon \; \mathcal{C}_{/X}, \mathcal{C}^{X/} \longrightarrow \mathcal{C} </annotation></semantics></math></div> <p>given by forgetting the morphisms to/from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>We here focus on this class of examples:</p> <div class="num_defn" id="CategoryOfPointedObjects"> <h6 id="definition_47">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category">category</a> with <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/coslice+category">coslice category</a> (def. <a class="maruku-ref" href="#SliceCategory"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> is the corresponding <em><a class="existingWikiWord" href="/nlab/show/category+of+pointed+objects">category of pointed objects</a></em>: its</p> <ul> <li> <p>objects are morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mover><mo>→</mo><mi>x</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">\ast \overset{x}{\to} X</annotation></semantics></math> (hence an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equipped with a choice of point; i.e. a <em><a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a></em>);</p> </li> <li> <p>morphisms are <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting triangles</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>x</mi></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>y</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \ast \\ & {}^{\mathllap{x}}\swarrow && \searrow^{\mathrlap{y}} \\ X && \overset{f}{\longrightarrow} && Y } </annotation></semantics></math></div> <p>(hence morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> which preserve the chosen points).</p> </li> </ul> </div> <div class="num_remark" id="PointedObjectsHaveZeroObject"> <h6 id="remark_24">Remark</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/category+of+pointed+objects">category of pointed objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math>, def. <a class="maruku-ref" href="#CategoryOfPointedObjects"></a>, the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> coincides with the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, both are given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\ast \in \mathcal{C}</annotation></semantics></math> itself, pointed in the unique way.</p> <p>In this situation one says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> is a <em><a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a></em> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> is a <em><a class="existingWikiWord" href="/nlab/show/pointed+category">pointed category</a></em>.</p> <p>It follows that also all <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{\mathcal{C}^{\ast/}}(X,Y)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> are canonically <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a>, pointed by the <em><a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><mn>0</mn><mover><mo>⟶</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \;\colon\; X \overset{\exists !}{\longrightarrow} 0 \overset{\exists !}{\longrightarrow} Y \,. </annotation></semantics></math></div></div> <div class="num_defn" id="BasePointAdjoined"> <h6 id="definition_48">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> with <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> and <a class="existingWikiWord" href="/nlab/show/finite+colimits">finite colimits</a>. Then the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">U \colon \mathcal{C}^{\ast/} \to \mathcal{C}</annotation></semantics></math> from its <a class="existingWikiWord" href="/nlab/show/category+of+pointed+objects">category of pointed objects</a>, def. <a class="maruku-ref" href="#CategoryOfPointedObjects"></a>, has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><munderover><mo>⊥</mo><munder><mo>⟶</mo><mi>U</mi></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow></mover></munderover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C}^{\ast/} \underoverset {\underset{U}{\longrightarrow}} {\overset{(-)_+}{\longleftarrow}} {\bot} \mathcal{C} </annotation></semantics></math></div> <p>given by forming the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>) with a base point (“adjoining a base point”).</p> </div> <div class="num_prop" id="LimitsAndColimitsOfPointedObjects"> <h6 id="proposition_37">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> with all <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. Then also the <a class="existingWikiWord" href="/nlab/show/category+of+pointed+objects">category of pointed objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math>, def. <a class="maruku-ref" href="#CategoryOfPointedObjects"></a>, has all limits and colimits.</p> <p>Moreover:</p> <ol> <li> <p>the limits are the limits of the underlying diagrams in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, with the base point of the limit induced by its universal property in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>the colimits are the limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> of the diagrams <em>with the basepoint adjoined</em>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_59">Proof</h6> <p>It is immediate to check the relevant <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a>. For details see at <em><a href="overcategory#LimitsAndColimits">slice category – limits and colimits</a></em>.</p> </div> <div class="num_example" id="WedgeSumAsCoproduct"> <h6 id="example_29">Example</h6> <p>Given two pointed objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,y)</annotation></semantics></math>, then:</p> <ol> <li> <p>their <a class="existingWikiWord" href="/nlab/show/product">product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> is simply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X\times Y, (x,y))</annotation></semantics></math>;</p> </li> <li> <p>their <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> has to be computed using the second clause in prop. <a class="maruku-ref" href="#LimitsAndColimitsOfPointedObjects"></a>: since the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> has to be adjoined to the diagram, it is given not by the coproduct in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, but by the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> of the form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>⟶</mo><mi>x</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>y</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>∨</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \ast &\overset{x}{\longrightarrow}& X \\ {}^{\mathllap{y}}\downarrow &(po)& \downarrow \\ Y &\longrightarrow& X \vee Y } \,. </annotation></semantics></math></div> <p>This is called the <em><a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a></em> operation on pointed objects.</p> </li> </ol> <p>Generally for a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i\}_{i \in I}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Top^{\ast/}</annotation></semantics></math></p> <ol> <li> <p>their <a class="existingWikiWord" href="/nlab/show/product">product</a> is formed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> as in example <a class="maruku-ref" href="#ProductTopologicalSpace"></a>, with the new basepoint canonically induced;</p> </li> <li> <p>their <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> is formed by the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> over the diagram with a basepoint adjoined, and is called the <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∨</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\vee_{i \in I} X_i</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_example"> <h6 id="example_30">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>, def. <a class="maruku-ref" href="#TopologicalCellComplex"></a> then for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (example <a class="maruku-ref" href="#QuotientSpaceAsPushout"></a>) of its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-skeleton by its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-skeleton is the <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a>, def. <a class="maruku-ref" href="#WedgeSumAsCoproduct"></a>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-spheres, one for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cell of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup><mo stretchy="false">/</mo><msup><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>≃</mo><munder><mo>∨</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>I</mi> <mi>n</mi></msub></mrow></munder><msup><mi>S</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X^n / X^{n-1} \simeq \underset{i \in I_n}{\vee} S^n \,. </annotation></semantics></math></div></div> <div class="num_defn" id="SmashProductOfPointedObjects"> <h6 id="definition_49">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category+of+pointed+objects">category of pointed objects</a> with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> and <a class="existingWikiWord" href="/nlab/show/finite+colimits">finite colimits</a>, the <em><a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a></em> is the <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo>×</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo>⟶</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> (-)\wedge(-) \;\colon\; \mathcal{C}^{\ast/} \times \mathcal{C}^{\ast/} \longrightarrow \mathcal{C}^{\ast/} </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∧</mo><mi>Y</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo>*</mo><munder><mo>⊔</mo><mrow><mi>X</mi><mo>⊔</mo><mi>Y</mi></mrow></munder><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \wedge Y \;\coloneqq\; \ast \underset{X\sqcup Y}{\sqcup} (X \times Y) \,, </annotation></semantics></math></div> <p>hence by the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>⊔</mo><mi>Y</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>id</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>∧</mo><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y \\ \downarrow && \downarrow \\ \ast &\longrightarrow& X \wedge Y } \,. </annotation></semantics></math></div> <p>In terms of the <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a> from def. <a class="maruku-ref" href="#WedgeSumAsCoproduct"></a>, this may be written concisely as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∧</mo><mi>Y</mi><mo>=</mo><mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><mrow><mi>X</mi><mo>∨</mo><mi>Y</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \wedge Y = \frac{X\times Y}{X \vee Y} \,. </annotation></semantics></math></div></div> <div class="num_remark" id="SmashProductOnTopNotAssociative"> <h6 id="remark_25">Remark</h6> <p>For a general category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> in def. <a class="maruku-ref" href="#SmashProductOfPointedObjects"></a>, the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> need not be <a class="existingWikiWord" href="/nlab/show/associativity">associative</a>, namely it fails to be associative if the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">(-)\times Z</annotation></semantics></math> does not preserve the <a class="existingWikiWord" href="/nlab/show/quotients">quotients</a> involved in the definition.</p> <p>In particular this may happen for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a>.</p> <p>A sufficient condition for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">(-) \times Z</annotation></semantics></math> to preserve quotients is that it is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> functor. This is the case in the smaller subcategory of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>, we come to this in prop. <a class="maruku-ref" href="#SmashProductInTopcgIsAssociative"></a> below.</p> </div> <p>These two operations are going to be ubiquituous in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>:</p> <table><thead><tr><th>symbol</th><th>name</th><th>category theory</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∨</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \vee Y</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∧</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \wedge Y</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math></td></tr> </tbody></table> <div class="num_example" id="WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces"> <h6 id="example_31">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X, Y \in Top</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>Y</mi> <mo>+</mo></msub><mo>∈</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X_+,Y_+ \in Top^{\ast/}</annotation></semantics></math>, def. <a class="maruku-ref" href="#BasePointAdjoined"></a>, then</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>+</mo></msub><mo>∨</mo><msub><mi>Y</mi> <mo>+</mo></msub><mo>≃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">X_+ \vee Y_+ \simeq (X \sqcup Y)_+</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>+</mo></msub><mo>∧</mo><msub><mi>Y</mi> <mo>+</mo></msub><mo>≃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">X_+ \wedge Y_+ \simeq (X \times Y)_+</annotation></semantics></math>.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof_60">Proof</h6> <p>By example <a class="maruku-ref" href="#WedgeSumAsCoproduct"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>+</mo></msub><mo>∨</mo><msub><mi>Y</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">X_+ \vee Y_+</annotation></semantics></math> is given by the colimit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> over the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mtd> <mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && && \ast \\ && & \swarrow && \searrow \\ X &\,\,& \ast && && \ast &\,\,& Y } \,. </annotation></semantics></math></div> <p>This is clearly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mo>*</mo><mo>⊔</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \sqcup \ast \sqcup Y</annotation></semantics></math>. Then, by definition <a class="maruku-ref" href="#SmashProductOfPointedObjects"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>X</mi> <mo>+</mo></msub><mo>∧</mo><msub><mi>Y</mi> <mo>+</mo></msub></mtd> <mtd><mo>≃</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>⊔</mo><mo>*</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>X</mi><mo>⊔</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>⊔</mo><mo>*</mo><mo stretchy="false">)</mo><mo>∨</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>⊔</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>⊔</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo>⊔</mo><mo>*</mo></mrow><mrow><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo>⊔</mo><mo>*</mo></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>⊔</mo><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} X_+ \wedge Y_+ & \simeq \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)} \\ & \simeq \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast} \\ & \simeq X \times Y \sqcup \ast \,. \end{aligned} </annotation></semantics></math></div></div> <div class="num_example" id="StandardReducedCyclinderInTop"> <h6 id="example_32">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo>=</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/} = Top^{\ast/}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a>. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mo>+</mo></msub><mo>∈</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> I_+ \in Top^{\ast/} </annotation></semantics></math></div> <p>denotes the standard interval object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I = [0,1]</annotation></semantics></math> from def. <a class="maruku-ref" href="#TopologicalInterval"></a>, with a djoint basepoint adjoined, def. <a class="maruku-ref" href="#BasePointAdjoined"></a>. Now for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy="false">}</mo><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X \wedge (I_+) = (X \times I)/(\{x_0\} \times I) </annotation></semantics></math></div> <p>is the <strong><a class="existingWikiWord" href="/nlab/show/reduced+cylinder">reduced cylinder</a></strong> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: the result of forming the ordinary cyclinder over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as in def. <a class="maruku-ref" href="#TopologicalInterval"></a>, and then identifying the interval over the basepoint of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the point.</p> <p>(Generally, any construction in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> properly adapted to pointed objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> is called the “reduced” version of the unpointed construction. Notably so for “<a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a>” which we come to <a href="#MappingCones">below</a>.)</p> <p>Just like the ordinary cylinder <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X\times I</annotation></semantics></math> receives a canonical injection from the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \sqcup X</annotation></semantics></math> formed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>, so the reduced cyclinder receives a canonical injection from the coproduct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \sqcup X</annotation></semantics></math> formed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Top^{\ast/}</annotation></semantics></math>, which is the <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a> from example <a class="maruku-ref" href="#WedgeSumAsCoproduct"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∨</mo><mi>X</mi><mo>⟶</mo><mi>X</mi><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \vee X \longrightarrow X \wedge (I_+) \,. </annotation></semantics></math></div></div> <div class="num_example" id="PointedMappingSpace"> <h6 id="example_33">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x),(Y,y)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact topological space</a>, then the <strong><a class="existingWikiWord" href="/nlab/show/pointed+mapping+space">pointed mapping space</a></strong> is the <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> of def. <a class="maruku-ref" href="#CompactOpenTopology"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>↪</mo><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>Y</mi></msup><mo>,</mo><msub><mi>const</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Maps((Y,y),(X,x))_\ast \hookrightarrow (X^Y, const_x) </annotation></semantics></math></div> <p>on those maps which preserve the basepoints, and pointed by the map constant on the basepoint of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>In particular, the <strong>standard topological pointed path space object</strong> on some pointed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (the pointed variant of def. <a class="maruku-ref" href="#TopologicalPathSpace"></a>) is the pointed mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo>,</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">Maps(I_+,X)_\ast</annotation></semantics></math>.</p> <p>The pointed consequence of prop. <a class="maruku-ref" href="#MappingTopologicalSpaceIsExponentialObject"></a> then gives that there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Maps</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{Top^{\ast/}}((Z,z) \wedge (Y,y), (X,x)) \simeq Hom_{Top^{\ast/}}((Z,z), Maps((Y,y),(X,x))_\ast) </annotation></semantics></math></div> <p>between basepoint-preserving continuous functions out of a <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a>, def. <a class="maruku-ref" href="#SmashProductOfPointedObjects"></a>, with pointed continuous functions of one variable into the pointed mapping space.</p> </div> <div class="num_example" id="FiberAndCofiberInPointedObjects"> <h6 id="example_34">Example</h6> <p>Given a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/category+of+pointed+objects">category of pointed objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math>, def. <a class="maruku-ref" href="#CategoryOfPointedObjects"></a>, with finite limits and colimits,</p> <ol> <li> <p>its <em><a class="existingWikiWord" href="/nlab/show/fiber">fiber</a></em> or <em><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a></em> is the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the point inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ fib(f) &\longrightarrow& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y } </annotation></semantics></math></div></li> <li> <p>its <em><a class="existingWikiWord" href="/nlab/show/cofiber">cofiber</a></em> or <em><a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></em> is the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of the point projection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>cofib</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& cofib(f) } \,. </annotation></semantics></math></div></li> </ol> </div> <div class="num_remark"> <h6 id="remark_26">Remark</h6> <p>In the situation of example <a class="maruku-ref" href="#FiberAndCofiberInPointedObjects"></a>, both the pullback as well as the pushout are equivalently computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. For the pullback this is the first clause of prop. <a class="maruku-ref" href="#LimitsAndColimitsOfPointedObjects"></a>. The second clause says that for computing the pushout in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, first the point is to be adjoined to the diagram, and then the colimit over the larger diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \ast \\ & \searrow \\ & & X &\overset{f}{\longrightarrow}& Y \\ & & \downarrow && \\ & & \ast && } </annotation></semantics></math></div> <p>be computed. But one readily checks that in this special case this does not affect the result. (The technical jargon is that the inclusion of the smaller diagram into the larger one in this case happens to be a <a class="existingWikiWord" href="/nlab/show/final+functor">final functor</a>.)</p> </div> <div class="num_prop" id="ModelStructureOnSliceCategory"> <h6 id="proposition_38">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/object">object</a>. Then both the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/X}</annotation></semantics></math> as well as the <a class="existingWikiWord" href="/nlab/show/coslice+category">coslice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{X/}</annotation></semantics></math>, def. <a class="maruku-ref" href="#SliceCategory"></a>, carry model structures themselves – the <strong><a class="existingWikiWord" href="/nlab/show/model+structure+on+a+slice+category">model structure on a (co-)slice category</a></strong>, where a morphism is a weak equivalence, fibration or cofibration iff its image under the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is so in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p>In particular the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a>, def. <a class="maruku-ref" href="#CategoryOfPointedObjects"></a>, in a model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> becomes itself a model category this way.</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a>, def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>, we call the <strong><a class="existingWikiWord" href="/nlab/show/pointed+category">pointed</a> <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C}^{\ast/})</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_61">Proof</h6> <p>This is immediate:</p> <p>By prop. <a class="maruku-ref" href="#LimitsAndColimitsOfPointedObjects"></a> the (co-)slice category has all limits and colimits. By definition of the weak equivalences in the (co-)slice, they satisfy <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a>, def. <a class="maruku-ref" href="#CategoryWithWeakEquivalences"></a>, because the do in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p>Similarly, the factorization and lifting is all induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>: Consider the coslice category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{X/}</annotation></semantics></math>, the case of the slice category is formally dual; then if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && X \\ & \swarrow && \searrow \\ A && \underset{f}{\longrightarrow} && B } </annotation></semantics></math></div> <p>commutes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and a factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> exists in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, it uniquely makes this diagram commute</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X \\ & \swarrow &\downarrow& \searrow \\ A &\longrightarrow& C & \longrightarrow& B } \,. </annotation></semantics></math></div> <p>Similarly, if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& C \\ \downarrow && \downarrow \\ B &\longrightarrow& D } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{X/}</annotation></semantics></math>, hence a commuting diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> as shown, with all objects equipped with compatible morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then inspection shows that any lift in the diagram necessarily respects the maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, too.</p> </div> <div class="num_example" id="HomotopyCategoryOfPointedModelStructureIsEnrichedInPointedSets"> <h6 id="example_35">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/slice+model+structure">pointed model structure</a> according to prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>, then the corresponding <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a> (def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>) is, by remark <a class="maruku-ref" href="#PointedObjectsHaveZeroObject"></a>, canonically <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a>, in that its <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><msup><mo stretchy="false">)</mo> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo>×</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Set</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [-,-]_\ast \;\colon\; Ho(\mathcal{C}^{\ast/})^\op \times Ho(\mathcal{C}^{\ast/}) \longrightarrow Set^{\ast/} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="ClassicalModelStructureOnPointedTopologicalSpaces"> <h6 id="definition_50">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>Quillen</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top^{\ast/}_{Quillen}</annotation></semantics></math> for the <strong><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+pointed+topological+spaces">classical model structure on pointed topological spaces</a></strong>, obtained from the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> (theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>) via the induced <a class="existingWikiWord" href="/nlab/show/coslice+model+structure">coslice model structure</a> of prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category</a>, def. <a class="maruku-ref" href="#HomotopyCategoryOfAModelCategory"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mo>≔</mo><mi>Ho</mi><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>Quillen</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho(Top^{\ast/}) \coloneqq Ho(Top_{Quillen}^{\ast/}) </annotation></semantics></math></div> <p>we call the <strong><a class="existingWikiWord" href="/nlab/show/classical+pointed+homotopy+category">classical pointed homotopy category</a></strong>.</p> </div> <div class="num_remark" id="NonDegenerateBasepointAsCofibrantObjects"> <h6 id="remark_27">Remark</h6> <p>The fibrant objects in the pointed model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/}</annotation></semantics></math>, prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>, are those that are fibrant as objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. But the cofibrant objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast}</annotation></semantics></math> are now those for which the basepoint inclusion is a cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo>=</mo><msubsup><mi>Top</mi> <mi>Quillen</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\ast/} = Top^{\ast/}_{Quillen}</annotation></semantics></math> from def. <a class="maruku-ref" href="#ClassicalModelStructureOnPointedTopologicalSpaces"></a>, then the corresponding cofibrant pointed topological spaces are tyically referred to as spaces <strong>with non-degenerate basepoints</strong> or . Notice that the point itself is cofibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math>, so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.</p> </div> <p>While the existence of the model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Top^{\ast/}</annotation></semantics></math> is immediate, via prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>, for the discussion of <a class="existingWikiWord" href="/nlab/show/topologically+enriched+functors">topologically enriched functors</a> (<a href="#ModelStructureOnTopEnrichedFunctors">below</a>) it is useful to record that this, too, is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> (def. <a class="maruku-ref" href="#CofibrantlyGeneratedModelCategory"></a>), as follows:</p> <div class="num_defn" id="GeneratingCofibrationsForPointedTopologicalSpaces"> <h6 id="definition_51">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub><mo>=</mo><mrow><mo>{</mo><msubsup><mi>S</mi> <mo>+</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>n</mi></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow></mover><msubsup><mi>D</mi> <mo>+</mo> <mi>n</mi></msubsup><mo>}</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I_{Top^{\ast/}} = \left\{ S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+ \right\} \;\; \subset Mor(Top^{\ast/}) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub><mo>=</mo><mrow><mo>{</mo><msubsup><mi>D</mi> <mo>+</mo> <mi>n</mi></msubsup><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow></mover><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><mo>}</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> J_{Top^{\ast/}} = \left\{ D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+ \right\} \;\;\; \subset Mor(Top^{\ast/}) \,, </annotation></semantics></math></div> <p>respectively, for the sets of morphisms obtained from the classical generating cofibrations, def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>, and the classical generating acyclic cofibrations, def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>, under adjoining of basepoints (def. <a class="maruku-ref" href="#BasePointAdjoined"></a>).</p> </div> <div class="num_theorem" id="CofibrantGenerationOfPointedTopologicalSpaces"> <h6 id="theorem_3">Theorem</h6> <p>The sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub></mrow><annotation encoding="application/x-tex">I_{Top^{\ast/}}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub></mrow><annotation encoding="application/x-tex">J_{Top^{\ast/}}</annotation></semantics></math> in def. <a class="maruku-ref" href="#GeneratingCofibrationsForPointedTopologicalSpaces"></a> exhibit the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+pointed+topological+spaces">classical model structure on pointed topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>Quillen</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top^{\ast/}_{Quillen}</annotation></semantics></math> of def. <a class="maruku-ref" href="#ClassicalModelStructureOnPointedTopologicalSpaces"></a> as a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a>, def. <a class="maruku-ref" href="#CofibrantlyGeneratedModelCategory"></a>.</p> </div> <p>(This is also a special case of a general statement about cofibrant generation of <a class="existingWikiWord" href="/nlab/show/coslice+model+structures">coslice model structures</a>, see <a href="model+structure+on+an+over+category#ModelStructureInheritsGoodProperties">this proposition</a>.)</p> <div class="proof"> <h6 id="proof_62">Proof</h6> <p>Due to the fact that in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub></mrow><annotation encoding="application/x-tex">J_{Top^{\ast/}}</annotation></semantics></math> a basepoint is freely adjoined, lemma <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a> goes through verbatim for the pointed case, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math> replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub></mrow><annotation encoding="application/x-tex">J_{Top^{\ast/}}</annotation></semantics></math>, as do the other two lemmas above that depend on <a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a> and lemma <a class="maruku-ref" href="#JTopRelativeCellComplexesAreWeakHomotopyEquivalences"></a>. With this, the rest of the proof follows by the same general abstract reasoning as <a href="#TheClassicalModelStructureOfTopologicalSpaces">above</a> in the proof of theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>.</p> </div> <h3 id="ModelStructureOnCompactlyGeneratedTopologicalSpaces">Model structure on compactly generated spaces</h3> <p>The category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> has the technical inconvenience that <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">X^Y</annotation></semantics></math> (def. <a class="maruku-ref" href="#CompactOpenTopology"></a>) satisfying the exponential property (prop. <a class="maruku-ref" href="#MappingTopologicalSpaceIsExponentialObject"></a>) exist in general only for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact topological space</a>, but fail to exist more generally. In other words: <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is not <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a>. But cartesian closure is necessary for some purposes of homotopy theory, for instance it ensures that</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> (def. <a class="maruku-ref" href="#SmashProductOfPointedObjects"></a>) on <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> is <a class="existingWikiWord" href="/nlab/show/associative">associative</a> (prop. <a class="maruku-ref" href="#SmashProductInTopcgIsAssociative"></a> below);</p> </li> <li> <p>there is a concept of <a class="existingWikiWord" href="/nlab/show/topologically+enriched+functors">topologically enriched functors</a> with values in topological spaces, to which we turn <a href="#ModelStructureOnTopEnrichedFunctors">below</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> preserves <a class="existingWikiWord" href="/nlab/show/products">products</a>.</p> </li> </ol> <p>The first two of these are crucial for the development of <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> in the <a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+1">next section</a>, the third is a great convenience in computations.</p> <p>Now, since the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of topological spaces only cares about the <a class="existingWikiWord" href="/nlab/show/CW+approximation">CW approximation</a> to any topological space (remark <a class="maruku-ref" href="#EveryTopologicalSpaceWeaklyEquivalentToACWComplex"></a>), it is plausible to ask for a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> which still contains all <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>, still has all <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>, still supports a model category structure constructed in the same way as above, but which in addition is <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a>, and preferably such that the model structure interacts well with the cartesian closure.</p> <p>Such a full subcategory exists, the category of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>. This we briefly describe now.</p> <p><strong>Literature</strong> (<a href="#Strickland09">Strickland 09</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_defn" id="kTop"> <h6 id="definition_52">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> <p>A subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> is called <strong>compactly closed</strong> (or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-closed</strong>) if for every <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon K \longrightarrow X</annotation></semantics></math> out of a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(A)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> <p>The space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is called <strong><a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated</a></strong> if its closed subsets exhaust (hence coincide with) the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-closed subsets.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> Top_{cg} \hookrightarrow Top </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> on the compactly generated topological spaces.</p> </div> <div class="num_defn" id="kfication"> <h6 id="definition_53">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mover><mo>⟶</mo><mi>k</mi></mover><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> Top \overset{k}{\longrightarrow} Top_{cg} \hookrightarrow Top </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> which sends any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = (S,\tau)</annotation></semantics></math> to the topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>k</mi><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S, k \tau)</annotation></semantics></math> with the same underlying set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, but with open subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>τ</mi></mrow><annotation encoding="application/x-tex">k \tau</annotation></semantics></math> the collection of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-open subsets with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>.</p> </div> <div class="num_lemma" id="ContinuousFunctionsOutOfCompactlyGeneratedFactorThroughCompactlyGeneratedClosureOfCodomain"> <h6 id="lemma_24">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X \in Top_{cg} \hookrightarrow Top</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Y\in Top</annotation></semantics></math>. Then <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> X \longrightarrow Y </annotation></semantics></math></div> <p>are also continuous when regarded as functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mi>k</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X \longrightarrow k(Y) </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> from def. <a class="maruku-ref" href="#kfication"></a>.</p> </div> <div class="proof"> <h6 id="proof_63">Proof</h6> <p>We need to show that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-closed subset, then the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f^{-1}(A) \subset X</annotation></semantics></math> is closed subset.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon K \longrightarrow X</annotation></semantics></math> be any continuous function out of a compact Hausdorff space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-closed by assumption, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>ϕ</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">(f \circ \phi)^{-1}(A) = \phi^{-1}(f^{-1}(A))\subset K</annotation></semantics></math> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(A)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-closed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. But by the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is compactly generated, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(A)</annotation></semantics></math> is already closed.</p> </div> <div class="num_cor" id="kTopIsCoreflectiveSubcategory"> <h6 id="corollary_5">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">X \in Top_{cg}</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{Top}(X,Y) \simeq Hom_{Top_{cg}}(X, k(Y)) \,. </annotation></semantics></math></div> <p>This means equivalently that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> (def. <a class="maruku-ref" href="#kfication"></a>) together with the inclusion from def. <a class="maruku-ref" href="#kTop"></a> forms an pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><munderover><mo>⊥</mo><munder><mo>⟵</mo><mi>k</mi></munder><mo>↪</mo></munderover><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top_{cg} \underoverset {\underset{k}{\longleftarrow}} {\hookrightarrow} {\bot} Top \,. </annotation></semantics></math></div> <p>This in turn means equivalently that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top_{cg} \hookrightarrow Top</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a> with coreflector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. In particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idemotent</a> in that there are <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural</a> <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> k(k(X))\simeq k(X) \,. </annotation></semantics></math></div> <p>Hence <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> exists and are computed as in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>. Also <a class="existingWikiWord" href="/nlab/show/limits">limits</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> exists, these are obtained by computing the limit in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> and then applying the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> to the result.</p> </div> <p>The following is a slight variant of def. <a class="maruku-ref" href="#CompactOpenTopology"></a>, appropriate for the context of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>.</p> <div class="num_defn" id="CompactlyGeneratedMappingSpaces"> <h6 id="definition_54">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">X, Y \in Top_{cg}</annotation></semantics></math> (def. <a class="maruku-ref" href="#kTop"></a>) the <strong>compactly generated mapping space</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>Y</mi></msup><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">X^Y \in Top_{cg}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a> whose underlying set is the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(Y,X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon Y \to X</annotation></semantics></math>, and for which a <a class="existingWikiWord" href="/nlab/show/topological+base">subbase</a> for its topology has elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">U^{\phi(K)}</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\phi \colon K \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> out of a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo></mrow></msup><mo>≔</mo><mrow><mo>{</mo><mi>f</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><mi>U</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U^{\phi(\kappa)} \coloneqq \left\{ f\in C(Y,X) | f(\phi(K)) \subset U \right\} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_28">Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is (compactly generated and) a <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, then the topology on the compactly generated mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">X^Y</annotation></semantics></math> in def. <a class="maruku-ref" href="#CompactlyGeneratedMappingSpaces"></a> agrees with the <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a> of def. <a class="maruku-ref" href="#CompactOpenTopology"></a>. Beware that it is common to say “compact-open topology” also for the topology of the compactly generated mapping space when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is not Hausdorff. In that case, however, the two definitions in general disagree.</p> </div> <div class="num_prop" id="CartesianClosureOfTopcg"> <h6 id="proposition_39">Proposition</h6> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> of def. <a class="maruku-ref" href="#kTop"></a> is <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a>:</p> <p>for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">X \in Top_{cg}</annotation></semantics></math> then the operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X\times (-) \times (-)\times X</annotation></semantics></math> of forming the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> (which by cor. <a class="maruku-ref" href="#kTopIsCoreflectiveSubcategory"></a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> applied to the usual <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a>) together with the operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">(-)^X</annotation></semantics></math> of forming the compactly generated <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> (def. <a class="maruku-ref" href="#CompactlyGeneratedMappingSpaces"></a>) forms a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>X</mi></msup></mrow></munder><mover><mo>⟵</mo><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></munderover><msub><mi>Top</mi> <mi>cg</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top_{cg} \underoverset {\underset{(-)^X}{\longrightarrow}} {\overset{X \times (-)}{\longleftarrow}} {\bot} Top_{cg} \,. </annotation></semantics></math></div></div> <p>For proof see for instance (<a href="#Strickland09">Strickland 09, prop. 2.12</a>).</p> <div class="num_cor" id="SmashHomAdjunctionOnPointedCompactlyGeneratedTopologicalSpaces"> <h6 id="corollary_6">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">X, Y \in Top_{cg}^{\ast/}</annotation></semantics></math>, the operation of forming the <a class="existingWikiWord" href="/nlab/show/pointed+mapping+space">pointed mapping space</a> (example <a class="maruku-ref" href="#PointedMappingSpace"></a>) inside the compactly generated mapping space of def. <a class="maruku-ref" href="#CompactlyGeneratedMappingSpaces"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>≔</mo><mi>fib</mi><mrow><mo>(</mo><msup><mi>X</mi> <mi>Y</mi></msup><mover><mo>⟶</mo><mrow><msub><mi>ev</mi> <mi>y</mi></msub></mrow></mover><mi>X</mi><mspace width="thickmathspace"></mspace><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> Maps(Y,X)_\ast \coloneqq fib\left( X^Y \overset{ev_y}{\longrightarrow} X \;, x \right) </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> operation on <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>*</mo></msub></mrow></munder><mover><mo>⟵</mo><mrow><mi>Y</mi><mo>∧</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></munderover><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top_{cg}^{\ast/} \underoverset {\underset{Maps(Y,-)_\ast}{\longrightarrow}} {\overset{Y \wedge (-)}{\longleftarrow}} {\bot} Top_{cg}^{\ast/} \,. </annotation></semantics></math></div></div> <div class="num_cor" id="MappingSpacesSendsColimitsInFirstArgumentToLimits"> <h6 id="corollary_7">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">X_\bullet \colon I \to Top^{\ast/}_{cg}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>, then the compactly generated <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> construction from def. <a class="maruku-ref" href="#CompactlyGeneratedMappingSpaces"></a> preserves <a class="existingWikiWord" href="/nlab/show/limits">limits</a> in its covariant argument and sends colimits in its contravariant argument to limits:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><msub><mi>Y</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>Y</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex"> Maps(X,\underset{\longleftarrow}{\lim}_i Y_i)_\ast \;\simeq\; \underset{\longleftarrow}{\lim}_i Maps(X, Y_i)_\ast </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Maps( \underset{\longrightarrow}{\lim}_i X_i, \; Y )_\ast \simeq \underset{\longleftarrow}{\lim}_i Maps( X_i, Y )_\ast \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_64">Proof</h6> <p>The first statement is an immediate implication of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">Maps(X,-)_\ast</annotation></semantics></math> being a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, according to cor. <a class="maruku-ref" href="#SmashHomAdjunctionOnPointedCompactlyGeneratedTopologicalSpaces"></a>.</p> <p>For the second statement, we use that by def. <a class="maruku-ref" href="#kTop"></a> a <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a> is uniquely determined if one knows all continuous functions out of compact Hausdorff spaces into it. Hence it is sufficient to show that there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow></msub><mrow><mo>(</mo><mi>K</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Maps</mi><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>)</mo></mrow><mo>≃</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow></msub><mrow><mo>(</mo><mi>K</mi><mo>,</mo><mspace width="thickmathspace"></mspace><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> Hom_{Top_{cg}^{\ast/}}\left( K,\; Maps( \underset{\longrightarrow}{\lim}_i X_i, \; Y )_\ast \right) \simeq Hom_{Top^{\ast/}_{cg}}\left( K, \; \underset{\longleftarrow}{\lim}_i Maps( X_i, Y )_\ast \right) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> any compact Hausdorff space.</p> <p>With this, the statement follows by cor. <a class="maruku-ref" href="#SmashHomAdjunctionOnPointedCompactlyGeneratedTopologicalSpaces"></a> and using that ordinary <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> take colimits in the first argument and limits in the second argument to limits:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Hom</mi> <mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow></msub><mrow><mo>(</mo><mi>K</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Maps</mi><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>)</mo></mrow></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow></msub><mrow><mo>(</mo><mi>K</mi><mo>∧</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow></msub><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>K</mi><mo>∧</mo><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mrow><mo>(</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow></msub><mo stretchy="false">(</mo><mi>K</mi><mo>∧</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><msub><mi>Hom</mi> <mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow></msub><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow></msub><mrow><mo>(</mo><mi>K</mi><mo>,</mo><mspace width="thickmathspace"></mspace><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mi>Maps</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} Hom_{Top^{\ast/}_{cg}} \left( K, \; Maps(\underset{\longrightarrow}{\lim}_i X_i,\; Y)_\ast \right) & \simeq Hom_{Top^{\ast/}_{cg}} \left( K \wedge \underset{\longrightarrow}{\lim}_i X_i,\; Y \right) \\ & \simeq Hom_{Top^{\ast/}_{cg}} \left( \underset{\longrightarrow}{\lim}_i (K \wedge X_i) ,\; Y \right) \\ & \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{Top^{\ast/}_{cg}} ( K \wedge X_i, \; Y ) \right) \\ & \simeq \underset{\longleftarrow}{\lim}_i Hom_{Top^{\ast/}_{cg}}( K, \; Maps(X_i,Y)_\ast ) \\ & \simeq Hom_{Top^{\ast/}_{cg}} \left( K,\; \underset{\longleftarrow}{\lim}_i Maps(X_i,Y)_\ast \right) \end{aligned} \,. </annotation></semantics></math></div></div> <p>Moreover, compact generation fixes the associativity of the smash product (remark <a class="maruku-ref" href="#SmashProductOnTopNotAssociative"></a>):</p> <div class="num_prop" id="SmashProductInTopcgIsAssociative"> <h6 id="proposition_40">Proposition</h6> <p>On <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> (def. <a class="maruku-ref" href="#CategoryOfPointedObjects"></a>) <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> (def. <a class="maruku-ref" href="#kTop"></a>) the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> (def. <a class="maruku-ref" href="#SmashProductOfPointedObjects"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>×</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>⟶</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> (-)\wedge (-) \;\colon\; Top_{cg}^{\ast/} \times Top_{cg}^{\ast/} \longrightarrow Top_{cg}^{\ast/} </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> and the <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a> is a <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> for it.</p> </div> <div class="proof"> <h6 id="proof_65">Proof</h6> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(-)\times X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> by prop. <a class="maruku-ref" href="#CartesianClosureOfTopcg"></a>, it presevers <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> and in particular <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> projections. Therefore with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>∈</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">X, Y, Z \in Top_{cg}^{\ast/}</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>∧</mo><mi>Z</mi></mtd> <mtd><mo>=</mo><mfrac><mrow><mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo><mo>⊔</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo>×</mo><mi>Y</mi></mrow></mfrac><mo>×</mo><mi>Z</mi></mrow><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">{</mo><mi>z</mi><mo stretchy="false">}</mo><mo>⊔</mo><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>×</mo><mi>Z</mi></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mfrac><mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>×</mo><mi>Z</mi></mrow><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo><mo>×</mo><mi>Z</mi><mo>⊔</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo>×</mo><mi>Y</mi><mo>×</mo><mi>Z</mi></mrow></mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>×</mo><mo stretchy="false">{</mo><mi>z</mi><mo stretchy="false">}</mo></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>×</mo><mi>Z</mi></mrow><mrow><mi>X</mi><mo>∨</mo><mi>Y</mi><mo>∨</mo><mi>Z</mi></mrow></mfrac></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} (X \wedge Y) \wedge Z & = \frac{ \frac{X\times Y}{X \times\{y\} \sqcup \{x\}\times Y} \times Z }{ (X \wedge Y)\times \{z\} \sqcup \{[x] = [y]\} \times Z} \\ & \simeq \frac{\frac{X \times Y \times Z}{X \times \{y\}\times Z \sqcup \{x\}\times Y \times Z}}{ X \times Y \times \{z\} } \\ &\simeq \frac{X\times Y \times Z}{ X \vee Y \vee Z} \end{aligned} \,. </annotation></semantics></math></div> <p>The analogous reasoning applies to yield also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∧</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>∧</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>≃</mo><mfrac><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>×</mo><mi>Z</mi></mrow><mrow><mi>X</mi><mo>∨</mo><mi>Y</mi><mo>∨</mo><mi>Z</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">X \wedge (Y\wedge Z) \simeq \frac{X\times Y \times Z}{ X \vee Y \vee Z}</annotation></semantics></math>.</p> <p>The second statement follows directly with prop. <a class="maruku-ref" href="#CartesianClosureOfTopcg"></a>.</p> </div> <div class="num_remark" id="PointedCompactlyGeneratedTopologicalSpacesIsSymmetricMonoidalClosed"> <h6 id="remark_29">Remark</h6> <p>Corollary <a class="maruku-ref" href="#SmashHomAdjunctionOnPointedCompactlyGeneratedTopologicalSpaces"></a> together with prop. <a class="maruku-ref" href="#SmashProductInTopcgIsAssociative"></a> says that under the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> the category of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed</a> <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> is a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> the <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>,</mo><mo>∧</mo><mo>,</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Top_{cg}^{\ast/}, \wedge, S^0) ,. </annotation></semantics></math></div> <p>Notice that by prop. <a class="maruku-ref" href="#CartesianClosureOfTopcg"></a> also unpointed compactly generated spaces under <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> form a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>, hence a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo>,</mo><mo>×</mo><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Top_{cg}, \times , \ast) \,. </annotation></semantics></math></div> <p>The fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top_{cg}^{\ast/}</annotation></semantics></math> is still closed symmetric monoidal but no longer Cartesian exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top_{cg}^{\ast/}</annotation></semantics></math> as being “more <a class="existingWikiWord" href="/nlab/show/linear+logic">linear</a>” than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>. The “full linearization” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> is the closed symmteric monoidal category of <a class="existingWikiWord" href="/nlab/show/structured+spectra">structured spectra</a> under <a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a> which we discuss in <a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+1">section 1</a>.</p> </div> <p>Due to the <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotency</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∘</mo><mi>k</mi><mo>≃</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">k \circ k \simeq k</annotation></semantics></math> (cor. <a class="maruku-ref" href="#kTopIsCoreflectiveSubcategory"></a>) it is useful to know plenty of conditions under which a given topological space is already compactly generated, for then applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> to it does not change it and one may continue working as in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>.</p> <div class="num_example" id="CWComplexIsCompactlyGenerated"> <h6 id="example_36">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> is <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated</a>.</p> </div> <div class="proof"> <h6 id="proof_66">Proof</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/a+CW-complex+is+a+Hausdorff+space">a CW-complex is a Hausdorff space</a>, by prop. <a class="maruku-ref" href="#HausdorffImpliessWeaklyHausdorff"></a> and prop. <a class="maruku-ref" href="#CharacterizationOfCompactClosedSetsInWeaklyHausdorffSpace"></a> its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-closed subsets are precisely those whose intersection with every <a class="existingWikiWord" href="/nlab/show/compact+subspace">compact subspace</a> is closed.</p> <p>Since a CW-complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> over attachments of standard <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">D^{n_i}</annotation></semantics></math> (its cells), by the characterization of colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> (<a href="Top#DescriptionOfLimitsAndColimitsInTop">prop.</a>) a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disks are compact, this implies one direction: if a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> intersected with all compact subsets is closed, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is closed.</p> <p>For the converse direction, since <a class="existingWikiWord" href="/nlab/show/a+CW-complex+is+a+Hausdorff+space">a CW-complex is a Hausdorff space</a> and since <a class="existingWikiWord" href="/nlab/show/compact+subspaces+of+Hausdorff+spaces+are+closed">compact subspaces of Hausdorff spaces are closed</a>, the intersection of a closed subset with a compact subset is closed.</p> </div> <p>For completeness we record further classes of examples:</p> <div class="num_example"> <h6 id="example_37">Example</h6> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> includes</p> <ol> <li> <p>all <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+spaces">locally compact topological spaces</a>,</p> </li> <li> <p>all <a class="existingWikiWord" href="/nlab/show/first-countable+topological+spaces">first-countable topological spaces</a>,</p> <p>hence in particular</p> <ol> <li> <p>all <a class="existingWikiWord" href="/nlab/show/metrizable+topological+spaces">metrizable topological spaces</a>,</p> </li> <li> <p>all <a class="existingWikiWord" href="/nlab/show/discrete+topological+spaces">discrete topological spaces</a>,</p> </li> <li> <p>all <a class="existingWikiWord" href="/nlab/show/codiscrete+topological+spaces">codiscrete topological spaces</a>.</p> </li> </ol> </li> </ol> </div> <p>(<a href="#Lewis78">Lewis 78, p. 148</a>)</p> <p>Recall that by corollary <a class="maruku-ref" href="#kTopIsCoreflectiveSubcategory"></a>, all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> of compactly generated spaces are again compactly generated.</p> <div class="num_example" id="ProductOfCWWithLocallyCompactCWIsCompactlyGenerated"> <h6 id="example_38">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> of a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> with a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> CW-complex, and more generally with a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a> CW-complex, is <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated</a>.</p> </div> <p>(<a href="CW+complex#HatcherTopologyOfCellComplexes">Hatcher “Topology of cell complexes”, theorem A.6</a>)</p> <p>More generally:</p> <div class="num_example" id="ProductOfCompactlyGeneratedWithLocallyCompactHausdorff"> <h6 id="proposition_41">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, then the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\times Y</annotation></semantics></math> is compactly generated.</p> </div> <p>e.g. (<a href="#Strickland09">Strickland 09, prop. 26</a>)</p> <p>Finally we check that the concept of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> does not change under passing to compactly generated spaces:</p> <div class="num_prop" id="kificationComparisonIsWeakHomotopyEquivalence"> <h6 id="proposition_42">Proposition</h6> <p>For every topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, the canonical function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">k(X) \longrightarrow X</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a>) is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>.</p> </div> <div class="proof"> <h6 id="proof_67">Proof</h6> <p>By example <a class="maruku-ref" href="#CWComplexIsCompactlyGenerated"></a>, example <a class="maruku-ref" href="#ProductOfCWWithLocallyCompactCWIsCompactlyGenerated"></a> and lemma <a class="maruku-ref" href="#ContinuousFunctionsOutOfCompactlyGeneratedFactorThroughCompactlyGeneratedClosureOfCodomain"></a>, continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^n \to k(X)</annotation></semantics></math> and their left homotopies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo>→</mo><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^n \times I \to k(X)</annotation></semantics></math> are in bijection with functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S^n \to X</annotation></semantics></math> and their homotopies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S^n \times I \to X</annotation></semantics></math>.</p> </div> <div class="num_theorem" id="ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"> <h6 id="theorem_4">Theorem</h6> <p>The restriction of the <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> from theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a> along the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top_{cg} \hookrightarrow Top</annotation></semantics></math> of def. <a class="maruku-ref" href="#kTop"></a> is still a model category structure, which is <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a> by the same sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>) The coreflection of cor. <a class="maruku-ref" href="#kTopIsCoreflectiveSubcategory"></a> is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> (def. <a class="maruku-ref" href="#QuillenEquivalence"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><munderover><mo>⊥</mo><munder><mo>⟵</mo><mi>k</mi></munder><mo>↪</mo></munderover><msub><mi>Top</mi> <mi>Quillen</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Top_{cg})_{Quillen} \underoverset {\underset{k}{\longleftarrow}} {\hookrightarrow} {\bot} Top_{Quillen} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_68">Proof</h6> <p>By example <a class="maruku-ref" href="#CWComplexIsCompactlyGenerated"></a>, the sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">I_{Top}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">J_{Top}</annotation></semantics></math> are indeed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mor</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mor(Top_{cg})</annotation></semantics></math>. By example <a class="maruku-ref" href="#ProductOfCWWithLocallyCompactCWIsCompactlyGenerated"></a> all arguments above about left homotopies between maps out of these basic cells go through verbatim in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>. Hence the three technical lemmas above depending on actual <a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, topology, lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a>, lemma <a class="maruku-ref" href="#JTopRelativeCellComplexesAreWeakHomotopyEquivalences"></a> and lemma <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a>, go through verbatim as before. Accordingly, since the remainder of the proof of theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> follows by general abstract arguments from these, it also still goes through verbatim for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> (repeatedly use the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> and the <a class="existingWikiWord" href="/nlab/show/retract+argument">retract argument</a> to establish the two weak factorization systems).</p> <p>Hence the (acyclic) cofibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> are identified with those in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math>, and so the inclusion is a part of a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> (def. <a class="maruku-ref" href="#QuillenAdjunction"></a>). To see that this is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> (def. <a class="maruku-ref" href="#QuillenEquivalence"></a>), it is sufficient to check that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a compactly generated space then a continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> (def. <a class="maruku-ref" href="#WeakHomotopyEquivalenceOfTopologicalSpaces"></a>) precisely if the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>k</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde f \colon X \to k(Y)</annotation></semantics></math> is a weak homotopy equivalence. But, by lemma <a class="maruku-ref" href="#ContinuousFunctionsOutOfCompactlyGeneratedFactorThroughCompactlyGeneratedClosureOfCodomain"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> is the same function as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, just considered with different codomain. Hence the result follows with prop. <a class="maruku-ref" href="#kificationComparisonIsWeakHomotopyEquivalence"></a>.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p><strong>Compactly generated weakly Hausdorff topological spaces</strong></p> <p>While the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top_{cg} \hookrightarrow Top</annotation></semantics></math> of def. <a class="maruku-ref" href="#kTop"></a> does satisfy the requirement that it gives a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a> with all <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> and containing all <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>, one may ask for yet smaller subcategories that still share all these properties but potentially exhibit further convenient properties still.</p> <p>A popular choice introduced in (<a href="weakly+Hausdorff+topological+space#McCord69">McCord 69</a>) is to add the further restriction to topopological spaces which are not only compactly generated but also <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+topological+space">weakly Hausdorff</a>. This was motivated from (<a href="compactly+generated+topological+space#Steenrod67">Steenrod 67</a>) where compactly generated Hausdorff spaces were used by the observation ((<a href="weakly+Hausdorff+topological+space#McCord69">McCord 69, section 2</a>)) that Hausdorffness is not preserved my many colimit operations, notably not by forming <a class="existingWikiWord" href="/nlab/show/quotient+spaces">quotient spaces</a>.</p> <p>On the other hand, in above we wouldn’t have imposed Hausdorffness in the first place. More intrinsic advantages of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cgwH</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cgwH}</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> are the following:</p> <ul> <li> <p>every <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cgwH</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top_{cgwH} \hookrightarrow Top</annotation></semantics></math> along a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a> inclusion in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> is again in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cgwH</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cgwH}</annotation></semantics></math></p> </li> <li> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cgwH</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cgwH}</annotation></semantics></math> quotient spaces are not only preserved by <a class="existingWikiWord" href="/nlab/show/cartesian+products">cartesian products</a> (as is the case for all compactly generated spaces due to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\times (-)</annotation></semantics></math> being a left adjoint, according to cor. <a class="maruku-ref" href="#kTopIsCoreflectiveSubcategory"></a>) but by all <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a></p> </li> <li> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cgwH</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cgwH}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/regular+monomorphisms">regular monomorphisms</a> are the <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a> inclusions</p> </li> </ul> <p>We will not need this here or in the following sections, but we briefly mention it for completenes:</p> <div class="num_defn" id="WeaklyHausdorff"> <h6 id="definition_55">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is called <strong><a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+topological+space">weakly Hausdorff</a></strong> if for every <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; K \longrightarrow X </annotation></semantics></math></div> <p>out of a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, its <a class="existingWikiWord" href="/nlab/show/image">image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f(K) \subset X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_prop" id="HausdorffImpliessWeaklyHausdorff"> <h6 id="proposition_43">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a> is a <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+space">weakly Hausdorff space</a>, def. <a class="maruku-ref" href="#WeaklyHausdorff"></a>.</p> </div> <div class="proof"> <h6 id="proof_69">Proof</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/compact+subspaces+of+Hausdorff+spaces+are+closed">compact subspaces of Hausdorff spaces are closed</a>.</p> </div> <div class="num_prop" id="CharacterizationOfCompactClosedSetsInWeaklyHausdorffSpace"> <h6 id="proposition_44">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+topological+space">weakly Hausdorff topological space</a>, def. <a class="maruku-ref" href="#WeaklyHausdorff"></a>, then a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-closed, def. <a class="maruku-ref" href="#kTop"></a>, precisely if for every subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">K \subset X</annotation></semantics></math> that is <a class="existingWikiWord" href="/nlab/show/compact+subspace">compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a> with respect to the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, then the <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>∩</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">K \cap A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>e.g. (<a href="#Strickland09">Strickland 09, lemma 1.4 (c)</a>)</p> <h3 id="TopologicalEnrichment">Topological enrichment</h3> <p>So far the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> which we established in theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>, as well as the <a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+functors">projective model structures on topologically enriched functors</a> induced from it in theorem <a class="maruku-ref" href="#ProjectiveModelStructureOnTopologicalFunctors"></a>, concern the <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a>, but not the <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a> (def. <a class="maruku-ref" href="#TopEnrichedCategory"></a>), i.e. the model structure so far has not been related to the topology on <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a>. The following statements say that in fact the model structure and the enrichment by topology on the hom-spaces are compatible in a suitable sense: we have an “<a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a>”. This implies in particular that the product/hom-adjunctions are <a class="existingWikiWord" href="/nlab/show/Quillen+adjunctions">Quillen adjunctions</a>, which is crucial for a decent discusson of the derived functors of the suspension/looping adjunction <a href="#TheSuspensionLoopingDiscussion">below</a>.</p> <div class="num_defn" id="PushoutProduct"> <h6 id="definition_56">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>Y</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i_1 \colon X_1 \to Y_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>Y</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">i_2 \colon X_2 \to Y_2</annotation></semantics></math> be morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>, def. <a class="maruku-ref" href="#kTop"></a>. Their <strong><a class="existingWikiWord" href="/nlab/show/pushout+product">pushout product</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>□</mo><msub><mi>i</mi> <mn>2</mn></msub><mo>≔</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i_1\Box i_2 \coloneqq ((id, i_2), (i_1,id)) </annotation></semantics></math></div> <p>is the universal morphism in the following diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>Y</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>Y</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><munder><mo>⊔</mo><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow></munder><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && X_1 \times X_2 \\ & {}^{\mathllap{(i_1,id)}}\swarrow && \searrow^{\mathrlap{(id,i_2)}} \\ Y_1 \times X_2 && (po) && X_1 \times Y_2 \\ & {}_{\mathllap{}}\searrow && \swarrow \\ && (Y_1 \times X_2) \underset{X_1 \times X_2}{\sqcup} (X_1 \times Y_2) \\ && \downarrow^{\mathrlap{((id, i_2), (i_1,id))}} \\ && Y_1 \times Y_2 } </annotation></semantics></math></div></div> <div class="num_example" id="PushoutProductOfTwoInclusions"> <h6 id="example_39">Example</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>↪</mo><msub><mi>Y</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i_1 \colon X_1 \hookrightarrow Y_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>↪</mo><msub><mi>Y</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">i_2 \colon X_2 \hookrightarrow Y_2</annotation></semantics></math> are inclusions, then their pushout product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>□</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">i_1 \Box i_2</annotation></semantics></math> from def. <a class="maruku-ref" href="#PushoutProduct"></a> is the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>∪</mo><mspace width="thickmathspace"></mspace><msub><mi>Y</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>)</mo></mrow><mo>↪</mo><msub><mi>Y</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( X_1 \times Y_2 \;\cup\; Y_1 \times X_2 \right) \hookrightarrow Y_1 \times Y_2 \,. </annotation></semantics></math></div> <p>For instance</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>↪</mo><mi>I</mi><mo>)</mo></mrow><mo>□</mo><mrow><mo>(</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>↪</mo><mi>I</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \{0\} \hookrightarrow I \right) \Box \left( \{0\} \hookrightarrow I \right) </annotation></semantics></math></div> <p>is the inclusion of two adjacent edges of a square into the square.</p> </div> <div class="num_example" id="PushoutProductWithInitialMorphism"> <h6 id="example_40">Example</h6> <p>The pushout product with an <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> morphism is just the ordinary <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>∅</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>□</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (\emptyset \to X) \Box (-) \simeq X \times (-) \,, </annotation></semantics></math></div> <p>i.e.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>∅</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>□</mo><mo stretchy="false">(</mo><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>A</mi><mover><mo>⟶</mo><mrow><mi>X</mi><mo>×</mo><mi>f</mi></mrow></mover><mi>X</mi><mo>×</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\emptyset \to X) \Box (A \overset{f}{\to} B) \simeq (X\times A \overset{X \times f}{\longrightarrow} X \times B ) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_70">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> with the <a class="existingWikiWord" href="/nlab/show/empty+set">empty</a> space is the empty space, hence the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>×</mo><mi>A</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover><mi>∅</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\emptyset \times A \overset{(id,f)}{\longrightarrow} \emptyset \times B</annotation></semantics></math> is an isomorphism, and so the pushout in the pushout product is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X \times A</annotation></semantics></math>. From this one reads off the universal map in question to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">X \times f</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi><mo>×</mo><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>A</mi></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mi>∅</mi><mo>×</mo><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>,</mo><mo>∃</mo><mo>!</mo><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \emptyset \times A \\ & {}^{\mathllap{}}\swarrow && \searrow^{\mathrlap{\simeq}} \\ X \times A && (po) && \emptyset \times B \\ & {}_{\mathllap{\simeq}}\searrow && \swarrow \\ && X \times A \\ && \downarrow^{\mathrlap{((id, f), \exists !)}} \\ && X \times B } \,. </annotation></semantics></math></div></div> <div class="num_example" id="PushoutProductOfITopwithITopAndJTop"> <h6 id="example_41">Example</h6> <p>With</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">{</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>↪</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>J</mi> <mi>Top</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">{</mo><msup><mi>D</mi> <mi>n</mi></msup><mover><mo>↪</mo><mrow><msub><mi>j</mi> <mi>n</mi></msub></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> I_{Top} \colon \{ S^{n-1} \overset{i_n}{\hookrightarrow} D^n\} \;\;\; and \;\;\; J_{Top} \colon \{ D^n \overset{j_n}{\hookrightarrow} D^n \times I\} </annotation></semantics></math></div> <p>the generating cofibrations (def. <a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>) and generating acyclic cofibrations (def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> (theorem <a class="maruku-ref" href="#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"></a>), then their <a class="existingWikiWord" href="/nlab/show/pushout-products">pushout-products</a> (def. <a class="maruku-ref" href="#PushoutProduct"></a>) are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>□</mo><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><mo>≃</mo><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>□</mo><msub><mi>j</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><mo>≃</mo><msub><mi>j</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} i_{n_1} \Box i_{n_2} & \simeq i_{n_1 + n_2} \\ i_{n_1} \Box j_{n_2} & \simeq j_{n_1 + n_2} \end{aligned} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_71">Proof</h6> <p>To see this, it is profitable to model <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a> and <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>, up to <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>, as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cubes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mo lspace="0em" rspace="thinmathspace">n</mo></msup><mo>≃</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mi>n</mi></msup><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^\n \simeq [0,1]^n \subset \mathbb{R}^n</annotation></semantics></math> and their boundaries <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>≃</mo><mo>∂</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^{n-1} \simeq \partial [0,1]^n</annotation></semantics></math> . For the idea of the proof, consider the situation in low dimensions, where one readily sees pictorially that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>□</mo><msub><mi>i</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∪</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>)</mo></mrow><mo>↪</mo><mo>□</mo></mrow><annotation encoding="application/x-tex"> i_1 \Box i_1 \;\colon\; \left(\;\; = \;\;\cup\;\; \vert\vert\;\;\right) \hookrightarrow \Box </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>□</mo><msub><mi>j</mi> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∪</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>)</mo></mrow><mo>↪</mo><mo>□</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> i_1 \Box j_0 \;\colon\; \left(\;\; = \;\;\cup\;\; \vert \;\; \right) \hookrightarrow \Box \,. </annotation></semantics></math></div> <p>Generally, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> may be represented as the space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-tuples of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math> as the suspace of tuples for which at least one of the coordinates is equal to 0 or to 1.</p> <p>Accordingly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>↪</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n_1} \times D^{n_2} \hookrightarrow D^{n_1 + n_2}</annotation></semantics></math> is the subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n_1+n_2)</annotation></semantics></math>-tuples, such that at least one of the first <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">n_1</annotation></semantics></math> coordinates is equal to 0 or 1, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>↪</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">D^{n_1} \times S^{n_2} \hookrightarrow D^{n_1+ n_2}</annotation></semantics></math> is the subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n_1 + n_2)</annotation></semantics></math>-tuples such that east least one of the last <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_2</annotation></semantics></math> coordinates is equal to 0 or to 1. Therefore</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>∪</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>≃</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^{n_1} \times D^{n_2} \cup D^{n_1} \times S^{n_2} \simeq S^{n_1 + n_2} \,. </annotation></semantics></math></div> <p>And of course it is clear that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>≃</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">D^{n_1} \times D^{n_2} \simeq D^{n_1 + n_2}</annotation></semantics></math>. This shows the first case.</p> <p>For the second, use that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">S^{n_1} \times D^{n_2} \times I</annotation></semantics></math> is contractible to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n_1} \times D^{n_2}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">D^{n_1} \times D^{n_2} \times I</annotation></semantics></math>, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n_1} \times D^{n_2}</annotation></semantics></math> is a subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">D^{n_1} \times D^{n_2}</annotation></semantics></math>.</p> </div> <div class="num_defn" id="PullbackPowering"> <h6 id="definition_57">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i \colon A \to B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p \colon X \to Y</annotation></semantics></math> be two morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>, def. <a class="maruku-ref" href="#kTop"></a>. Their <strong>pullback powering</strong> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mrow><mo>□</mo><mi>i</mi></mrow></msup><mo>≔</mo><mo stretchy="false">(</mo><msup><mi>p</mi> <mi>B</mi></msup><mo>,</mo><msup><mi>X</mi> <mi>i</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> p^{\Box i} \coloneqq (p^B, X^i) </annotation></semantics></math></div> <p>being the universal morphism in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>X</mi> <mi>B</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msup><mi>p</mi> <mi>B</mi></msup><mo>,</mo><msup><mi>X</mi> <mi>i</mi></msup><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>Y</mi> <mi>B</mi></msup><munder><mo>×</mo><mrow><msup><mi>Y</mi> <mi>A</mi></msup></mrow></munder><msup><mi>X</mi> <mi>A</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msup><mi>Y</mi> <mi>B</mi></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msup><mi>X</mi> <mi>A</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>Y</mi> <mi>i</mi></msup></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msup><mi>p</mi> <mi>A</mi></msup></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>Y</mi> <mi>A</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && X^B \\ && \downarrow^{\mathrlap{(p^B, X^i)}} \\ && Y^B \underset{Y^A}{\times} X^A \\ & \swarrow && \searrow \\ Y^B && (pb) && X^A \\ & {}_{\mathllap{Y^i}}\searrow && \swarrow_{\mathrlap{p^A}} \\ && Y^A } </annotation></semantics></math></div></div> <div class="num_prop" id="JoyalTierneyCalculus"> <h6 id="proposition_45">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo>,</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">i_1, i_2 , p</annotation></semantics></math> be three morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>, def. <a class="maruku-ref" href="#kTop"></a>. Then for their <a class="existingWikiWord" href="/nlab/show/pushout-products">pushout-products</a> (def. <a class="maruku-ref" href="#PushoutProduct"></a>) and pullback-powerings (def. <a class="maruku-ref" href="#PullbackPowering"></a>) the following <a class="existingWikiWord" href="/nlab/show/lifting+properties">lifting properties</a> are equivalent (“<a class="existingWikiWord" href="/nlab/show/Joyal-Tierney+calculus">Joyal-Tierney calculus</a>”):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><msub><mi>i</mi> <mn>1</mn></msub><mo>□</mo><msub><mi>i</mi> <mn>2</mn></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><mi>p</mi></mtd></mtr> <mtr><mtd><mo>⇔</mo></mtd> <mtd><msub><mi>i</mi> <mn>1</mn></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><msup><mi>p</mi> <mrow><mo>□</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><mo>⇔</mo></mtd> <mtd><msub><mi>i</mi> <mn>2</mn></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><msup><mi>p</mi> <mrow><mo>□</mo><msub><mi>i</mi> <mn>1</mn></msub></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & i_1 \Box i_2 & \text{has LLP against} & p \\ \Leftrightarrow & i_1 & \text{has LLP against} & p^{\Box i_2} \\ \Leftrightarrow & i_2 & \text{has LLP against} & p^{\Box i_1} } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_72">Proof</h6> <p>We claim that by the <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>, and carefully collecting terms, one finds a natural bijection between <a class="existingWikiWord" href="/nlab/show/commuting+squares">commuting squares</a> and their <a class="existingWikiWord" href="/nlab/show/lifts">lifts</a> as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><msup><mi>X</mi> <mi>B</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>p</mi> <mrow><mo>□</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><msup><mi>Y</mi> <mi>B</mi></msup><munder><mo>×</mo><mrow><msup><mi>Y</mi> <mi>A</mi></msup></mrow></munder><msup><mi>X</mi> <mi>A</mi></msup></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>Q</mi><mo>×</mo><mi>B</mi><munder><mo>⊔</mo><mrow><mi>Q</mi><mo>×</mo><mi>A</mi></mrow></munder><mi>P</mi><mo>×</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>□</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{p^{\Box i_2}}} \\ P &\underset{(g_1,g_2)}{\longrightarrow}& Y^B \underset{Y^A}{\times} X^A } \;\;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\;\; \array{ Q \times B \underset{Q \times A}{\sqcup} P \times A &\overset{(\tilde f, \tilde g_2)}{\longrightarrow}& X \\ {}^{\mathllap{i_1 \Box i_2}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B & \underset{\tilde g_1}{\longrightarrow} & Y } \,, </annotation></semantics></math></div> <p>where the tilde denotes product/hom-<a class="existingWikiWord" href="/nlab/show/adjuncts">adjuncts</a>, for instance</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>P</mi><mover><mo>⟶</mo><mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></mover><msup><mi>Y</mi> <mi>B</mi></msup></mrow><mrow><mi>P</mi><mo>×</mo><mi>B</mi><mover><mo>⟶</mo><mrow><msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mrow></mover><mi>Y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \frac{ P \overset{g_1}{\longrightarrow} Y^B }{ P \times B \overset{\tilde g_1}{\longrightarrow} Y } </annotation></semantics></math></div> <p>etc.</p> <p>To see this in more detail, observe that both squares above each represent two squares from the two components into the fiber product and out of the pushout, respectively, as well as one more square exhibiting the compatibility condition on these components:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><msup><mi>X</mi> <mi>B</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>p</mi> <mrow><mo>□</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><msup><mi>Y</mi> <mi>B</mi></msup><munder><mo>×</mo><mrow><msup><mi>Y</mi> <mi>A</mi></msup></mrow></munder><msup><mi>X</mi> <mi>A</mi></msup></mtd></mtr></mtable></mrow></mtd></mtr> <mtr><mtd><mo>≃</mo></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><msup><mi>X</mi> <mi>B</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>p</mi> <mi>B</mi></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></munder></mtd> <mtd><msup><mi>Y</mi> <mi>B</mi></msup></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><msup><mi>X</mi> <mi>B</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>X</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></munder></mtd> <mtd><msup><mi>X</mi> <mi>A</mi></msup></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mi>X</mi> <mi>A</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>p</mi> <mi>A</mi></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Y</mi> <mi>B</mi></msup></mtd> <mtd><munder><mo>⟶</mo><mrow><msup><mi>Y</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow></msup></mrow></munder></mtd> <mtd><msup><mi>Y</mi> <mi>A</mi></msup></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>}</mo></mrow></mtd></mtr> <mtr><mtd><mo>↔</mo></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>Q</mi><mo>×</mo><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>Q</mi><mo>×</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Q</mi><mo>×</mo><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub></mrow></munder></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>P</mi><mo>×</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>}</mo></mrow></mtd></mtr> <mtr><mtd><mo>≃</mo></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>Q</mi><mo>×</mo><mi>B</mi><munder><mo>⊔</mo><mrow><mi>Q</mi><mo>×</mo><mi>A</mi></mrow></munder><mi>P</mi><mo>×</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>□</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mrow></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \;\;\;\; \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{p^{\Box i_2}}} \\ P &\underset{(g_1,g_2)}{\longrightarrow}& Y^B \underset{Y^A}{\times} X^A } \\ \simeq & \;\;\;\; \left\{ \;\;\;\; \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{p^B}} \\ P &\underset{g_1}{\longrightarrow}& Y^B } \;\;\;\;\; \,, \;\;\;\;\; \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{X^{i_2}}} \\ P &\underset{g_1}{\longrightarrow}& X^A } \;\;\;\;\; \,, \;\;\;\;\; \array{ P &\overset{g_2}{\longrightarrow}& X^A \\ {}^{\mathllap{g_1}}\downarrow && \downarrow^{\mathrlap{p^A}} \\ Y^B &\underset{Y^{i_2}}{\longrightarrow}& Y^A } \;\;\;\;\; \right\} \\ \leftrightarrow & \;\;\;\; \left\{ \;\;\;\;\; \array{ Q \times B &\overset{\tilde f}{\longrightarrow}& X \\ {}^{\mathllap{(i_1,id)}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B &\underset{\tilde g_2}{\longrightarrow}& Y } \;\;\;\;\; \,, \;\;\;\;\; \array{ Q \times A &\overset{(id,i_2)}{\longrightarrow}& Q \times B \\ {}^{\mathllap{(i_1,id)}}\downarrow && \downarrow^{\mathrlap{\tilde f}} \\ P \times A &\underset{\tilde g_2}{\longrightarrow}& X } \;\;\;\;\; \,, \;\;\;\;\; \array{ P \times A &\overset{\tilde g_2}{\longrightarrow}& X \\ {}^{\mathllap{(id,i_2)}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B &\underset{\tilde g_1}{\longrightarrow}& Y } \;\;\;\;\; \right\} \\ \simeq & \;\;\;\; \array{ Q \times B \underset{Q \times A}{\sqcup} P \times A &\overset{(\tilde f, \tilde g_2)}{\longrightarrow}& X \\ {}^{\mathllap{i_1 \Box i_2}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B & \underset{\tilde g_1}{\longrightarrow} & Y } \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_prop" id="PushoutProductInTopCGSendsCofCofToCof"> <h6 id="proposition_46">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/pushout-product">pushout-product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> (def. <a class="maruku-ref" href="#kTop"></a>) of two classical cofibrations is a classical cofibration:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Cof</mi> <mi>cl</mi></msub><mo>□</mo><msub><mi>Cof</mi> <mi>cl</mi></msub><mo>⊂</mo><msub><mi>Cof</mi> <mi>cl</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Cof_{cl} \Box Cof_{cl} \subset Cof_{cl} \,. </annotation></semantics></math></div> <p>If one of them is acyclic, then so is the pushout-product:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Cof</mi> <mi>cl</mi></msub><mo>□</mo><mo stretchy="false">(</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Cof</mi> <mi>cl</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Cof</mi> <mi>cl</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Cof_{cl} \Box (W_{cl} \cap Cof_{cl}) \subset W_{cl}\cap Cof_{cl} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_73">Proof</h6> <p>Regarding the first point:</p> <p>By example <a class="maruku-ref" href="#PushoutProductOfITopwithITopAndJTop"></a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub><mo>□</mo><msub><mi>I</mi> <mi>Top</mi></msub><mo>⊂</mo><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex"> I_{Top} \Box I_{Top} \subset I_{Top} </annotation></semantics></math></div> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><msub><mi>I</mi> <mi>Top</mi></msub><mo>□</mo><msub><mi>I</mi> <mi>Top</mi></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>cl</mi></msub></mtd></mtr> <mtr><mtd><mo>⇔</mo></mtd> <mtd><msub><mi>I</mi> <mi>Top</mi></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>cl</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo>□</mo><msub><mi>I</mi> <mi>Top</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><mo>⇒</mo></mtd> <mtd><msub><mi>Cof</mi> <mi>cl</mi></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>cl</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo>□</mo><msub><mi>I</mi> <mi>Top</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><mo>⇔</mo></mtd> <mtd><msub><mi>I</mi> <mi>Top</mi></msub><mo>□</mo><msub><mi>Cof</mi> <mi>cl</mi></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>cl</mi></msub></mtd></mtr> <mtr><mtd><mo>⇔</mo></mtd> <mtd><msub><mi>I</mi> <mi>Top</mi></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>cl</mi></msub><msup><mo stretchy="false">)</mo> <mrow><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><mo>⇒</mo></mtd> <mtd><msub><mi>Cof</mi> <mi>cl</mi></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>cl</mi></msub><msup><mo stretchy="false">)</mo> <mrow><msub><mi>Cof</mi> <mi>cl</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><mo>⇔</mo></mtd> <mtd><msub><mi>Cof</mi> <mi>cl</mi></msub><mo>□</mo><msub><mo lspace="0em" rspace="thinmathspace">Cof</mo> <mi>cl</mi></msub></mtd> <mtd><mtext>has LLP against</mtext></mtd> <mtd><msub><mi>W</mi> <mi>cl</mi></msub><mo>∩</mo><msub><mi>Fib</mi> <mi>cl</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ & I_{Top} \Box I_{Top} & \text{has LLP against} & W_{cl} \cap Fib_{cl} \\ \Leftrightarrow & I_{Top} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{\Box I_{Top}} \\ \Rightarrow & Cof_{cl} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{\Box I_{Top}} \\ \Leftrightarrow & I_{Top} \Box Cof_{cl} & \text{has LLP against} & W_{cl} \cap Fib_{cl} \\ \Leftrightarrow & I_{Top} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{Cof_{cl}} \\ \Rightarrow & Cof_{cl} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{Cof_{cl}} \\ \Leftrightarrow & Cof_{cl} \Box \Cof_{cl} & \text{has LLP against} & W_{cl} \cap Fib_{cl} } \,, </annotation></semantics></math></div> <p>where all logical equivalences used are those of prop. <a class="maruku-ref" href="#JoyalTierneyCalculus"></a> and where all implications appearing are by the closure property of lifting problems, prop. <a class="maruku-ref" href="#ClosurePropertiesOfInjectiveAndProjectiveMorphisms"></a>.</p> <p>Regarding the second point: By example <a class="maruku-ref" href="#PushoutProductOfITopwithITopAndJTop"></a> we moreover have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>Top</mi></msub><mo>□</mo><msub><mi>J</mi> <mi>Top</mi></msub><mo>⊂</mo><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex"> I_{Top} \Box J_{Top} \subset J_{Top} </annotation></semantics></math></div> <p>and the conclusion follows by the same kind of reasoning.</p> </div> <div class="num_remark"> <h6 id="remark_30">Remark</h6> <p>In <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> theory the property in proposition <a class="maruku-ref" href="#PushoutProductInTopCGSendsCofCofToCof"></a> is referred to as saying that the model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> from theorem <a class="maruku-ref" href="#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"></a></p> <ol> <li> <p>is a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> with respect to the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>;</p> </li> <li> <p>is an <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a>, over itself.</p> </li> </ol> </div> <p>A key point of what this entails is the following:</p> <div class="num_prop" id="HomProductAdjunctionForCofibrantObjectInTopCGIsQuillen"> <h6 id="proposition_47">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">X \in (Top_{cg})_{Quillen}</annotation></semantics></math> cofibrant (a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>) then the product-hom-adjunction for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> (prop. <a class="maruku-ref" href="#CartesianClosureOfTopcg"></a>) is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>X</mi></msup></mrow></munder><mover><mo>⟵</mo><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></munderover><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Top_{cg})_{Quillen} \underoverset \underset{(-)^X}{\longrightarrow} \overset{X \times (-)}{\longleftarrow} {\bot} (Top_{cg})_{Quillen} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_74">Proof</h6> <p>By example <a class="maruku-ref" href="#PushoutProductWithInitialMorphism"></a> we have that the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> functor is equivalently the <a class="existingWikiWord" href="/nlab/show/pushout+product">pushout product</a> functor with the initial morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>∅</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>□</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \times (-) \simeq (\emptyset \to X) \Box (-) \,. </annotation></semantics></math></div> <p>By assumption <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>∅</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\emptyset \to X)</annotation></semantics></math> is a cofibration, and hence prop. <a class="maruku-ref" href="#PushoutProductInTopCGSendsCofCofToCof"></a> says that this is a left Quillen functor.</p> </div> <p>The statement and proof of prop. <a class="maruku-ref" href="#HomProductAdjunctionForCofibrantObjectInTopCGIsQuillen"></a> has a direct analogue in <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a></p> <div class="num_prop" id="HomProductAdjunctionForCofibrantObjectInPointedTopCGIsQuillen"> <h6 id="proposition_48">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">X \in (Top^{\ast/}_{cg})_{Quillen}</annotation></semantics></math> cofibrant with respect to the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+pointed+topological+spaces">classical model structure on pointed</a> <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> (theorem <a class="maruku-ref" href="#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"></a>, prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>) (hence a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a> with non-degenerate basepoint, remark <a class="maruku-ref" href="#NonDegenerateBasepointAsCofibrantObjects"></a>) then the pointed product-hom-adjunction from corollary <a class="maruku-ref" href="#SmashHomAdjunctionOnPointedCompactlyGeneratedTopologicalSpaces"></a> is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> (def. <a class="maruku-ref" href="#QuillenAdjunction"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>*</mo></msub></mrow></munder><mover><mo>⟵</mo><mrow><mi>X</mi><mo>∧</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></munderover><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Top^{\ast/}_{cg})_{Quillen} \underoverset \underset{Maps(X,-)_\ast}{\longrightarrow} \overset{X \wedge (-)}{\longleftarrow} {\bot} (Top^{\ast/}_{cg})_{Quillen} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_75">Proof</h6> <p>Let now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>□</mo> <mo>∧</mo></msub></mrow><annotation encoding="application/x-tex">\Box_\wedge</annotation></semantics></math> denote the <strong>smash pushout product</strong> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo>□</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(-)^{\Box(-)}</annotation></semantics></math> the <strong>smash pullback powering</strong> defined as in def. <a class="maruku-ref" href="#PushoutProduct"></a> and def. <a class="maruku-ref" href="#PullbackPowering"></a>, but with <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> replaced by <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> (def. <a class="maruku-ref" href="#SmashProductOfPointedObjects"></a>) and compactly generated <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> replaced by pointed mapping spaces (def. <a class="maruku-ref" href="#PointedMappingSpace"></a>).</p> <p>By theorem <a class="maruku-ref" href="#CofibrantGenerationOfPointedTopologicalSpaces"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg}^{\ast/})_{Quillen}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>Top</mi></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">I_{Top^{\ast/}} = (I_{Top})_+</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>J</mi> <mi>Top</mi></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">J_{Top^{\ast/}}= (J_{Top})_+</annotation></semantics></math>. Example <a class="maruku-ref" href="#WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces"></a> gives that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>I</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">i_n \in I_{Top}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>J</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">j_n \in J_{Top}</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><msub><mo>□</mo> <mo>∧</mo></msub><mo stretchy="false">(</mo><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex"> (i_{n_1})_+ \Box_\wedge (i_{n_2})_+ \simeq (i_{n_1 + n_2})_+ </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><msub><mo>∧</mo> <mo>∧</mo></msub><mo stretchy="false">(</mo><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>i</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (i_{n_1})_+ \wedge_\wedge (i_{n_2})_+ \simeq (i_{n_1 + n_2})_+ \,. </annotation></semantics></math></div> <p>Hence the pointed analog of prop. <a class="maruku-ref" href="#PushoutProductInTopCGSendsCofCofToCof"></a> holds and therefore so does the pointed analog of the conclusion in prop. <a class="maruku-ref" href="#HomProductAdjunctionForCofibrantObjectInTopCGIsQuillen"></a>.</p> </div> <h3 id="ModelStructureOnTopEnrichedFunctors">Model structure on topological functors</h3> <p>With classical topological homotopy theory in hand (theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>, theorem <a class="maruku-ref" href="#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"></a>), it is straightforward now to generalize this to a homotopy theory of <em>topological diagrams</em>. This is going to be the basis for the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a>, because spectra may be identified with certain topological diagrams (<a href="Introduction+to+Stable+homotopy+theory+--+1-1#SequentialSpectraAsDiagramSpectra">prop.</a>).</p> <p>Technically, “topological diagram” here means “<a class="existingWikiWord" href="/nlab/show/Top">Top</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>”. We now discuss what this means and then observe that as an immediate corollary of theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a> we obtain a model category structure on topological diagrams.</p> <p>As a by-product, we obtain the model category theory of <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a> in topological spaces, which will be useful.</p> <p>In the following we say <em><a class="existingWikiWord" href="/nlab/show/Top">Top</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></em> and <em><a class="existingWikiWord" href="/nlab/show/Top">Top</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a></em> etc. for what often is referred to as “<a class="existingWikiWord" href="/nlab/show/topological+category">topological category</a>” and “<a class="existingWikiWord" href="/nlab/show/topological+functor">topological functor</a>” etc. As discussed there, these latter terms are ambiguous.</p> <p><strong>Literature</strong> (<a class="existingWikiWord" href="/nlab/show/Categorical+Homotopy+Theory">Riehl, chapter 3</a>) for basics of <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>; (<a href="#Piacenza91">Piacenza 91</a>) for the <a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+functors">projective model structure on topological functors</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_defn" id="TopEnrichedCategory"> <h6 id="definition_58">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>, hence:</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/class">class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Obj(\mathcal{C})</annotation></semantics></math>, called the <strong>class of <a class="existingWikiWord" href="/nlab/show/objects">objects</a></strong>;</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a, b\in Obj(\mathcal{C})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a> (def. <a class="maruku-ref" href="#kTop"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}(a,b)\in Top_{cg} \,, </annotation></semantics></math></div> <p>called the <strong>space of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a></strong> or the <strong><a class="existingWikiWord" href="/nlab/show/hom-space">hom-space</a></strong> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>;</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a,b,c\in Obj(\mathcal{C})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∘</mo> <mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>×</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \circ_{a,b,c} \;\colon\; \mathcal{C}(a,b)\times \mathcal{C}(b,c) \longrightarrow \mathcal{C}(a,c) </annotation></semantics></math></div> <p>out of the <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> (by cor. <a class="maruku-ref" href="#kTopIsCoreflectiveSubcategory"></a>: the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a>), called the <em><a class="existingWikiWord" href="/nlab/show/composition">composition</a></em> operation;</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \in Obj(\mathcal{C})</annotation></semantics></math> a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mi>a</mi></msub><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Id_a\in \mathcal{C}(a,a)</annotation></semantics></math>, called the <em><a class="existingWikiWord" href="/nlab/show/identity">identity</a></em> morphism on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></p> </li> </ol> <p>such that the composition is <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unital</a>.</p> <p>Similarly a <strong>pointed topologically enriched category</strong> is such a structure with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top^{\ast/}_{cg}</annotation></semantics></math> (def. <a class="maruku-ref" href="#CategoryOfPointedObjects"></a>) and with the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> replaced by the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> (def. <a class="maruku-ref" href="#SmashProductOfPointedObjects"></a>) of pointed topological spaces.</p> </div> <div class="num_remark" id="UnderlyingCategoryOfTopEnrichedCategory"> <h6 id="remark_31">Remark</h6> <p>Given a (pointed) <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a> as in def. <a class="maruku-ref" href="#TopEnrichedCategory"></a>, then forgetting the topology on the <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a> (along the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U \colon Top_{cg} \to Set</annotation></semantics></math>) yields an ordinary <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{C}}(a,b) = U(\mathcal{C}(a,b)) \,. </annotation></semantics></math></div> <p>It is in this sense that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a category with <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a>, and hence “<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a>”.</p> </div> <p>The archetypical example is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> itself:</p> <div class="num_example" id="TopkAsATopologicallyEnrichedCategory"> <h6 id="example_42">Example</h6> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> (def. <a class="maruku-ref" href="#kTop"></a>) canonically obtains the structure of a <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a>, def. <a class="maruku-ref" href="#TopEnrichedCategory"></a>, with <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a> given by the compactly generated <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> (def. <a class="maruku-ref" href="#CompactlyGeneratedMappingSpaces"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>Y</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex"> Top_{cg}(X,Y) \coloneqq Y^X </annotation></semantics></math></div> <p>and with <a class="existingWikiWord" href="/nlab/show/composition">composition</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup><mo>×</mo><msup><mi>Z</mi> <mi>Y</mi></msup><mo>⟶</mo><msup><mi>Z</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex"> Y^X \times Z^Y \longrightarrow Z^X </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> under the (product<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> mapping-space)-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> from prop. <a class="maruku-ref" href="#CartesianClosureOfTopcg"></a> of the <a class="existingWikiWord" href="/nlab/show/evaluation+morphisms">evaluation morphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>Y</mi> <mi>X</mi></msup><mo>×</mo><msup><mi>Z</mi> <mi>Y</mi></msup><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>ev</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover><mi>Y</mi><mo>×</mo><msup><mi>Z</mi> <mi>Y</mi></msup><mover><mo>⟶</mo><mi>ev</mi></mover><mi>Z</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \times Y^X \times Z^Y \overset{(ev, id)}{\longrightarrow} Y \times Z^Y \overset{ev}{\longrightarrow} Z \,. </annotation></semantics></math></div> <p>Similarly, <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed</a> <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>k</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top_k^{\ast/}</annotation></semantics></math> form a pointed topologically enriched category, using the <a class="existingWikiWord" href="/nlab/show/pointed+mapping+spaces">pointed mapping spaces</a> from example <a class="maruku-ref" href="#PointedMappingSpace"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top^{\ast/}_{cg}(X,Y) \coloneqq Maps(X,Y)_\ast \,. </annotation></semantics></math></div></div> <div class="num_defn" id="TopologicallyEnrichedFunctor"> <h6 id="definition_59">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/topologically+enriched+functor">topologically enriched functor</a> between two <a class="existingWikiWord" href="/nlab/show/topologically+enriched+categories">topologically enriched categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex"> F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} </annotation></semantics></math></div> <p>is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, hence:</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F_0 \;\colon\; Obj(\mathcal{C}) \longrightarrow Obj(\mathcal{D}) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/objects">objects</a>;</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a,b \in Obj(\mathcal{C})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>𝒟</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>F</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F_{a,b} \;\colon\; \mathcal{C}(a,b) \longrightarrow \mathcal{D}(F_0(a), F_0(b)) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a>,</p> </li> </ol> <p>such that this preserves <a class="existingWikiWord" href="/nlab/show/composition">composition</a> and <a class="existingWikiWord" href="/nlab/show/identity">identity</a> morphisms in the evident sense.</p> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of topologically enriched functors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>F</mi><mo>⇒</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; F \Rightarrow G </annotation></semantics></math></div> <p>is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a>: for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in Obj(\mathcal{C})</annotation></semantics></math> a choice of morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub><mo>∈</mo><mi>𝒟</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_c \in \mathcal{D}(F(c),G(c))</annotation></semantics></math> such that for each pair of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>,</mo><mi>d</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c,d \in \mathcal{C}</annotation></semantics></math> the two continuous functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub><mo>∘</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>𝒟</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta_d \circ F(-) \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d)) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>c</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>𝒟</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d)) </annotation></semantics></math></div> <p>agree.</p> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, \mathcal{D}]</annotation></semantics></math> for the resulting category of topologically enriched functors.</p> </div> <div class="num_remark" id="TopologicallyEnrichedNaturalTransformationIsTransformationOfUnderlyingFunctors"> <h6 id="remark_32">Remark</h6> <p>The condition on an <a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a> in def. <a class="maruku-ref" href="#TopologicallyEnrichedFunctor"></a> is just that on an ordinary <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> on the underlying unenriched functors, saying that for every morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>c</mi><mo>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">f \colon c \to d</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>c</mi></msub></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>d</mi></msub></mrow></munder></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\;\;\; \mapsto \;\;\;\; \array{ \mathcal{C}(c,c) \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ \mathcal{C}(c,d) \times X &\underset{\eta_d}{\longrightarrow}& F(d) } \,. </annotation></semantics></math></div></div> <div class="num_example" id="TopologicallyEnrichedFunctorsToTopk"> <h6 id="example_43">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a>, def. <a class="maruku-ref" href="#TopEnrichedCategory"></a> then a <a class="existingWikiWord" href="/nlab/show/topologically+enriched+functor">topologically enriched functor</a> (def. <a class="maruku-ref" href="#TopologicallyEnrichedFunctor"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex"> F \;\colon\; \mathcal{C} \longrightarrow Top_{cg} </annotation></semantics></math></div> <p>to the archetypical topologically enriched category from example <a class="maruku-ref" href="#TopkAsATopologicallyEnrichedCategory"></a> may be thought of as a topologically enriched <a class="existingWikiWord" href="/nlab/show/copresheaf">copresheaf</a>, at least if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/small+category">small</a> (in that its <a class="existingWikiWord" href="/nlab/show/class">class</a> of objects is a proper <a class="existingWikiWord" href="/nlab/show/set">set</a>).</p> <p>Hence the category of topologically enriched functors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [\mathcal{C}, Top_{cg}] </annotation></semantics></math></div> <p>according to def. <a class="maruku-ref" href="#TopologicallyEnrichedFunctor"></a> may be thought of as the (<a class="existingWikiWord" href="/nlab/show/copresheaf">co-</a>)<a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> in the realm of topological enriched categories.</p> <p>A functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F \in [\mathcal{C}, Top_{cg}]</annotation></semantics></math> is equivalently</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>a</mi></msub><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">F_a \in Top_{cg}</annotation></semantics></math> for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \in Obj(\mathcal{C})</annotation></semantics></math>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>a</mi></msub><mo>×</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>F</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex"> F_a \times \mathcal{C}(a,b) \longrightarrow F_b </annotation></semantics></math></div> <p>for all pairs of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a,b \in Obj(\mathcal{C})</annotation></semantics></math></p> </li> </ol> <p>such that composition is respected, in the evident sense.</p> <p>For every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math>, there is a topologically enriched <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(c)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(c,-)</annotation></semantics></math> which sends objects to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex"> y(c)(d) = \mathcal{C}(c,d) \in Top_{cg} </annotation></semantics></math></div> <p>and whose action on morphisms is, under the above identification, just the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </div> <div class="num_prop" id="TopologicallyEnrichedCopresheavesHaveAllLimitsAndColimits"> <h6 id="proposition_49">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a>, def. <a class="maruku-ref" href="#TopEnrichedCategory"></a> then the <a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top_{cg}]</annotation></semantics></math> from example <a class="maruku-ref" href="#TopologicallyEnrichedFunctorsToTopk"></a> has all <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>, and they are computed objectwise:</p> <p>if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>I</mi><mo>⟶</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> F_\bullet \;\colon\; I \longrightarrow [\mathcal{C}, Top_{cg}] </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c\in \mathcal{C}</annotation></semantics></math> is any object, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex"> (\underset{\longleftarrow}{\lim}_i F_i)(c) \simeq \underset{\longleftarrow}{\lim}_i (F_i(c)) \;\;\in Top_{cg} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\underset{\longrightarrow}{\lim}_i F_i)(c) \simeq \underset{\longrightarrow}{\lim}_i (F_i(c)) \;\; \in Top_{cg} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_76">Proof</h6> <p>First consider the underlying diagram of functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>F</mi> <mi>i</mi> <mo>∘</mo></msubsup></mrow><annotation encoding="application/x-tex">F_i^\circ</annotation></semantics></math> where the topology on the <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> has been forgotten. Then one finds</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><msubsup><mi>F</mi> <mi>i</mi> <mo>∘</mo></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mo stretchy="false">(</mo><msubsup><mi>F</mi> <mi>i</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> (\underset{\longleftarrow}{\lim}_i F^\circ_i)(c) \simeq \underset{\longleftarrow}{\lim}_i (F^\circ_i(c)) \;\;\in Set </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><msubsup><mi>F</mi> <mi>i</mi> <mo>∘</mo></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><mo stretchy="false">(</mo><msubsup><mi>F</mi> <mi>i</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> (\underset{\longrightarrow}{\lim}_i F^\circ _i)(c) \simeq \underset{\longrightarrow}{\lim}_i (F^\circ_i(c)) \;\; \in Set </annotation></semantics></math></div> <p>by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of limits and colimits. (Given a morphism of diagrams then a unique compatible morphism between their limits or colimits, respectively, is induced as the universal factorization of the morphism of diagrams regarded as a cone or cocone, respectvely, over the codomain or domain diagram, respectively).</p> <p>Hence it only remains to see that equipped with topology, these limits and colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> become limits and colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>. That is just the statement of prop. <a class="maruku-ref" href="#DescriptionOfLimitsAndColimitsInTop"></a> with corollary <a class="maruku-ref" href="#kTopIsCoreflectiveSubcategory"></a>.</p> </div> <div class="num_defn" id="TensoringAndPoweringOfTopologicallyEnrichedCopresheaves"> <h6 id="definition_60">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a>, def. <a class="maruku-ref" href="#TopEnrichedCategory"></a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top_{cg}]</annotation></semantics></math> its category of topologically enriched copresheaves from example <a class="maruku-ref" href="#TopologicallyEnrichedFunctorsToTopk"></a>.</p> <ol> <li> <p>Define a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo><mo>×</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (-)\cdot(-) \;\colon\; [\mathcal{C}, Top_{cg}] \times Top_{cg} \longrightarrow [\mathcal{C}, Top_{cg}] </annotation></semantics></math></div> <p>by forming objectwise <a class="existingWikiWord" href="/nlab/show/cartesian+products">cartesian products</a> (hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/product+topological+spaces">product topological spaces</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⋅</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>c</mi><mo>↦</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \cdot X \;\colon\; c \mapsto F(c) \times X \,. </annotation></semantics></math></div> <p>This is called the <strong><a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C},Top_{cg}]</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#TensoringAndPoweringOfTopologicallyEnrichedCopresheaves"></a>).</p> </li> <li> <p>Define a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>×</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo><mo>⟶</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (-)^{(-)} \;\colon\; (Top_{cg})^{op} \times [\mathcal{C}, Top_{cg}] \longrightarrow [\mathcal{C}, Top_{cg}] </annotation></semantics></math></div> <p>by forming objectwise compactly generated <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> (def. <a class="maruku-ref" href="#CompactlyGeneratedMappingSpaces"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>X</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>c</mi><mo>↦</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mi>X</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^X \;\colon\; c \mapsto F(c)^X \,. </annotation></semantics></math></div> <p>This is called the <strong><a class="existingWikiWord" href="/nlab/show/powering">powering</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top_{cg}]</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>.</p> </li> </ol> <p>Analogously, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a pointed <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a>, def. <a class="maruku-ref" href="#TopEnrichedCategory"></a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top_{cg}^{\ast/}]</annotation></semantics></math> its category of pointed topologically enriched copresheaves from example <a class="maruku-ref" href="#TopologicallyEnrichedFunctorsToTopk"></a>, then:</p> <ol> <li> <p>Define a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">]</mo><mo>×</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>⟶</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (-)\wedge(-) \;\colon\; [\mathcal{C}, Top^{\ast/}_{cg}] \times Top^{\ast/}_{cg} \longrightarrow [\mathcal{C}, Top^{\ast/}_{cg}] </annotation></semantics></math></div> <p>by forming objectwise <a class="existingWikiWord" href="/nlab/show/smash+products">smash products</a> (def. <a class="maruku-ref" href="#SmashProductOfPointedObjects"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∧</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>c</mi><mo>↦</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>∧</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \wedge X \;\colon\; c \mapsto F(c) \wedge X \,. </annotation></semantics></math></div> <p>This is called the <strong>smash <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C},Top^{\ast/}_{cg}]</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top^{\ast/}_{cg}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#TensoringAndPoweringOfTopologicallyEnrichedCopresheaves"></a>).</p> </li> <li> <p>Define a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>×</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">]</mo><mo>⟶</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Maps(-,-)_\ast \;\colon\; Top^{\ast/}_{cg} \times [\mathcal{C}, Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C}, Top^{\ast/}_{cg}] </annotation></semantics></math></div> <p>by forming objectwise <a class="existingWikiWord" href="/nlab/show/pointed+mapping+spaces">pointed mapping spaces</a> (example <a class="maruku-ref" href="#PointedMappingSpace"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>X</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>c</mi><mo>↦</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^X \;\colon\; c \mapsto Maps(X,F(c))_\ast \,. </annotation></semantics></math></div> <p>This is called the <strong>pointed <a class="existingWikiWord" href="/nlab/show/powering">powering</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top_{cg}]</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>.</p> </li> </ol> </div> <p>There is a full blown <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a>. The following records a slightly simplified version which is all that is needed here:</p> <div class="num_prop" id="TopologicallyEnrichedYonedaLemma"> <h6 id="proposition_50">Proposition</h6> <p><strong>(topologically enriched Yoneda-lemma)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a>, def. <a class="maruku-ref" href="#TopEnrichedCategory"></a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top_{cg}]</annotation></semantics></math> for its category of topologically enriched (co-)presheaves, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c\in Obj(\mathcal{C})</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>k</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">y(c) = \mathcal{C}(c,-) \in [\mathcal{C}, Top_k]</annotation></semantics></math> for the topologically enriched functor that it represents, all according to example <a class="maruku-ref" href="#TopologicallyEnrichedFunctorsToTopk"></a>. Recall the <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>F</mi><mo>⋅</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(F,X) \mapsto F \cdot X</annotation></semantics></math> from def. <a class="maruku-ref" href="#TensoringAndPoweringOfTopologicallyEnrichedCopresheaves"></a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c\in Obj(\mathcal{C})</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">X \in Top_{cg}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F \in [\mathcal{C}, Top_{cg}]</annotation></semantics></math>, there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> between</p> <ol> <li> <p>morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>X</mi><mo>⟶</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">y(c) \cdot X \longrightarrow F</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top_{cg}]</annotation></semantics></math>;</p> </li> <li> <p>morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \longrightarrow F(c)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>.</p> </li> </ol> <p>In short:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>X</mi><mo>⟶</mo><mi>F</mi></mrow><mrow><mi>X</mi><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \frac{ y(c)\cdot X \longrightarrow F }{ X \longrightarrow F(c) } </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_77">Proof</h6> <p>Given a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>X</mi><mo>⟶</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\eta \colon y(c) \cdot X \longrightarrow F</annotation></semantics></math> consider its component</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta_c \;\colon\; \mathcal{C}(c,c)\times X \longrightarrow F(c) </annotation></semantics></math></div> <p>and restrict that to the identity morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>c</mi></msub><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">id_c \in \mathcal{C}(c,c)</annotation></semantics></math> in the first argument</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>c</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta_c(id_c,-) \;\colon\; X \longrightarrow F(c) \,. </annotation></semantics></math></div> <p>We claim that just this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>c</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_c(id_c,-)</annotation></semantics></math> already uniquely determines all components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta_d \;\colon\; \mathcal{C}(c,d)\times X \longrightarrow F(d) </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d \in Obj(\mathcal{C})</annotation></semantics></math>: By definition of the transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicallyEnrichedFunctor"></a>), the two functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>c</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> F(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \longrightarrow F(d)^{\mathcal{C}(c,c) \times X} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub><mo>∘</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> \eta_d \circ \mathcal{C}(c,-) \times X \;\colon\; \mathcal{C}(c,d) \longrightarrow F(d)^{\mathcal{C}(c,c) \times X} </annotation></semantics></math></div> <p>agree. This means (remark <a class="maruku-ref" href="#TopologicallyEnrichedNaturalTransformationIsTransformationOfUnderlyingFunctors"></a>) that they may be thought of jointly as a function with values in commuting squares in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> of this form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>c</mi></msub></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>d</mi></msub></mrow></munder></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> f \;\;\;\; \mapsto \;\;\;\; \array{ \mathcal{C}(c,c) \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ \mathcal{C}(c,d) \times X &\underset{\eta_d}{\longrightarrow}& F(d) } </annotation></semantics></math></div> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in \mathcal{C}(c,d)</annotation></semantics></math>, consider the restriction of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub><mo>∘</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> \eta_d \circ \mathcal{C}(c,f) \in F(d)^{\mathcal{C}(c,c) \times X} </annotation></semantics></math></div> <p>to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>c</mi></msub><mo>∈</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">id_c \in \mathcal{C}(c,c)</annotation></semantics></math>, hence restricting the above commuting squares to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mo stretchy="false">{</mo><msub><mi>id</mi> <mi>c</mi></msub><mo stretchy="false">}</mo><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>c</mi></msub></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">}</mo><mo>×</mo><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>d</mi></msub></mrow></munder></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> f \;\;\;\; \mapsto \;\;\;\; \array{ \{id_c\} \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ \{f\} \times X &\underset{\eta_d}{\longrightarrow}& F(d) } </annotation></semantics></math></div> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">\eta_d</annotation></semantics></math> is fixed to be the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>c</mi></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta_d(f,x) = F(f)\circ \eta_c(id_c,x) </annotation></semantics></math></div> <p>and this is a continuous function since all the operations it is built from are continuous.</p> <p>Conversely, given a continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \colon X \longrightarrow F(c)</annotation></semantics></math>, define for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>α</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta_d \colon (f,x) \mapsto F(f) \circ \alpha \,. </annotation></semantics></math></div> <p>Running the above analysis backwards shows that this determines a transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\eta \colon y(c)\times X \to F</annotation></semantics></math>.</p> </div> <div class="num_defn" id="GeneratingCofibrationsForProjectiveStructureOnFunctors"> <h6 id="definition_61">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> topologically enriched category, def. <a class="maruku-ref" href="#TopEnrichedCategory"></a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>I</mi> <mi>Top</mi> <mi>𝒞</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mrow><mo>{</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>}</mo></mrow> <mfrac linethickness="0"><mrow><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>,</mo></mrow></mrow><mrow><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></msub></mrow><annotation encoding="application/x-tex"> I_{Top}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c)\cdot (S^{n-1} \overset{\iota_n}{\longrightarrow} D^n) \right\}_{{n \in \mathbb{N},} \atop {c \in Obj(\mathcal{C})}} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Top</mi> <mi>𝒞</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mrow><mo>{</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><mo stretchy="false">)</mo><mo>}</mo></mrow> <mfrac linethickness="0"><mrow><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>,</mo></mrow></mrow><mrow><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></msub></mrow><annotation encoding="application/x-tex"> J_{Top}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c)\cdot (D^n \overset{(id, \delta_0)}{\longrightarrow} D^n \times I) \right\}_{{n \in \mathbb{N},} \atop {c \in Obj(\mathcal{C})}} </annotation></semantics></math></div> <p>for the sets of morphisms given by <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> (def. <a class="maruku-ref" href="#TensoringAndPoweringOfTopologicallyEnrichedCopresheaves"></a>) the representable functors (example <a class="maruku-ref" href="#TopologicallyEnrichedFunctorsToTopk"></a>) with the generating cofibrations (def.<a class="maruku-ref" href="#TopologicalGeneratingCofibrations"></a>) and acyclic generating cofibrations (def. <a class="maruku-ref" href="#TopologicalGeneratingAcyclicCofibrations"></a>), respectively, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> (theorem <a class="maruku-ref" href="#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"></a>).</p> <p>These are going to be called the <strong><a class="existingWikiWord" href="/nlab/show/generating+cofibrations">generating cofibrations</a></strong> and <strong>acyclic generating cofibrations</strong> for the <em>projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on topologically enriched functors</a></em> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p>Analgously, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a pointed topologically enriched category, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>I</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow> <mi>𝒞</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mrow><mo>{</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><msubsup><mi>S</mi> <mo>+</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>n</mi></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow></mover><msubsup><mi>D</mi> <mo>+</mo> <mi>n</mi></msubsup><mo stretchy="false">)</mo><mo>}</mo></mrow> <mfrac linethickness="0"><mrow><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>,</mo></mrow></mrow><mrow><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></msub></mrow><annotation encoding="application/x-tex"> I_{Top^{\ast/}}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c) \wedge (S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+) \right\}_{{n \in \mathbb{N},} \atop {c \in Obj(\mathcal{C})}} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow> <mi>𝒞</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mrow><mo>{</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><msubsup><mi>D</mi> <mo>+</mo> <mi>n</mi></msubsup><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow></mover><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><mo stretchy="false">)</mo><mo>}</mo></mrow> <mfrac linethickness="0"><mrow><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>,</mo></mrow></mrow><mrow><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></msub></mrow><annotation encoding="application/x-tex"> J_{Top^{\ast/}}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c) \wedge (D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+) \right\}_{{n \in \mathbb{N},} \atop {c \in Obj(\mathcal{C})}} </annotation></semantics></math></div> <p>for the analogous construction applied to the pointed generating (acyclic) cofibrations of def. <a class="maruku-ref" href="#GeneratingCofibrationsForPointedTopologicalSpaces"></a>.</p> </div> <div class="num_defn" id="ClassesOfMorphismsInTheProjectiveModelStructureOnTopEnrichedFunctors"> <h6 id="definition_62">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> (pointed) <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, def. <a class="maruku-ref" href="#TopEnrichedCategory"></a>, say that a morphism in the category of (pointed) topologically enriched copresheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top_{cg}]</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C},Top_{cg}^{\ast/}]</annotation></semantics></math>), example <a class="maruku-ref" href="#TopologicallyEnrichedFunctorsToTopk"></a>, hence a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> between topologically enriched functors, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\eta \colon F \to G</annotation></semantics></math> is</p> <ul> <li> <p>a <strong>projective weak equivalence</strong>, if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c\in Obj(\mathcal{C})</annotation></semantics></math> the component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_c \colon F(c) \to G(c)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> (def. <a class="maruku-ref" href="#WeakHomotopyEquivalenceOfTopologicalSpaces"></a>);</p> </li> <li> <p>a <strong>projective fibration</strong> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c\in Obj(\mathcal{C})</annotation></semantics></math> the component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>c</mi></msub><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_c \colon F(c) \to G(c)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a> (def. <a class="maruku-ref" href="#SerreFibration"></a>);</p> </li> <li> <p>a <strong>projective cofibration</strong> if it is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> (rmk. <a class="maruku-ref" href="#RetractsOfMorphisms"></a>) of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>I</mi> <mi>Top</mi> <mi>𝒞</mi></msubsup></mrow><annotation encoding="application/x-tex">I_{Top}^{\mathcal{C}}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> (def. <a class="maruku-ref" href="#TopologicalCCellComplex"></a>, def. <a class="maruku-ref" href="#GeneratingCofibrationsForProjectiveStructureOnFunctors"></a>).</p> </li> </ul> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [\mathcal{C}, (Top_{cg})_{Quillen}]_{proj} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [\mathcal{C}, (Top^{\ast/}_{cg})_{Quillen}]_{proj} </annotation></semantics></math></div> <p>for the categories of topologically enriched functors equipped with these classes of morphisms.</p> </div> <div class="num_theorem" id="ProjectiveModelStructureOnTopologicalFunctors"> <h6 id="theorem_5">Theorem</h6> <p>The classes of morphisms in def. <a class="maruku-ref" href="#ClassesOfMorphismsInTheProjectiveModelStructureOnTopEnrichedFunctors"></a> constitute a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top_{cg}]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}, Top^{\ast/}_{cg}]</annotation></semantics></math>, called the <strong><a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+enriched+functors">projective model structure on enriched functors</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [\mathcal{C}, (Top_{cg})_{Quillen}]_{proj} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>𝒞</mi><mo>,</mo><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [\mathcal{C}, (Top^{\ast/}_{cg})_{Quillen}]_{proj} </annotation></semantics></math></div> <p>These are <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a>, def. <a class="maruku-ref" href="#CofibrantlyGeneratedModelCategory"></a>, with set of generating (acyclic) cofibrations the sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>I</mi> <mi>Top</mi> <mi>𝒞</mi></msubsup></mrow><annotation encoding="application/x-tex">I_{Top}^{\mathcal{C}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Top</mi> <mi>𝒞</mi></msubsup></mrow><annotation encoding="application/x-tex">J_{Top}^{\mathcal{C}}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>I</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow> <mi>𝒞</mi></msubsup></mrow><annotation encoding="application/x-tex">I_{Top^{\ast/}}^{\mathcal{C}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow> <mi>𝒞</mi></msubsup></mrow><annotation encoding="application/x-tex">J_{Top^{\ast/}}^{\mathcal{C}}</annotation></semantics></math> from def. <a class="maruku-ref" href="#GeneratingCofibrationsForProjectiveStructureOnFunctors"></a>, respectively.</p> </div> <p>(<a href="#Piacenza91">Piacenza 91, theorem 5.4</a>)</p> <div class="proof"> <h6 id="proof_78">Proof</h6> <p>By prop. <a class="maruku-ref" href="#TopologicallyEnrichedCopresheavesHaveAllLimitsAndColimits"></a> the category has all limits and colimits, hence it remains to check the model structure</p> <p>But via the enriched Yoneda lemma (prop. <a class="maruku-ref" href="#TopologicallyEnrichedYonedaLemma"></a>) it follows that proving the model structure reduces objectwise to the proof of theorem <a class="maruku-ref" href="#TopQuillenModelStructure"></a>, theorem <a class="maruku-ref" href="#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"></a>. In particular, the technical lemmas <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a>, <a class="maruku-ref" href="#JTopRelativeCellComplexesAreWeakHomotopyEquivalences"></a> and <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a> generalize immediately to the present situation, with the evident small change of wording:</p> <p>For instance, the fact that a morphism of topologically enriched functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\eta \colon F \to G</annotation></semantics></math> that has the right lifting property against the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>I</mi> <mi>Top</mi> <mi>𝒞</mi></msubsup></mrow><annotation encoding="application/x-tex">I_{Top}^{\mathcal{C}}</annotation></semantics></math> is a projective weak equivalence, follows by noticing that for fixed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\eta \colon F \to G</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a> prop. <a class="maruku-ref" href="#TopologicallyEnrichedYonedaLemma"></a> gives a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> of commuting diagrams (and their fillers) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>F</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>⋅</mo><msub><mi>ι</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>η</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>G</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>η</mi> <mi>c</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ y(c) \cdot S^{n-1} &\longrightarrow& F \\ {}^{\mathllap{(id\cdot \iota_n)}}\downarrow && \downarrow^{\mathrlap{\eta}} \\ y(c) \cdot D^n &\longrightarrow& G } \right) \;\;\;\leftrightarrow\;\;\; \left( \array{ S^{n-1} &\longrightarrow& F(c) \\ \downarrow && \downarrow^{\mathrlap{\eta_c}} \\ D^n &\longrightarrow& G(c) } \right) \,, </annotation></semantics></math></div> <p>and hence the statement follows with part A) of the proof of lemma <a class="maruku-ref" href="#AcyclicSerreFibrationsAreTheJTopFibrations"></a>.</p> <p>With these three lemmas in hand, the remaining formal part of the proof goes through verbatim as <a href="#VerificationOfTopQuillen">above</a>: repeatedly use the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> (prop. <a class="maruku-ref" href="#SmallObjectArgument"></a>) and the <a class="existingWikiWord" href="/nlab/show/retract+argument">retract argument</a> (prop. <a class="maruku-ref" href="#RetractArgument"></a>) to establish the two <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a>. (While again the structure of a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> is evident.)</p> </div> <div class="num_example" id="PreExcisiveFunctors"> <h6 id="example_44">Example</h6> <p>Given examples <a class="maruku-ref" href="#TopkAsATopologicallyEnrichedCategory"></a> and <a class="maruku-ref" href="#TopologicallyEnrichedFunctorsToTopk"></a>, the next evident example of a pointed <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a> besides <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top^{\ast/}_{cg}</annotation></semantics></math> itself is the functor category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [Top_{cg}^{\ast/}, Top_{cg}^{\ast/}] \,. </annotation></semantics></math></div> <p>The only technical problem with this is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top^{\ast/}_{cg}</annotation></semantics></math> is not a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> (it has a <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a> of objects), which means that the existence of all limits and colimits via prop. <a class="maruku-ref" href="#TopologicallyEnrichedCopresheavesHaveAllLimitsAndColimits"></a> may (and does) fail.</p> <p>But so we just restrict to a small topologically enriched subcategory. A good choice is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mrow><mi>cg</mi><mo>,</mo><mi>fin</mi></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>↪</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> Top^{\ast/}_{cg,fin} \hookrightarrow Top^{\ast/}_{cg} </annotation></semantics></math></div> <p>of topological spaces homoemorphic to <a class="existingWikiWord" href="/nlab/show/finite+CW-complexes">finite CW-complexes</a>. The resulting projective model category (via theorem <a class="maruku-ref" href="#ProjectiveModelStructureOnTopologicalFunctors"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>Top</mi> <mrow><mi>cg</mi><mo>,</mo><mi>fin</mi></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [Top_{cg,fin}^{\ast/}\;,\; (Top^{\ast/}_{cg})_{Quillen}]_{proj} </annotation></semantics></math></div> <p>is also also known as the <strong>strict <a class="existingWikiWord" href="/nlab/show/model+structure+for+excisive+functors">model structure for excisive functors</a></strong>. (This terminology is the special case for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> of the terminology “<a class="existingWikiWord" href="/nlab/show/n-excisive+functors">n-excisive functors</a>” as used in “<a class="existingWikiWord" href="/nlab/show/Goodwillie+calculus">Goodwillie calculus</a>”, a homotopy-theoretic analog of <a class="existingWikiWord" href="/nlab/show/differential+calculus">differential calculus</a>.) After enlarging its class of weak equivalences while keeping the cofibrations fixed, this will become <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to a <a class="existingWikiWord" href="/nlab/show/model+structure+for+spectra">model structure for spectra</a>. This we discuss in <a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory+--+1-2">part 1.2</a>, in the section <a href="Introduction+to+Stable+homotopy+theory+--+1-2#OnPreExcisiveFunctors">on pre-excisive functors</a>.</p> </div> <p>One consequence of theorem <a class="maruku-ref" href="#ProjectiveModelStructureOnTopologicalFunctors"></a> is the model category theoretic incarnation of the theory of <em><a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a></em>.</p> <p>Observe that ordinary <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> (def. <a class="maruku-ref" href="#Limits"></a>) are equivalently characterized in terms of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a>:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/category">category</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>𝒞</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[I,\mathcal{C}]</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>. We may think of its objects as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>-shaped <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and of its morphisms as homomorphisms of these diagrams. There is a canonical functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mi>I</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mover><mo>⟶</mo><mrow></mrow></mover><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>𝒞</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> const_I \;\colon\; \mathcal{C} \overset{}{\longrightarrow} [I,\mathcal{C}] </annotation></semantics></math></div> <p>which sends each object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> to the diagram that is constant on this object. Inspection of the definition of the <a class="existingWikiWord" href="/nlab/show/universal+properties">universal properties</a> of <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> on one hand, and of <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> and <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> functors on the other hand, shows that</p> <ol> <li> <p>precisely when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> has all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>, then the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">const_I</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> functor, which is the operation of forming these colimits:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>𝒞</mi><mo stretchy="false">]</mo><munderover><mo>⊥</mo><munder><mo>⟵</mo><mrow><msub><mi>const</mi> <mi>I</mi></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>I</mi></msub></mrow></mover></munderover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> [I,\mathcal{C}] \underoverset {\underset{const_I}{\longleftarrow}} {\overset{\underset{\longrightarrow}{\lim}_I}{\longrightarrow}} {\bot} \mathcal{C} </annotation></semantics></math></div></li> <li> <p>precisely when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> has all <a class="existingWikiWord" href="/nlab/show/limits">limits</a> of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>, then the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">const_I</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> functor, which is the operation of forming these limits.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>𝒞</mi><mo stretchy="false">]</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>I</mi></msub></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>const</mi> <mi>I</mi></msub></mrow></mover></munderover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> [I,\mathcal{C}] \underoverset {\underset{\underset{\longleftarrow}{\lim}_I}{\longrightarrow}} {\overset{const_I}{\longleftarrow}} {\bot} \mathcal{C} </annotation></semantics></math></div></li> </ol> <div class="num_prop" id="ColimitIsLeftQuillenOfProjectiveModelStructureOnFunctors"> <h6 id="proposition_51">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a> (def. <a class="maruku-ref" href="#TopEnrichedCategory"></a>). Then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>I</mi></msub><mo>⊣</mo><msub><mi>const</mi> <mi>I</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\underset{\longrightarrow}{\lim}_I \dashv const_I)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><munderover><mo>⊥</mo><munder><mo>⟵</mo><mrow><msub><mi>const</mi> <mi>I</mi></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>I</mi></msub></mrow></mover></munderover><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> [I,(Top_{cg})_{Quillen}]_{proj} \underoverset {\underset{const_I}{\longleftarrow}} {\overset{\underset{\longrightarrow}{\lim}_I}{\longrightarrow}} {\bot} (Top_{cg})_{Quillen} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> (def. <a class="maruku-ref" href="#QuillenAdjunction"></a>) between the <a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+enriched+functors">projective model structure on topological functors</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>, from theorem <a class="maruku-ref" href="#ProjectiveModelStructureOnTopologicalFunctors"></a>, and the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> from theorem <a class="maruku-ref" href="#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces"></a>.</p> <p>Similarly, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a>, then for the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+pointed+topological+spaces">classical model structure on pointed topological spaces</a> (prop. <a class="maruku-ref" href="#ModelStructureOnSliceCategory"></a>, theorem <a class="maruku-ref" href="#CofibrantGenerationOfPointedTopologicalSpaces"></a>) the adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><munderover><mo>⊥</mo><munder><mo>⟵</mo><mi>const</mi></munder><mover><mo>⟶</mo><munder><mi>lim</mi><mo>⟶</mo></munder></mover></munderover><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> [I,(Top^{\ast/}_{cg})_{Quillen}]_{proj} \underoverset {\underset{const}{\longleftarrow}} {\overset{\underset{\longrightarrow}{\lim}}{\longrightarrow}} {\bot} (Top^{\ast/}_{cg})_{Quillen} </annotation></semantics></math></div> <p>is a Quillen adjunction.</p> </div> <div class="proof"> <h6 id="proof_79">Proof</h6> <p>Since the fibrations and weak equivalences in the projective model structure (def. <a class="maruku-ref" href="#ClassesOfMorphismsInTheProjectiveModelStructureOnTopEnrichedFunctors"></a>) on the functor category are objectwise those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top^{\ast/}_{cg})_{Quillen}</annotation></semantics></math>, respectively, it is immediate that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">const_I</annotation></semantics></math> preserves these. In particular it preserves fibrations and acyclic fibrations and so the claim follows (prop. <a class="maruku-ref" href="#ConditionsOnQuillenAdjunctionAreIndeedEquivalent"></a>).</p> </div> <div class="num_defn" id="LeftDerivedFunctorOfColimitFunctor"> <h6 id="definition_63">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a>)</strong></p> <p>In the situation of prop. <a class="maruku-ref" href="#ColimitIsLeftQuillenOfProjectiveModelStructureOnFunctors"></a> we say that the <a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a> (def. <a class="maruku-ref" href="#LeftAndRightDerivedFunctorsOnModelCategories"></a>) of the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> functor is the <strong><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>hocolim</mi> <mi>I</mi></msub><mo>≔</mo><mi>𝕃</mi><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>I</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>Top</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> hocolim_I \coloneqq \mathbb{L}\underset{\longrightarrow}{\lim}_I \;\colon\; Ho([I,Top]) \longrightarrow Ho(Top) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>hocolim</mi> <mi>I</mi></msub><mo>≔</mo><mi>𝕃</mi><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>I</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> hocolim_I \coloneqq \mathbb{L}\underset{\longrightarrow}{\lim}_I \;\colon\; Ho([I,Top^{\ast/}]) \longrightarrow Ho(Top^{\ast/}) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_33">Remark</h6> <p>Since every object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> and in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top^{\ast/}_{cg})_{Quillen}</annotation></semantics></math> is fibrant, the <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> of any diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math>, according to def. <a class="maruku-ref" href="#LeftDerivedFunctorOfColimitFunctor"></a>, is (up to <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>) the result of forming the ordinary <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of any <a class="existingWikiWord" href="/nlab/show/projectively+cofibrant+diagram">projectively cofibrant</a> replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>X</mi><mo stretchy="false">^</mo></mover> <mo>•</mo></msub><mover><mo>→</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>proj</mi></msub></mrow></mover><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\hat X_\bullet \overset{\in W_{proj}}{\to} X_\bullet</annotation></semantics></math>.</p> </div> <div class="num_example" id="ProjectiveModelStructureOnNSequencesOfTopologicalSpaces"> <h6 id="example_45">Example</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℕ</mi> <mo>≤</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{N}^{\leq}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> (def. <a class="maruku-ref" href="#PosetsWosetTosetsAndOrdinals"></a>) of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>, hence for the <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> (with at most one morphism from any given object to any other given object) that looks like</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℕ</mi> <mo>≤</mo></msup><mo>=</mo><mrow><mo>{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo>→</mo><mn>2</mn><mo>→</mo><mn>3</mn><mo>→</mo><mi>⋯</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{N}^{\leq} = \left\{ 0 \to 1 \to 2 \to 3 \to \cdots \right\} \,. </annotation></semantics></math></div> <p>Regard this as a <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a> with the, necessarily, <a class="existingWikiWord" href="/nlab/show/discrete+topological+space">discrete topology</a> on its <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a>.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/topologically+enriched+functor">topologically enriched functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℕ</mi> <mo>≤</mo></msup><mo>⟶</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex"> X_\bullet \;\colon\; \mathbb{N}^{\leq} \longrightarrow Top_{cg} </annotation></semantics></math></div> <p>is just a plain functor and is equivalently a sequence of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> (morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math>) of the form (also called a <em><a class="existingWikiWord" href="/nlab/show/cotower">cotower</a></em>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow></mover><msub><mi>X</mi> <mn>1</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><msub><mi>X</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover><msub><mi>X</mi> <mn>3</mn></msub><mo>⟶</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} X_3 \longrightarrow \cdots \,. </annotation></semantics></math></div> <p>It is immediate to check that those sequences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> which are cofibrant in the projective model structure (theorem <a class="maruku-ref" href="#ProjectiveModelStructureOnTopologicalFunctors"></a>) are precisely those for which</p> <ol> <li> <p>all component morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f_i</annotation></semantics></math> are cofibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top^{\ast/}_{cg})_{Quillen}</annotation></semantics></math>, respectively, hence <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> (remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>) of <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> inclusions (def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>);</p> </li> <li> <p>the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math>, and hence all other objects, are cofibrant, hence are <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> of <a class="existingWikiWord" href="/nlab/show/cell+complexes">cell complexes</a> (def. <a class="maruku-ref" href="#TopologicalCellComplex"></a>).</p> </li> </ol> </div> <p>By example <a class="maruku-ref" href="#ProjectiveModelStructureOnNSequencesOfTopologicalSpaces"></a> it is immediate that the operation of forming colimits sends projective (acyclic) cofibrations between sequences of topological spaces to (acyclic) cofibrations in the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+pointed+topological+spaces">classical model structure on pointed topological spaces</a>. On those projectively cofibrant sequences where every map is not just a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> inclusion, but a plain relative cell complex inclusion, more is true:</p> <div class="num_prop" id="PropertiesOfColimitOverSequencesOfRelativeCellComplexes"> <h6 id="proposition_52">Proposition</h6> <p>In the <a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+functors">projective model structures</a> on <a class="existingWikiWord" href="/nlab/show/cotowers">cotowers</a> in topological spaces, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>ℕ</mi> <mo>≤</mo></msup><mo>,</mo><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[\mathbb{N}^{\leq}, (Top_{cg})_{Quillen}]_{proj}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>ℕ</mi> <mo>≤</mo></msup><mo>,</mo><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[\mathbb{N}^{\leq}, (Top^{\ast/}_{cg})_{Quillen}]_{proj}</annotation></semantics></math> from def. <a class="maruku-ref" href="#ProjectiveModelStructureOnNSequencesOfTopologicalSpaces"></a>, the following holds:</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> functor preserves fibrations between sequences of <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> inclusions;</p> </li> <li> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/finite+category">finite category</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>ℕ</mi> <mo>≤</mo></msup><mo>,</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">D_\bullet(-) \colon I \to [\mathbb{N}^{\leq}, Top_{cg}]</annotation></semantics></math> be a finite <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of sequences of relative cell complexes. Then there is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>n</mi></msub><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi>W</mi> <mi>cl</mi></msub></mrow></mover><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>n</mi></msub><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \underset{\longrightarrow}{\lim}_{n} \left( \underset{\longleftarrow}{\lim}_i D_n(i) \right) \overset{\in W_{cl}}{\longrightarrow} \underset{\longleftarrow}{\lim}_i \left( \underset{\longrightarrow}{\lim}_{n} D_n(i) \right) </annotation></semantics></math></div> <p>from the colimit over the limit sequnce to the limit of the colimits of sequences.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_80">Proof</h6> <p>Regarding the first statement:</p> <p>Use that both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top_{cg})_{Quillen}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">(Top^{\ast/}_{cg})_{Quillen}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+categories">cofibrantly generated model categories</a> (theorem <a class="maruku-ref" href="#CofibrantGenerationOfPointedTopologicalSpaces"></a>) whose generating acyclic cofibrations have <a class="existingWikiWord" href="/nlab/show/compact+topological+spaces">compact topological spaces</a> as <a class="existingWikiWord" href="/nlab/show/domains">domains</a> and <a class="existingWikiWord" href="/nlab/show/codomains">codomains</a>. The colimit over a sequence of relative cell complexes (being a <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a>) yields another <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a>, and hence lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a> says that every morphism out of the domain or codomain of a generating acyclic cofibration into this colimit factors through a finite stage inclusion. Since a projective fibration is a degreewise fibration, we have the <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> at that finite stage, and hence also the lifting property against the morphisms of colimits.</p> <p>Regarding the second statement:</p> <p>This is a model category theoretic version of a standard fact of plain <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, which says that in the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of sets, <a href="commutativity+of+limits+and+colimits#FilteredColimitsCommuteWithFiniteLimits">filtered colimits commute with finite limits</a> in that there is an isomorphism of sets of the form which we have to prove is a weak homotopy equivalence of topological spaces. But now using that weak homotopy equivalences are detected by forming <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> (def. <a class="maruku-ref" href="#HomotopyGroupsOftopologicalSpaces"></a>), hence <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> out of <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/spheres">spheres</a> are <a class="existingWikiWord" href="/nlab/show/compact+topological+spaces">compact topological spaces</a>, lemma <a class="maruku-ref" href="#CompactSubsetsAreSmallInCellComplexes"></a> says that homming out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-spheres commutes over the colimits in question. Moreover, generally homming out of anything commutes over <a class="existingWikiWord" href="/nlab/show/limits">limits</a>, in particular <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> (every <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> is <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a> in the second variable). Therefore we find isomorphisms of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mrow><mo>(</mo><msup><mi>S</mi> <mi>q</mi></msup><mo>,</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>n</mi></msub><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>n</mi></msub><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mi>Hom</mi><mrow><mo>(</mo><msup><mi>S</mi> <mi>q</mi></msup><mo>,</mo><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mover><mo>⟶</mo><mo>∼</mo></mover><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>n</mi></msub><mi>Hom</mi><mrow><mo>(</mo><msup><mi>S</mi> <mi>q</mi></msup><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mo>≃</mo><mi>Hom</mi><mrow><mo>(</mo><msup><mi>S</mi> <mi>q</mi></msup><mo>,</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>i</mi></msub><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>n</mi></msub><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> Hom\left( S^q, \underset{\longrightarrow}{\lim}_{n} \left( \underset{\longleftarrow}{\lim}_i D_n(i) \right) \right) \simeq \underset{\longrightarrow}{\lim}_{n} \left( \underset{\longleftarrow}{\lim}_i Hom\left(S^q, D_n(i)\right) \right) \overset{\sim}{\longrightarrow} \underset{\longleftarrow}{\lim}_i \left( \underset{\longrightarrow}{\lim}_{n} Hom\left(S^q D_n(i)\right) \right) \simeq Hom\left( S^q, \underset{\longleftarrow}{\lim}_i \left( \underset{\longrightarrow}{\lim}_{n} D_n(i) \right) \right) </annotation></semantics></math></div> <p>and similarly for the <a class="existingWikiWord" href="/nlab/show/left+homotopies">left homotopies</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>q</mi></msup><mo>×</mo><mi>I</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(S^q \times I,-)</annotation></semantics></math> (and similarly for the pointed case). This implies the claimed isomorphism on homotopy groups.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h2 id="SimplicialHomotopyTheory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Groupoids II): Simplicial homotopy theory</h2> <p>With <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> and <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> we have seen two kinds of objects which support concepts of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, such as a concept of <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> between them (<a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a> on the one hand, and <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> on the other). In some sense these two cases are opposite extremes in the more general context of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> have homotopical structure (e.g. <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a>) in arbitrary high degree, i.e. they may be <a class="existingWikiWord" href="/nlab/show/homotopy+n-types">homotopy n-types</a> for arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, but they are fully <em>abelian</em> in that there is never any <a class="existingWikiWord" href="/nlab/show/nonabelian+group">nonabelian group</a> structure in a chain complex, not is there any non-trivial <a class="existingWikiWord" href="/nlab/show/action">action</a> of the homology groups of a chain complex on each other;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> have more general non-abelian structure, for every (<a class="existingWikiWord" href="/nlab/show/nonabelian+group">nonabelian</a>) <a class="existingWikiWord" href="/nlab/show/group">group</a> there is a groupoid which has this as its <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, but this fundamental group (in degree 1) is already the highest homotopical structure they carry, groupoids are necessarily <a class="existingWikiWord" href="/nlab/show/homotopy+1-types">homotopy 1-types</a>.</p> </li> </ul> <p>On the other hand, both groupoids and chain complexes naturally have incarnations in the joint context of <em><a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a></em>. We now discuss how their common joint generalization is given by those simplicial sets whose simplices have a sensible notion of composition and inverses, the <em><a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a></em>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>topological</mi></mrow><mrow><mi>spaces</mi></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mi>higher</mi></mrow><mrow><mi>path</mi></mrow></mfrac></mrow><mrow><mi>groupoid</mi></mrow></mfrac></mpadded></msup></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>groupoids</mi></mtd> <mtd><mover><mo>⟶</mo><mfrac linethickness="0"><mrow><mi>Grothendieck</mi></mrow><mrow><mi>nerve</mi></mrow></mfrac></mover></mtd> <mtd><mfrac linethickness="0"><mrow><mrow><mstyle mathvariant="bold"><mi>Kan</mi></mstyle><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>complexes</mi></mstyle></mrow></mrow><mrow><mrow><mo>≃</mo><mn>∞</mn><mo>−</mo><mi>groupoids</mi></mrow></mrow></mfrac></mtd> <mtd><mover><mo>⟵</mo><mfrac linethickness="0"><mrow><mi>Dold</mi><mo>−</mo><mi>Kan</mi></mrow><mrow><mi>correspondence</mi></mrow></mfrac></mover></mtd> <mtd><mfrac linethickness="0"><mrow><mi>chain</mi></mrow><mrow><mi>complexes</mi></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mfrac linethickness="0"><mrow><mi>included</mi></mrow><mrow><mi>in</mi></mrow></mfrac></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>simplicial</mi></mrow><mrow><mi>sets</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && {topological \atop spaces} \\ && \downarrow^{\mathrlap{{higher \atop path}\atop groupoid}} & \\ groupoids &\stackrel{{Grothendieck \atop nerve}}{\longrightarrow}& { {\mathbf{Kan}\;\mathbf{complexes}} \atop {\simeq \infty-groupoids} } &\stackrel{{Dold-Kan \atop correspondence}}{\longleftarrow}& {chain \atop complexes} \\ && \downarrow^{\mathrlap{included \atop in}} \\ && {simplicial \atop sets} } </annotation></semantics></math></div> <p>Kan complexes serve as a standard powerful model on which the complete formulation of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> (without geometry) may be formulated.</p> <h3 id="simplicial_sets">Simplicial sets</h3> <p>The concept of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is secretly well familiar already in basic <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>: it reflects just the abstract structure carried by the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complexes">singular simplicial complexes</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, as in the definition of <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> and <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a>.</p> <p>Conversely, every simplicial set may be <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometrically realized</a> as a topological space. These two <a class="existingWikiWord" href="/nlab/show/adjoint">adjoint</a> operations turn out to exhibit the homotopy theory of simplicial sets as being equivalent (<a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a>) to the homotopy theory of topological spaces. For some purposes, working in <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a> is preferable over working with topological homotopy theory.</p> <div class="num_defn" id="TopologicalSimplex"> <h6 id="definition_64">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/topological+simplex">topological simplex</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, the <strong><a href="simplex#TopologicalSimplex">topological n-simplex</a></strong> is, up to <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>, the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> whose underlying set is the subset</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup><mo>≔</mo><mo stretchy="false">{</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>∈</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">|</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></munderover><msub><mi>x</mi> <mi>i</mi></msub><mo>=</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mo>∀</mo><mi>i</mi><mo>.</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo stretchy="false">}</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \Delta^n \coloneqq \{ \vec x \in \mathbb{R}^{n+1} | \sum_{i = 0 }^n x_i = 1 \; and \; \forall i . x_i \geq 0 \} \subset \mathbb{R}^{n+1} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n+1}</annotation></semantics></math>, and whose topology is the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> induces from the canonical topology in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n+1}</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_46">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> this is the <a class="existingWikiWord" href="/nlab/show/point">point</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>0</mn></msup><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\Delta^0 = *</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> this is the standard <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></msup><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta^1 = [0,1]</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> this is the filled triangle.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math> this is the filled tetrahedron.</p> </div> <div class="num_defn" id="FaceInclusionInBarycentricCoords"> <h6 id="definition_65">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">n</mo><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\n \geq 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq k \leq n</annotation></semantics></math>, the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-face (inclusion)</strong> of the topological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex, def. <a class="maruku-ref" href="#TopologicalSimplex"></a>, is the subspace inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>k</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>↪</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \delta_k : \Delta^{n-1} \hookrightarrow \Delta^n </annotation></semantics></math></div> <p>induced under the coordinate presentation of def. <a class="maruku-ref" href="#TopologicalSimplex"></a>, by the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1} </annotation></semantics></math></div> <p>which “omits” the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th canonical coordinate:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mn>0</mn><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_n) \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_47">Example</h6> <p>The inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mn>0</mn></msub><mo>:</mo><msup><mi>Δ</mi> <mn>0</mn></msup><mo>→</mo><msup><mi>Δ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> \delta_0 : \Delta^0 \to \Delta^1 </annotation></semantics></math></div> <p>is the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>↪</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \{1\} \hookrightarrow [0,1] </annotation></semantics></math></div> <p>of the “right” end of the standard interval. The other inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mn>1</mn></msub><mo>:</mo><msup><mi>Δ</mi> <mn>0</mn></msup><mo>→</mo><msup><mi>Δ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> \delta_1 : \Delta^0 \to \Delta^1 </annotation></semantics></math></div> <p>is that of the “left” end <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>↪</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\{0\} \hookrightarrow [0,1]</annotation></semantics></math>.</p> </div> <p><img src="http://ncatlab.org/nlab/files/faceanddegeneracymaps.jpg" width="500" /></p> <p>(graphics taken from <a href="https://ncatlab.org/nlab/show/simplicial+set#Friedman08">Friedman 08</a>)</p> <div class="num_defn" id="DegeneracyProjectionsInBarycentricCoords"> <h6 id="definition_66">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq k \lt n</annotation></semantics></math> the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th degenerate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n)</annotation></semantics></math>-simplex (projection)</strong> is the surjective map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>k</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \sigma_k : \Delta^{n} \to \Delta^{n-1} </annotation></semantics></math></div> <p>induced under the barycentric coordinates of def. <a class="maruku-ref" href="#TopologicalSimplex"></a> under the surjection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{n+1} \to \mathbb{R}^n </annotation></semantics></math></div> <p>which sends</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>k</mi></msub><mo>+</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="SingularSimplex"> <h6 id="definition_67">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/singular+simplex">singular simplex</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, a <strong>singular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \sigma : \Delta^n \to X </annotation></semantics></math></div> <p>from the topological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex, def. <a class="maruku-ref" href="#TopologicalSimplex"></a>, to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>≔</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X) </annotation></semantics></math></div> <p>for the set of singular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p><img src="http://ncatlab.org/nlab/files/singularsimplices.jpg" width="500" /></p> <p>(graphics taken from <a href="https://ncatlab.org/nlab/show/simplicial+set#Friedman08">Friedman 08</a>)</p> <p>The sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(Sing X)_\bullet</annotation></semantics></math> here are closely related by an interlocking system of maps that make them form what is called a <em><a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a></em>, and as such the collection of these sets of singular simplices is called the <em><a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. We discuss the definition of simplicial sets now and then come back to this below in def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a>.</p> <p>Since the topological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^n</annotation></semantics></math> from def. <a class="maruku-ref" href="#TopologicalSimplex"></a> sit inside each other by the face inclusions of def. <a class="maruku-ref" href="#FaceInclusionInBarycentricCoords"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>k</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \delta_k : \Delta^{n-1} \to \Delta^{n} </annotation></semantics></math></div> <p>and project onto each other by the degeneracy maps, def. <a class="maruku-ref" href="#DegeneracyProjectionsInBarycentricCoords"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>k</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \sigma_k : \Delta^{n+1} \to \Delta^n </annotation></semantics></math></div> <p>we dually have functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>k</mi></msub><mo>≔</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><msub><mi>δ</mi> <mi>k</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> d_k \coloneqq Hom_{Top}(\delta_k, X) : (Sing X)_n \to (Sing X)_{n-1} </annotation></semantics></math></div> <p>that send each singular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex to its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-face and functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>k</mi></msub><mo>≔</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><msub><mi>σ</mi> <mi>k</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> s_k \coloneqq Hom_{Top}(\sigma_k,X) : (Sing X)_{n} \to (Sing X)_{n+1} </annotation></semantics></math></div> <p>that regard an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex as beign a degenerate (“thin”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-simplex. All these sets of simplices and face and degeneracy maps between them form the following structure.</p> <div class="num_defn" id="sSet"> <h6 id="definition_68">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>)</strong></p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is</p> <ul> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S_n \in Set</annotation></semantics></math> – the <strong>set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplices">simplices</a></strong>;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/injective+map">injective map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>i</mi></msub><mo>:</mo><mover><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>¯</mo></mover><mo>→</mo><mover><mi>n</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\delta_i : \overline{n-1} \to \overline{n}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/totally+ordered+sets">totally ordered sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>n</mi><mo stretchy="false">¯</mo></mover><mo>≔</mo><mo stretchy="false">{</mo><mn>0</mn><mo><</mo><mn>1</mn><mo><</mo><mi>⋯</mi><mo><</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\bar n \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}</annotation></semantics></math></p> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_i : S_{n} \to S_{n-1}</annotation></semantics></math> – the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th <strong>face map</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/surjective+map">surjective map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mo>:</mo><mover><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>¯</mo></mover><mo>→</mo><mover><mi>n</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\sigma_i : \overline{n+1} \to \bar n</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/totally+ordered+sets">totally ordered sets</a></p> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\sigma_i : S_{n} \to S_{n+1}</annotation></semantics></math> – the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th <strong>degeneracy map</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices;</p> </li> </ul> <p>such that these functions satisfy the following identities, called the_<a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a><em>:</em></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>d</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> d_i \circ d_j = d_{j-1} \circ d_i</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \lt j</annotation></semantics></math>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>s</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>s</mi> <mi>j</mi></msub><mo>∘</mo><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">s_i \circ s_j = s_j \circ s_{i-1}</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>></mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \gt j</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>s</mi> <mi>j</mi></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>s</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo><</mo><mi>j</mi></mtd></mtr> <mtr><mtd><mi>id</mi></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mi>j</mi><mspace width="thickmathspace"></mspace><mi>or</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mi>j</mi><mo>+</mo><mn>1</mn></mtd></mtr> <mtr><mtd><msub><mi>s</mi> <mi>j</mi></msub><mo>∘</mo><msub><mi>d</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mi>if</mi><mi>i</mi><mo>></mo><mi>j</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex">d_i \circ s_j = \left\{ \array{ s_{j-1} \circ d_i & if \; i \lt j \\ id & if \; i = j \; or \; i = j+1 \\ s_j \circ d_{i-1} & if i \gt j+1 } \right. </annotation></semantics></math></p> </li> </ol> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S, T</annotation></semantics></math> two simplicial sets, a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> of simplicial sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>T</mi></mrow><annotation encoding="application/x-tex">S \overset{f}{\longrightarrow} T</annotation></semantics></math> is for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>N</mi></msub><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow></mover><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_N \overset{f_n}{\longrightarrow} T_n</annotation></semantics></math> between sets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices, such that these functions are compatible with all the face and degeneracy maps.</p> <p>This defines a <a class="existingWikiWord" href="/nlab/show/category">category</a> (Def. <a class="maruku-ref" href="#Categories"></a>) <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>.</p> </div> <p>It is straightforward to check by explicit inspection that the evident injection and restriction maps between the sets of <a class="existingWikiWord" href="/nlab/show/singular+simplices">singular simplices</a> make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(Sing X)_\bullet</annotation></semantics></math> into a simplicial set. However for working with this, it is good to streamline a little:</p> <div class="num_defn" id="SimplexCategory"> <h6 id="definition_69">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a>)</strong></p> <p>The <strong><a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> on the free categories of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo>→</mo><mn>2</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} [0] & \coloneqq \{0\} \\ [1] & \coloneqq \{0 \to 1\} \\ [2] & \coloneqq \{0 \to 1 \to 2\} \\ \vdots \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_34">Remark</h6> <p>This is called the “simplex category” because we are to think of the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> as being the “<a class="existingWikiWord" href="/nlab/show/spine">spine</a>” of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a>. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> we think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">0 \to 1 \to 2</annotation></semantics></math> as the “spine” of the triangle. This becomes clear if we don’t just draw the morphisms that <em>generate</em> the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math>, but draw also all their composites. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> we have_</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\} \,. </annotation></semantics></math></div></div> <div class="num_prop" id="SimplicialSetsAsPresheavesOnTheSimplexCategory"> <h6 id="proposition_53">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> are <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> on the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a>)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> S \;\colon\; \Delta^{op} \to Set </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> (Example <a class="maruku-ref" href="#OppositeCategory"></a>) of the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> (Def. <a class="maruku-ref" href="#SimplexCategory"></a>) to the <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</a>, hence a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> (Example <a class="maruku-ref" href="#CategoryOfPresheaves"></a>), is canonically identified with a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, def. <a class="maruku-ref" href="#SimplicialSet"></a>.</p> <p>Via this identification, the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> (Def. <a class="maruku-ref" href="#sSet"></a>) is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> on the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sSet</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sSet \;=\; [\Delta^{op}, Set] \,. </annotation></semantics></math></div> <p>In particular this means that <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> is a <a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a> for <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> (Example <a class="maruku-ref" href="#ExamplesOfCosmoi"></a>), by Prop. <a class="maruku-ref" href="#PropertiesOfSheafToposes"></a>.</p> </div> <div class="proof"> <h6 id="proof_81">Proof</h6> <p>One checks by inspection that the <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a> characterize precisely the behaviour of the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta^{op}([n],[n+1])</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta^{op}([n],[n-1])</annotation></semantics></math>.</p> </div> <p>This makes the following evident:</p> <div class="num_example" id="StandardCosimplicialTopologicalSpace"> <h6 id="example_48">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/topological+simplices">topological simplices</a> from def. <a class="maruku-ref" href="#TopologicalSimplex"></a> arrange into a <em><a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial object</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a></em>, namely a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mo>•</mo></msup><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta^\bullet : \Delta \to Top \,. </annotation></semantics></math></div></div> <p>With this now the structure of a simplicial set on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(Sing X)_\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#SingularSimplex"></a>, is manifest: it is just the <em><a class="existingWikiWord" href="/nlab/show/nerve">nerve</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">\Delta^\bullet</annotation></semantics></math>, namely:</p> <div class="num_defn" id="SingularSimplicialComplex"> <h6 id="definition_70">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> its <strong><a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">simplicial set of singular simplicies</a></strong> (often called the <strong><a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a></strong>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> (Sing X)_\bullet : \Delta^{op} \to Set </annotation></semantics></math></div> <p>is given by composition of the functor from example <a class="maruku-ref" href="#StandardCosimplicialTopologicalSpace"></a> with the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Sing X) : [n] \mapsto Hom_{Top}( \Delta^n , X ) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_35">Remark</h6> <p>It turns out – this is the content of the <em><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</em> (<a href="model+structure+on+simplicial+sets#Quillen67">Quillen 67</a>) – that <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is entirely captured by its singular simplicial complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math>. Moreover, the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math> is a model for the same <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> as that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, but with the special property that it is canonically a <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a> – a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>. Better yet, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math> is itself already good cell complex, namely a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>. We come to this below.</p> </div> <h3 id="simplicial_homotopy">Simplicial homotopy</h3> <p>The concept of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> of morphisms between simplicial sets proceeds in direct analogy with that in <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>.</p> <div class="num_defn" id="LeftHomotopyOfSimplicialSets"> <h6 id="definition_71">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, def. <a class="maruku-ref" href="#SimplicialSet"></a>, its <em>simplicial <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></em> is the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X\times \Delta[1]</annotation></semantics></math> (formed in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>, Prop. <a class="maruku-ref" href="#SimplicialSetsAsPresheavesOnTheSimplexCategory "></a>).</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; f \Rightarrow g </annotation></semantics></math></div> <p>between two morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f,g\;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; X \times \Delta[1] \longrightarrow Y </annotation></semantics></math></div> <p>such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><msup><mi>Δ</mi> <mn>1</mn></msup></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>x</mi></msub><mo>,</mo><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \\ {}^{\mathllap{(id_X,d_1)}}\downarrow & \searrow^{\mathllap{f}} \\ X \times \Delta^1 &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id_x, d_0)}}\uparrow & \nearrow_{\mathllap{g}} \\ X } \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, def. <a class="maruku-ref" href="#SimplicialSet"></a>, its <em>simplicial <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></em> is the <a class="existingWikiWord" href="/nlab/show/function+complex">function complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{\Delta[1]}</annotation></semantics></math> (formed in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>, Prop. <a class="maruku-ref" href="#SimplicialSetsAsPresheavesOnTheSimplexCategory"></a>).</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; f \Rightarrow g </annotation></semantics></math></div> <p>between two morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f,g\;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><msup><mi>Y</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> \eta \colon X \longrightarrow Y^{\Delta[1]} </annotation></semantics></math></div> <p>such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msup><mi>Y</mi> <mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><msup><mi>Y</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>Y</mi> <mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Y \\ & {}^{\mathllap{f}}\nearrow & \uparrow^{\mathrlap{Y^{d_1}}} \\ X &\stackrel{\eta}{\longrightarrow}& Y^{\Delta[1]} \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{Y^{d_0}}} \\ && Y } \,. </annotation></semantics></math></div></div> <div class="num_prop" id="LeftHomotopyIsEquivalence"> <h6 id="proposition_54">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, def. <a class="maruku-ref" href="#KanComplexes"></a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, then left homotopy, def. <a class="maruku-ref" href="#LeftHomotopyOfSimplicialSets"></a>, regarded as a <a class="existingWikiWord" href="/nlab/show/relation">relation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>∼</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mi>f</mi><mover><mo>⇒</mo><mo>∃</mo></mover><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (f\sim g) \Leftrightarrow (f \stackrel{\exists}{\Rightarrow} g) </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/hom+set">hom set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{sSet}(X,Y)</annotation></semantics></math>, is an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a>.</p> </div> <div class="num_defn" id="HomotopyEquivalenceOfSimplicialSets"> <h6 id="definition_72">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> in <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>)</strong></p> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> is a left/right <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> if there exists a morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟵</mo><mi>Y</mi><mo lspace="verythinmathspace">:</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">X \longleftarrow Y \colon g</annotation></semantics></math> and left/right homotopies (def. <a class="maruku-ref" href="#LeftHomotopyOfSimplicialSets"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo>⇒</mo><msub><mi>id</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>f</mi><mo>∘</mo><mi>g</mi><mo>⇒</mo><msub><mi>id</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex"> g \circ f \Rightarrow id_X\,,\;\;\;\; f\circ g \Rightarrow id_Y </annotation></semantics></math></div></div> <p>The the basic invariants of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>/<a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> in <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a> are their <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a>, to which we turn now.</p> <p>Given that a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> is a special <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> that <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">behaves like</a> a combinatorial model for a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the <em>simplicial homotopy groups</em> of a Kan complex are accordingly the combinatorial analog of the <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>: instead of being maps from topological <a class="existingWikiWord" href="/nlab/show/spheres">spheres</a> modulo maps from topological disks, they are maps from the <a class="existingWikiWord" href="/nlab/show/boundary+of+a+simplex">boundary of a simplex</a> modulo those from the <a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> itself.</p> <p>Accordingly, the definition of the discussion of simplicial homotopy groups is essentially literally the same as that of <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of topological spaces. One technical difference is for instance that the definition of the group structure is slightly more non-immediate for simplicial homotopy groups than for topological homotopy groups (see below).</p> <div class="num_defn" id="UnderlyingSetsOfSimplicialHomotopyGroups"> <h6 id="definition_73">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, then its <strong>0th <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">simplicial homotopy group</a></strong> (or <strong>set of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a></strong>) is the set of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of vertices modulo the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><msub><mi>X</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \stackrel{(d_1,d_0)}{\longrightarrow} X_0 \times X_0</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>X</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0(X) \colon X_0/X_1 \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x \in X_0</annotation></semantics></math> a vertex and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math>, then the underlying <a class="existingWikiWord" href="/nlab/show/set">set</a> of the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">simplicial homotopy group</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> – denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math> – is, the set of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\alpha]</annotation></semantics></math> of morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \alpha \colon \Delta^n \to X </annotation></semantics></math></div> <p>from the simplicial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^n</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, such that these take the <a class="existingWikiWord" href="/nlab/show/boundary+of+a+simplex">boundary of the simplex</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, i.e. such that they fit into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>x</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>α</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \partial \Delta[n] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] &\stackrel{\alpha}{\longrightarrow}& X } \,, </annotation></semantics></math></div> <p>where two such maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>α</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha, \alpha'</annotation></semantics></math> are taken to be equivalent is they are related by a <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy">simplicial homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mi>α</mi></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msup></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>α</mi><mo>′</mo></mrow></msub></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Delta[n] \\ \downarrow^{i_0} & \searrow^{\alpha} \\ \Delta[n] \times \Delta[1] &\stackrel{\eta}{\longrightarrow}& X \\ \uparrow^{i_1} & \nearrow_{\alpha'} \\ \Delta[n] } </annotation></semantics></math></div> <p>that fixes the boundary in that it fits into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>x</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \partial \Delta[n] \times \Delta[1] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] \times \Delta[1] &\stackrel{\eta}{\longrightarrow}& X } \,. </annotation></semantics></math></div></div> <p>These sets are taken to be equipped with the following group structure.</p> <div class="num_defn" id="ProductOnSimplicialHomotopyGroups"> <h6 id="definition_74">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x\in X_0</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f,g \colon \Delta[n] \to X</annotation></semantics></math> two representatives of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math> as in def. <a class="maruku-ref" href="#UnderlyingSetsOfSimplicialHomotopyGroups"></a>, consider the following <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>≔</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>s</mi> <mn>0</mn></msub><mo>∘</mo><msub><mi>s</mi> <mn>0</mn></msub><mo>∘</mo><mi>⋯</mi><mo>∘</mo><msub><mi>s</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>for</mi><mspace width="thickmathspace"></mspace><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn></mtd></mtr> <mtr><mtd><mi>f</mi></mtd> <mtd><mi>for</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mi>g</mi></mtd> <mtd><mi>for</mi><mspace width="thickmathspace"></mspace><mi>i</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> v_i \coloneqq \left\{ \array{ s_0 \circ s_0 \circ \cdots \circ s_0 (x) & for \; 0 \leq i \leq n-2 \\ f & for \; i = n-1 \\ g & for \; i = n+1 } \right. </annotation></semantics></math></div> <p>This corresponds to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Lambda^{n+1}[n] \to X</annotation></semantics></math> from a <a class="existingWikiWord" href="/nlab/show/horn">horn</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. By the <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> this morphism has an <a class="existingWikiWord" href="/nlab/show/extension">extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> through the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>θ</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^{n+1}[n] & \longrightarrow & X \\ \downarrow & \nearrow_{\mathrlap{\theta}} \\ \Delta[n+1] } </annotation></semantics></math></div> <p>From the <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a> one finds that the boundary of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex arising as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th boundary piece <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi></mrow><annotation encoding="application/x-tex">d_n \theta</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is constant on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi><mo>=</mo><msub><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><mi>θ</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> d_i d_{n} \theta = d_{n-1} d_i \theta = x </annotation></semantics></math></div> <p>So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi></mrow><annotation encoding="application/x-tex">d_n \theta</annotation></semantics></math> represents an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math> and we define a product operation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo><mo>⋅</mo><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">]</mo><mo>≔</mo><mo stretchy="false">[</mo><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [f]\cdot [g] \coloneqq [d_n \theta] \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, p. 26</a>)</p> <div class="num_remark"> <h6 id="remark_36">Remark</h6> <p>All the degenerate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">v_{0 \leq i \leq n-2}</annotation></semantics></math> in def. <a class="maruku-ref" href="#ProductOnSimplicialHomotopyGroups"></a> are just there so that the gluing of the two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> to each other can be regarded as forming the boundary of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-simplex except for one face. By the Kan extension property that missing face exists, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mi>θ</mi></mrow><annotation encoding="application/x-tex">d_n \theta</annotation></semantics></math>. This is a choice of gluing composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>.</p> </div> <div class="num_lemma"> <h6 id="lemma_25">Lemma</h6> <p>The product on homotopy group elements in def. <a class="maruku-ref" href="#ProductOnSimplicialHomotopyGroups"></a> is well defined, in that it is independent of the choice of representatives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> and of the extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math>.</p> </div> <p>e.g. (<a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, lemma 7.1</a>)</p> <div class="num_lemma"> <h6 id="lemma_26">Lemma</h6> <p>The product operation in def. <a class="maruku-ref" href="#ProductOnSimplicialHomotopyGroups"></a> yields a <a class="existingWikiWord" href="/nlab/show/group">group</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(X,x)</annotation></semantics></math>, which is <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math>.</p> </div> <p>e.g. (<a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, theorem 7.2</a>)</p> <div class="num_remark"> <h6 id="remark_37">Remark</h6> <p>The first homotopy group, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math>, is also called the <em><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_defn" id="WeakHomotopyEquivalence"> <h6 id="definition_75">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>KanCplx</mi><mo>↪</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">X,Y \in KanCplx \hookrightarrow sSet</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>, then a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \colon X \longrightarrow Y </annotation></semantics></math></div> <p>is called a <strong><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></strong> if it induces <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on all <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a>, i.e. if</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(f) \colon \pi_0(X) \longrightarrow \pi_0(Y)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> of sets;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(f,x) \colon \pi_n(X,x) \longrightarrow \pi_n(Y,f(x))</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x\in X_0</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math>.</p> </li> </ol> </div> <h3 id="KanComplexes">Kan complexes</h3> <p>Recall the definition of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> from <a href="#SingularSimplicialSet">above</a>. Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mstyle mathvariant="bold"><mi>Δ</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>∈</mo><mi>Simp</mi><mi>Set</mi></mrow><annotation encoding="application/x-tex"> \Delta[n] = \mathbf{\Delta}( -, [n]) \in Simp Set </annotation></semantics></math></div> <p>be the standard simplicial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> in <a class="existingWikiWord" href="/nlab/show/SimpSet">SimpSet</a>.</p> <div class="num_defn" id="SimplicialHorn"> <h6 id="definition_76">Definition</h6> <p>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq i \leq n</annotation></semantics></math>, the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,i)</annotation></semantics></math>-horn</strong> or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,i)</annotation></semantics></math>-box</strong> is the subsimplicial set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \Lambda^i[n] \hookrightarrow \Delta[n] </annotation></semantics></math></div> <p>which is the <a class="existingWikiWord" href="/nlab/show/union">union</a> of all faces <em>except</em> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">i^{th}</annotation></semantics></math> one.</p> <p>This is called an <strong>outer horn</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k = 0</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k = n</annotation></semantics></math>. Otherwise it is an <strong>inner horn</strong>.</p> </div> <div class="num_remark"> <h6 id="remark_38">Remark</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> is a <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a>, <a class="existingWikiWord" href="/nlab/show/unions">unions</a> of <a class="existingWikiWord" href="/nlab/show/subobjects">subobjects</a> make sense and they are calculated objectwise, thus in this case dimensionwise. This way it becomes clear what the structure of a horn as a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Lambda^k[n]: \Delta^{op} \to Set</annotation></semantics></math> must therefore be: it takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[m]</annotation></semantics></math> to the collection of ordinal maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f: [m] \to [n]</annotation></semantics></math> which do not have the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in the image.</p> </div> <div class="num_example"> <h6 id="example_49">Example</h6> <p>The inner horn, def. <a class="maruku-ref" href="#SimplicialHorn"></a> of the <a class="existingWikiWord" href="/nlab/show/simplex">2-simplex</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>2</mn></msup><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow &\Downarrow& \searrow \\ 0 &&\to&& 2 } \right\} </annotation></semantics></math></div> <p>with <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><msup><mi>Δ</mi> <mn>2</mn></msup><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\partial \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\} </annotation></semantics></math></div> <p>looks like</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>1</mn> <mn>2</mn></msubsup><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Lambda^2_1 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\} \,. </annotation></semantics></math></div> <p>The two outer horns look like</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>0</mn> <mn>2</mn></msubsup><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\Lambda^2_0 = \left\{ \array{ && 1 \\ & \nearrow && \\ 0 &&\to&& 2 } \right\} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>2</mn> <mn>2</mn></msubsup><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\Lambda^2_2 = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\} </annotation></semantics></math></div> <p>respectively.</p> <p><img src="http://ncatlab.org/nlab/files/2horns.jpg" width="500" /></p> <p>(graphics taken from <a href="https://ncatlab.org/nlab/show/simplicial+set#Friedman08">Friedman 08</a>)</p> </div> <div class="num_defn" id="KanComplexe"> <h6 id="definition_77">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>)</strong></p> <p>A <em><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></em> is a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> that satisfies the <em>Kan condition</em>,</p> <ul> <li> <p>which says that all <a class="existingWikiWord" href="/nlab/show/horns">horns</a> of the simplicial set have <em>fillers</em>/extend to <a class="existingWikiWord" href="/nlab/show/simplices">simplices</a>;</p> </li> <li> <p>which means equivalently that the unique homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex">S \to pt</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/point">point</a> (the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>) is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>;</p> </li> <li> <p>which means equivalently that for all <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>pt</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^i[n] &\to& S \\ \downarrow && \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow && \\ \Delta[n] } </annotation></semantics></math></div> <p>there exists a diagonal morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>pt</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \\ \Delta[n] } </annotation></semantics></math></div> <p>completing this to a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a>;</p> </li> <li> <p>which in turn means equivalently that the map from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,i)</annotation></semantics></math>-horns is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>S</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>↠</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>S</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Delta[n], S]\, \twoheadrightarrow \,[\Lambda^i[n],S] \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_prop"> <h6 id="proposition_55">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, its <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(X)</annotation></semantics></math>, def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a>, is a Kan complex, def. <a class="maruku-ref" href="#KanComplexes"></a>.</p> </div> <div class="proof"> <h6 id="proof_82">Proof</h6> <p>The inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><msub><mrow><msup><mi>Λ</mi> <mi>n</mi></msup></mrow> <mi>Top</mi></msub></mrow> <mi>k</mi></msub><mo>↪</mo><msubsup><mi>Δ</mi> <mi>Top</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">{{\Lambda^n}_{Top}}_k \hookrightarrow \Delta^n_{Top}</annotation></semantics></math> of topological horns into topological simplices are <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a>, in that there are <a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Δ</mi> <mi>Top</mi> <mi>n</mi></msubsup><mo>→</mo><msub><mrow><msub><mrow><msup><mi>Λ</mi> <mi>n</mi></msup></mrow> <mi>Top</mi></msub></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k</annotation></semantics></math> given by “squashing” a topological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex onto parts of its boundary, such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mrow><msub><mrow><msup><mi>Λ</mi> <mi>n</mi></msup></mrow> <mi>Top</mi></msub></mrow> <mi>k</mi></msub><mo>→</mo><msubsup><mi>Δ</mi> <mi>Top</mi> <mi>n</mi></msubsup><mo>→</mo><msub><mrow><msub><mrow><msup><mi>Λ</mi> <mi>n</mi></msup></mrow> <mi>Top</mi></msub></mrow> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>Id</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ({{\Lambda^n}_{Top}}_k \to \Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k) = Id \,. </annotation></semantics></math></div> <p>Therefore the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta^n, \Pi(X)] \to [\Lambda^n_k,\Pi(X)]</annotation></semantics></math> is an epimorphism, since it is equal to to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Top</mi><mo stretchy="false">(</mo><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Top(\Delta^n, X) \to Top(\Lambda^n_k, X)</annotation></semantics></math> which has a right inverse <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Top</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Top(\Lambda^n_k, X) \to Top(\Delta^n, X)</annotation></semantics></math>.</p> </div> <p>More generally:</p> <div class="num_defn" id="KanFibration"> <h6 id="definition_78">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>)</strong></p> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>⟶</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\phi \colon S \longrightarrow T</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> is called a <em><a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a></em> if it has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> again all <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusions, def. <a class="maruku-ref" href="#Horn"></a>, hence if for every <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ϕ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>T</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^i[n] &\longrightarrow& S \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T } </annotation></semantics></math></div> <p>there exists a lift</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ϕ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>T</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Lambda^i[n] &\longrightarrow& S \\ \downarrow &\nearrow& \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T } \,. </annotation></semantics></math></div></div> <p>This is the simplicial incarnation of the concept of <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a> of topological spaces:</p> <div class="num_defn" id="SerreFibration"> <h6 id="definition_79">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a> if for all <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and for every <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ C \times I &\longrightarrow& Y } </annotation></semantics></math></div> <p>there exists a lift</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C &\longrightarrow& X \\ \downarrow &\nearrow& \downarrow^{\mathrlap{f}} \\ C \times I &\longrightarrow& Y } \,. </annotation></semantics></math></div></div> <div class="num_prop" id="SingDetextsAndReflectsFibrations"> <h6 id="proposition_56">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a>, precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Sing</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Sing</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(f) \colon Sing(X) \longrightarrow Sing(Y)</annotation></semantics></math> (def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a>) is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, def. <a class="maruku-ref" href="#KanFibration"></a>.</p> </div> <p>The proof uses the basic tool of <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> to which we get to below in prop. <a class="maruku-ref" href="#NerveAndRealizationAdjunction"></a>.</p> <div class="proof"> <h6 id="proof_83">Proof</h6> <p>First observe that the left <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> against all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↪</mo><mi>C</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">C \hookrightarrow C \times I</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> is equivalent to left lifting against <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">|</mo></mrow><mo>↪</mo><mrow><mo stretchy="false">|</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert \Lambda^i[n]\vert} \hookrightarrow {\vert \Delta[n]\vert}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusions. Then apply the <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>sSet</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Sing</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Top({\vert-\vert},-) \simeq sSet(-,Sing(-))</annotation></semantics></math>, given by the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> of prop. <a class="maruku-ref" href="#NerveAndRealizationAdjunction"></a> and example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>, to the lifting diagrams.</p> </div> <div class="num_lemma" id="PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence"> <h6 id="lemma_27">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p \colon X \longrightarrow Y</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, def. <a class="maruku-ref" href="#KanFibration"></a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f_1,f_2 \colon A \longrightarrow X</annotation></semantics></math> be two morphisms. If there is a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> (def. <a class="maruku-ref" href="#LeftHomotopyOfSimplicialSets"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>⇒</mo><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_1 \Rightarrow f_2</annotation></semantics></math> between these maps, then there is a fiberwise <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, def. <a class="maruku-ref" href="#HomotopyEquivalence"></a>, between the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> fibrations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>X</mi><mo>≃</mo><msubsup><mi>f</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>X</mi></mrow><annotation encoding="application/x-tex">f_1^\ast X \simeq f_2^\ast X</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, chapter I, lemma 10.6</a>)</p> <p>While <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> have the advantage of being purely combinatorial structures, the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> of any given <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a> is in general a huge simplicial set which does not lend itself to detailed inspection. The following is about small models.</p> <div class="num_defn" id="MinimalKanFibration"> <h6 id="definition_80">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>⟶</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\phi \colon S \longrightarrow T</annotation></semantics></math>, def. <a class="maruku-ref" href="#KanFibration"></a>, is called a <strong><a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a></strong> if for any two cells in the same fiber with the same <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> if they are homotopic relative their boundary, then they are already equal.</p> <p>More formally, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is minimal precisely if for every <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>h</mi></mover></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ϕ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>T</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (\partial \Delta[n]) \times \Delta[1] &\stackrel{p_1}{\longrightarrow}& \partial \Delta[n] \\ \downarrow && \downarrow \\ \Delta[n] \times \Delta[1] &\stackrel{h}{\longrightarrow}& S \\ \downarrow^{\mathrlap{p_1}} && \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T } </annotation></semantics></math></div> <p>then the two composites</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mover><munder><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow></mover></mover><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mover><mo>⟶</mo><mi>h</mi></mover><mi>S</mi></mrow><annotation encoding="application/x-tex"> \Delta[n] \stackrel{\overset{d_0}{\longrightarrow}}{\underset{d_1}{\longrightarrow}} \Delta[n] \times \Delta[1] \stackrel{h}{\longrightarrow} S </annotation></semantics></math></div> <p>are equal.</p> </div> <div class="num_prop" id="PullbackPreservesMinimalFibration"> <h6 id="proposition_57">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> (in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>) of a <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a>, along any morphism is again a mimimal Kan fibration.</p> </div> <p>… <a class="existingWikiWord" href="/nlab/show/anodyne+extensions">anodyne extensions</a>…</p> <p>(<a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, chapter I, section 4</a>, <a href="classical+model+structure+on+simplicial+sets#JoyalTierney05">Joyal-Tierney 05, section 31</a>)</p> <div class="num_prop" id="KanFibrationHasMinimalStrongDeformationRetract"> <h6 id="proposition_58">Proposition</h6> <p>For every <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, def. <a class="maruku-ref" href="#KanFibration"></a>, there exists a fiberwise <a class="existingWikiWord" href="/nlab/show/strong+deformation+retract">strong deformation retract</a> to a <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a>.</p> </div> <p>(e.g. <a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, chapter I, prop. 10.3</a>, <a href="classical+model+structure+on+simplicial+sets#JoyalTierney05">Joyal-Tierney 05, theorem 3.3.1, theorem 3.3.3</a>).</p> <div class="proof"> <h6 id="proof_idea">Proof idea</h6> <p>Choose representatives by <a class="existingWikiWord" href="/nlab/show/induction">induction</a>, use that in the induction step one needs lifts of <a class="existingWikiWord" href="/nlab/show/anodyne+extensions">anodyne extensions</a> against a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, which exist.</p> </div> <div class="num_lemma" id="FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso"> <h6 id="lemma_28">Lemma</h6> <p>A morphism between <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibrations">minimal Kan fibrations</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a>, which is fiberwise a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, def. <a class="maruku-ref" href="#HomotopyEquivalenceOfSimplicialSets"></a>, is already an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <p>(e.g. <a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, chapter I, lemma 10.4</a>)</p> <div class="proof"> <h6 id="proof_idea_2">Proof idea</h6> <p>Show the statement degreewise. In the <a class="existingWikiWord" href="/nlab/show/induction">induction</a> one needs to lift <a class="existingWikiWord" href="/nlab/show/anodyne+extensions">anodyne extensions</a> agains a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>.</p> </div> <div class="num_lemma" id="MinimalKanFibrationAreFiberBundles"> <h6 id="lemma_29">Lemma</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a>, over a <a class="existingWikiWord" href="/nlab/show/connected">connected</a> base is a simplicial <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, locally trivial over every simplex of the base.</p> </div> <p>(e.g. <a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, chapter I, corollary 10.8</a>)</p> <div class="proof"> <h6 id="proof_84">Proof</h6> <p>By assumption of the base being connected, the classifying maps for the fibers over any two vertices are connected by a <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> of <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a>, hence by lemma <a class="maruku-ref" href="#PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence"></a> the fibers are connected by <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> and then by prop. <a class="maruku-ref" href="#PullbackPreservesMinimalFibration"></a> and lemma <a class="maruku-ref" href="#FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso"></a> they are already isomorphic. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> for this <a class="existingWikiWord" href="/nlab/show/typical+fiber">typical fiber</a>.</p> <p>Moreover, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n] \to \Delta[0] \to \Delta[n]</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mi>id</mi></mover><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n] \stackrel{id}{\to} \Delta[n]</annotation></semantics></math> and so applying lemma <a class="maruku-ref" href="#PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence"></a> and prop. <a class="maruku-ref" href="#FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso"></a> once more yields that the fiber over each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\Delta[n]\times F</annotation></semantics></math>.</p> </div> <h4 id="GroupoidsAsKanComplexes">Groupoids as Kan complexes</h4> <div class="num_defn" id="Groupoid"> <h6 id="definition_81">Definition</h6> <p>A (<a class="existingWikiWord" href="/nlab/show/small+category">small</a>) <em><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> is</p> <ul> <li> <p>a pair of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>0</mn></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}_0 \in Set </annotation></semantics></math> (the set of <a class="existingWikiWord" href="/nlab/show/objects">objects</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}_1 \in Set</annotation></semantics></math> (the set of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>)</p> </li> <li> <p>equipped with <a class="existingWikiWord" href="/nlab/show/functions">functions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>𝒢</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow></msub><msub><mi>𝒢</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mo>∘</mo></mover></mtd> <mtd><msub><mi>𝒢</mi> <mn>1</mn></msub></mtd> <mtd><mover><mover><munder><mo>⟶</mo><mi>s</mi></munder><mover><mo>←</mo><mi>i</mi></mover></mover><mover><mo>⟶</mo><mi>t</mi></mover></mover></mtd> <mtd><msub><mi>𝒢</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 &\stackrel{\circ}{\longrightarrow}& \mathcal{G}_1 & \stackrel{\overset{t}{\longrightarrow}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\longrightarrow}}}& \mathcal{G}_0 }\,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> on the left is that over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>t</mi></mover><msub><mi>𝒢</mi> <mn>0</mn></msub><mover><mo>←</mo><mi>s</mi></mover><msub><mi>𝒢</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1</annotation></semantics></math>,</p> </li> </ul> <p>such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> takes values in <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a>;</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∘</mo><mi>i</mi><mo>=</mo><mi>s</mi><mo>∘</mo><mi>i</mi><mo>=</mo><msub><mi>id</mi> <mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> t \circ i = s \circ i = id_{\mathcal{G}_0}, \;\;\; </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math> defines a partial <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation which is <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unital</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(\mathcal{G}_0)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/identities">identities</a>; in particular</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s (g \circ f) = s(f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t (g \circ f) = t(g)</annotation></semantics></math>;</p> </li> <li> <p>every morphism has an <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> under this composition.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark_39">Remark</h6> <p>This data is visualized as follows. The set of morphisms is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>=</mo><mrow><mo>{</mo><msub><mi>ϕ</mi> <mn>0</mn></msub><mover><mo>→</mo><mi>k</mi></mover><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathcal{G}_1 = \left\{ \phi_0 \stackrel{k}{\to} \phi_1 \right\} </annotation></semantics></math></div> <p>and the set of pairs of composable morphisms is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>2</mn></msub><mo>≔</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><munder><mo>×</mo><mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow></munder><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>ϕ</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>ϕ</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>k</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>k</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>ϕ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{G}_2 \coloneqq \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 = \left\{ \array{ && \phi_1 \\ & {}^{\mathllap{k_1}}\nearrow && \searrow^{\mathrlap{k_2}} \\ \phi_0 && \stackrel{k_2 \circ k_1}{\to} && \phi_2 } \right\} \,. </annotation></semantics></math></div> <p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>,</mo><mo>∘</mo><mo lspace="verythinmathspace">:</mo><msub><mi>𝒢</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>𝒢</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1, p_2, \circ \colon \mathcal{G}_2 \to \mathcal{G}_1</annotation></semantics></math> are those which send, respectively, these triangular diagrams to the left morphism, or the right morphism, or the bottom morphism.</p> </div> <div class="num_example" id="SetAsGroupoid"> <h6 id="example_50">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/set">set</a>, it becomes a groupoid by taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to be the set of objects and adding only precisely the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> morphism from each object to itself</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>X</mi><munderover><mover><mo>⟵</mo><mi>id</mi></mover><munder><mo>⟶</mo><mi>id</mi></munder><mover><mo>⟶</mo><mi>id</mi></mover></munderover><mi>X</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( X \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longrightarrow}} { \overset{id}{\longleftarrow} } X \right) \,. </annotation></semantics></math></div></div> <div class="num_example" id="DeloopingGroupoid"> <h6 id="example_51">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group">group</a>, its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(\mathbf{B}G)_\bullet</annotation></semantics></math> has</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">(\mathbf{B}G)_0 = \ast</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">(\mathbf{B}G)_1 = G</annotation></semantics></math>.</p> </li> </ul> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> two groups, group homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">f \colon G \to K</annotation></semantics></math> are in <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> with groupoid homomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>f</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>→</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{B}f)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}K)_\bullet \,. </annotation></semantics></math></div> <p>In particular a <a class="existingWikiWord" href="/nlab/show/group+character">group character</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \colon G \to U(1)</annotation></semantics></math> is equivalently a groupoid homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>c</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>→</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{B}c)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}U(1))_\bullet \,. </annotation></semantics></math></div> <p>Here, for the time being, all groups are <a class="existingWikiWord" href="/nlab/show/discrete+groups">discrete groups</a>. Since the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> also has a standard structure of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, and since later for the discussion of Chern-Simons type theories this will be relevant, we will write from now on</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex"> \flat U(1) \in Grp </annotation></semantics></math></div> <p>to mean explicitly the <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> underlying the circle group. (Here “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>” denotes the “<a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a>”.)</p> </div> <div class="num_example" id="ActionGroupoid"> <h6 id="example_52">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X </annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/set">set</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\rho \colon X \times G \to X</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (a <a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a>), the <strong><a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a></strong> or <strong><a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is the groupoid</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ρ</mi></msub><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>X</mi><mo>×</mo><mi>G</mi><mover><munder><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munder><mover><mo>⟶</mo><mi>ρ</mi></mover></mover><mi>X</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> X//_\rho G = \left( X \times G \stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} X \right) </annotation></semantics></math></div> <p>with composition induced by the product in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Hence this is the groupoid whose objects are the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and where <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>g</mi></mover><msub><mi>x</mi> <mn>2</mn></msub><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> x_1 \stackrel{g}{\to} x_2 = \rho(x_1)(g) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x_1, x_2 \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math>.</p> </div> <p>As an important special case we have:</p> <div class="num_example" id="BGGroupoidAsActionGroupoid"> <h6 id="example_53">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/discrete">discrete</a> group and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> the trivial action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> (the singleton set), the corresponding <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> according to def. <a class="maruku-ref" href="#ActionGroupoid"></a> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> according to def. <a class="maruku-ref" href="#DeloopingGroupoid"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>=</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\ast //G)_\bullet = (\mathbf{B}G)_\bullet \,. </annotation></semantics></math></div> <p>Another canonical action is the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on itself by right multiplication. The corresponding action groupoid we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>≔</mo><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{E}G)_\bullet \coloneqq G//G \,. </annotation></semantics></math></div> <p>The constant map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">G \to \ast</annotation></semantics></math> induces a canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ G//G & \simeq & \mathbf{E}G \\ \downarrow && \downarrow \\ \ast //G & \simeq & \mathbf{B}G } \,. </annotation></semantics></math></div> <p>This is known as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>. See below in <a class="maruku-ref" href="#PullbackOfEGGroupoidAsHomotopyFiberProduct"></a> for more on this.</p> </div> <div class="num_example"> <h6 id="example_54">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/interval">interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is the groupoid with</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mn>0</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I_0 = \{a,b\}</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mn>1</mn></msub><mo>=</mo><mo stretchy="false">{</mo><msub><mi mathvariant="normal">id</mi> <mi>a</mi></msub><mo>,</mo><msub><mi mathvariant="normal">id</mi> <mi>b</mi></msub><mo>,</mo><mi>a</mi><mo>→</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I_1 = \{\mathrm{id}_a, \mathrm{id}_b, a \to b \}</annotation></semantics></math>.</li> </ul> </div> <div class="num_example" id="FundamentalGroupoid"> <h6 id="example_55">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, its <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(\Sigma)</annotation></semantics></math> is</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">\Pi_1(\Sigma)_0 = </annotation></semantics></math> points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">\Pi_1(\Sigma)_1 = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> modulo <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> that leaves the endpoints fixed.</li> </ul> </div> <div class="num_example" id="PathSpaceGroupoid"> <h6 id="example_56">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> any groupoid, there is the <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒢</mi> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{G}^I_\bullet</annotation></semantics></math> with</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒢</mi> <mn>0</mn> <mi>I</mi></msubsup><mo>=</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>ϕ</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>k</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>ϕ</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{G}^I_0 = \mathcal{G}_1 = \left\{ \array{ \phi_0 \\ \downarrow^{\mathrlap{k}} \\ \phi_1 } \right\}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒢</mi> <mn>1</mn> <mi>I</mi></msubsup><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{G}^I_1 = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting squares</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> = <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>ϕ</mi> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><msub><mover><mi>ϕ</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>k</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>k</mi><mo stretchy="false">˜</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>ϕ</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mover><mi>ϕ</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\{ \array{ \phi_0 &\stackrel{h_0}{\to}& \tilde \phi_0 \\ {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{\tilde k}} \\ \phi_1 &\stackrel{h_1}{\to}& \tilde \phi_1 } \right\} \,. </annotation></semantics></math></p> </li> </ul> <p>This comes with two canonical homomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒢</mi> <mo>•</mo> <mi>I</mi></msubsup><mover><munder><mo>⟶</mo><mrow><msub><mi>ev</mi> <mn>0</mn></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>ev</mi> <mn>1</mn></msub></mrow></mover></mover><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> \mathcal{G}^I_\bullet \stackrel{\overset{ev_1}{\longrightarrow}}{\underset{ev_0}{\longrightarrow}} \mathcal{G}_\bullet </annotation></semantics></math></div> <p>which are given by endpoint evaluation, hence which send such a commuting square to either its top or its bottom hirizontal component.</p> </div> <div class="num_defn"> <h6 id="definition_82">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>g</mi> <mo>•</mo></msub><mo>:</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>𝒦</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">f_\bullet, g_\bullet : \mathcal{G}_\bullet \to \mathcal{K}_\bullet</annotation></semantics></math> two morphisms between groupoids, a <em><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow g</annotation></semantics></math> (a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>) is a homomorphism of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mo>•</mo></msub><mo>:</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><mo>→</mo><msubsup><mi>𝒦</mi> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">\eta_\bullet : \mathcal{G}_\bullet \to \mathcal{K}^I_\bullet</annotation></semantics></math> (with <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> the <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒦</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{K}_\bullet</annotation></semantics></math> as in example <a class="maruku-ref" href="#PathSpaceGroupoid"></a>) such that it fits into the diagram as depicted here on the right:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝒢</mi></mtd> <mtd><msup><mo>⇓</mo> <mpadded width="0"><mi>η</mi></mpadded></msup></mtd> <mtd><mi>𝒦</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow></mpadded></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>𝒦</mi> <mo>•</mo></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mo>•</mo></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>𝒢</mi> <mo>•</mo></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>η</mi> <mo>•</mo></msub></mrow></mover></mtd> <mtd><msubsup><mi>𝒦</mi> <mo>•</mo> <mi>I</mi></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>𝒦</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow^{\mathrlap{f_\bullet}} \\ \mathcal{G} &\Downarrow^{\mathrlap{\eta}}& \mathcal{K} \\ & \searrow \nearrow_{\mathrlap{g_\bullet}} } \;\;\;\; \coloneqq \;\;\;\; \array{ && \mathcal{K}_\bullet \\ & {}^{\mathllap{f_\bullet}}\nearrow & \uparrow^{\mathrlap{(ev_1)_\bullet}} \\ \mathcal{G}_\bullet &\stackrel{\eta_\bullet}{\to}& \mathcal{K}^I_\bullet \\ & {}_{\mathllap{g_\bullet}}\searrow & \downarrow^{\mathrlap{(ev_0)_\bullet}} \\ && \mathcal{K} } \,. </annotation></semantics></math></div></div> <div class="num_defn" id="GroupoidsAsHomotopy1Types"> <h6 id="definition_notation">Definition (Notation)</h6> <p>Here and in the following, the convention is that we write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> (with the subscript decoration) when we regard groupoids with just homomorphisms (<a class="existingWikiWord" href="/nlab/show/functors">functors</a>) between them,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math> (without the subscript decoration) when we regard groupoids with homomorphisms (<a class="existingWikiWord" href="/nlab/show/functors">functors</a>) between them and <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> (<a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a>) between these</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mi>g</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow^{\mathrlap{f}} \\ X &\Downarrow& Y \\ & \searrow \nearrow_{g} } \,. </annotation></semantics></math></div></li> </ul> <p>The unbulleted version of groupoids are also called <em><a class="existingWikiWord" href="/nlab/show/homotopy+1-types">homotopy 1-types</a></em> (or often just their <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>-<a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> are called this way.) Below we generalize this to arbitrary homotopy types (def. <a class="maruku-ref" href="#KanComplexesAsHomotopyTypes"></a>).</p> </div> <div class="num_example" id="MappingGroupoid"> <h6 id="example_57">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math> two groupoids, the <a class="existingWikiWord" href="/nlab/show/internal+hom">mapping groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,Y]</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Y^X</annotation></semantics></math> is</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">[X,Y]_0 = </annotation></semantics></math> homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">]</mo> <mn>1</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">[X,Y]_1 = </annotation></semantics></math> homotopies between such.</li> </ul> </div> <div class="num_defn" id="HomotopyEquivalenceOfGroupoids"> <h6 id="definition_83">Definition</h6> <p>A (<a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy-</a>) <em><a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a></em> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mo>→</mo><mi>𝒦</mi></mrow><annotation encoding="application/x-tex">\mathcal{G} \to \mathcal{K}</annotation></semantics></math> which has a left and right <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> up to <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>.</p> </div> <div class="num_example" id="BZIsPiSOne"> <h6 id="example_58">Example</h6> <p>The map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mover><mo>→</mo><mrow></mrow></mover><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}\mathbb{Z} \stackrel{}{\to} \Pi(S^1) </annotation></semantics></math></div> <p>which picks any point and sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{Z}</annotation></semantics></math> to the loop based at that point which winds around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times, is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a>.</p> </div> <div class="num_prop" id="DiscreteGroupoidIsDijointUnioonOfDeloopings"> <h6 id="proposition_59">Proposition</h6> <p>Assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> in the ambient <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>, every groupoid is equivalent to a disjoint union of <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoids, example <a class="maruku-ref" href="#DeloopingGroupoid"></a> – a <em><a class="existingWikiWord" href="/nlab/show/skeleton">skeleton</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_40">Remark</h6> <p>The statement of prop. <a class="maruku-ref" href="#DiscreteGroupoidIsDijointUnioonOfDeloopings"></a> becomes false as when we pass to groupoids that are equipped with <a class="existingWikiWord" href="/nlab/show/geometry">geometric</a> structure. This is the reason why for discrete geometry all <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons</a>-type field theories (namely <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a>-type theories) fundamentally involve just groups (and higher groups), while for nontrivial geometry there are genuine groupoid theories, for instance the <a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-models">AKSZ sigma-models</a>. But even so, <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> is usefully discussed in terms of groupoid technology, in particular since the choice of equivalence in prop. <a class="maruku-ref" href="#DiscreteGroupoidIsDijointUnioonOfDeloopings"></a> is not canonical.</p> </div> <div class="num_defn" id="HomotopyFiberProductOfGroupoids"> <h6 id="definition_84">Definition</h6> <p>Given two morphisms of groupoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>B</mi><mover><mo>←</mo><mi>g</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\rightarrow} B \stackrel{g}{\leftarrow} Y</annotation></semantics></math> their <em><a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><munder><mo>×</mo><mi>B</mi></munder><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><munder><mo>→</mo><mi>g</mi></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X \underset{B}{\times} Y &\stackrel{}{\to}& X \\ \downarrow &\swArrow& \downarrow^{\mathrlap{f}} \\ Y &\underset{g}{\to}& B } </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mo>•</mo></msub><munder><mo>×</mo><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></munder><msubsup><mi>B</mi> <mo>•</mo> <mi>I</mi></msubsup><munder><mo>×</mo><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></munder><msub><mi>Y</mi> <mo>•</mo></msub></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msubsup><mi>B</mi> <mo>•</mo> <mi>I</mi></msubsup></mtd> <mtd><munder><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></munder></mtd> <mtd><msub><mi>B</mi> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>Y</mi> <mo>•</mo></msub></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow></munder></mtd> <mtd><msub><mi>B</mi> <mo>•</mo></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X_\bullet \underset{B_\bullet}{\times} B^I_\bullet \underset{B_\bullet}{\times} Y_\bullet &\to& &\to& X_\bullet \\ \downarrow && && \downarrow^{\mathrlap{f_\bullet}} \\ && B^I_\bullet &\underset{(ev_0)_\bullet}{\to}& B_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ Y_\bullet &\underset{g_\bullet}{\to}& B_\bullet } \,, </annotation></semantics></math></div> <p>hence the ordinary iterated <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> over the <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> groupoid, as indicated.</p> </div> <div class="num_remark" id="FiberProductsOfGroupoidsComponentwise"> <h6 id="remark_41">Remark</h6> <p>An ordinary fiber product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><munder><mo>×</mo><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></munder><msub><mi>Y</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet \underset{B_\bullet}{\times}Y_\bullet</annotation></semantics></math> of groupoids is given simply by the fiber product of the underlying sets of objects and morphisms:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><munder><mo>×</mo><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></munder><msub><mi>Y</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>=</mo><msub><mi>X</mi> <mi>i</mi></msub><munder><mo>×</mo><mrow><msub><mi>B</mi> <mi>i</mi></msub></mrow></munder><msub><mi>Y</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X_\bullet \underset{B_\bullet}{\times}Y_\bullet)_i = X_i \underset{B_i}{\times} Y_i \,. </annotation></semantics></math></div></div> <div class="num_example" id="PullbackOfEGGroupoidAsHomotopyFiberProduct"> <h6 id="example_59">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group">group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> a map into its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>, the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a> of example <a class="maruku-ref" href="#BGGroupoidAsActionGroupoid"></a> is equivalently the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the point over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>≃</mo><mi>X</mi><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> P \simeq X \underset{\mathbf{B}G}{\times} \ast \,. </annotation></semantics></math></div> <p>Namely both squares in the following diagram are pullback squares</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mo>*</mo> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msubsup><mo stretchy="false">)</mo> <mo>•</mo> <mi>I</mi></msubsup></mtd> <mtd><munder><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></munder></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mo>•</mo></msub></mtd> <mtd><munder><mo>→</mo><mrow></mrow></munder></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P &\to& \mathbf{E}G &\to& \ast_\bullet \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& (\mathbf{B}G)_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ X_\bullet &\underset{}{\to}& (\mathbf{B}G)_\bullet } \,. </annotation></semantics></math></div> <p>(This is the first example of the more general phenomenon of <a class="existingWikiWord" href="/nlab/show/universal+principal+infinity-bundles">universal principal infinity-bundles</a>.)</p> </div> <div class="num_example" id="LoopSpaceGroupoid"> <h6 id="example_60">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\ast \to X</annotation></semantics></math> a point in it, we call</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>≔</mo><mo>*</mo><munder><mo>×</mo><mi>X</mi></munder><mo>*</mo></mrow><annotation encoding="application/x-tex"> \Omega X \coloneqq \ast \underset{X}{\times} \ast </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a group and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid from example <a class="maruku-ref" href="#DeloopingGroupoid"></a>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>≃</mo><mi>Ω</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>=</mo><mo>*</mo><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G \simeq \Omega \mathbf{B}G = \ast \underset{\mathbf{B}G}{\times} \ast \,. </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> of its own <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>, as it should be.</p> </div> <div class="proof"> <h6 id="proof_85">Proof</h6> <p>We are to compute the ordinary limiting cone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mrow></munder><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>I</mi></msup><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mrow></munder><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast \underset{\mathbf{B}G_\bullet}{\times} (\mathbf{B}G^I)_\bullet \underset{\mathbf{B}G_\bullet}{\times} \ast</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msubsup><mo stretchy="false">)</mo> <mo>•</mo> <mi>I</mi></msubsup></mtd> <mtd><munder><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></munder></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>→</mo><mrow></mrow></munder></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ &\to& &\to& \ast \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& \mathbf{B}G_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ \ast &\underset{}{\to}& \mathbf{B}G_\bullet } \,, </annotation></semantics></math></div> <p>In the middle we have the groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msubsup><mo stretchy="false">)</mo> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{B}G)^I_\bullet</annotation></semantics></math> whose objects are elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and whose morphisms starting at some element are labeled by pairs of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>2</mn></msub><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">h_1, h_2 \in G</annotation></semantics></math> and end at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mn>1</mn></msub><mo>⋅</mo><mi>g</mi><mo>⋅</mo><msub><mi>h</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">h_1 \cdot g \cdot h_2</annotation></semantics></math>. Using remark <a class="maruku-ref" href="#FiberProductsOfGroupoidsComponentwise"></a> the limiting cone is seen to precisely pick those morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub><msubsup><mo stretchy="false">)</mo> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{B}G_\bullet)^I_\bullet</annotation></semantics></math> such that these two elements are constant on the neutral element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>h</mi> <mn>2</mn></msub><mo>=</mo><mi>e</mi><mo>=</mo><msub><mi>id</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">h_1 = h_2 = e = id_{\ast}</annotation></semantics></math>, hence it produces just the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> regarded as a groupoid with only identity morphisms, as in example <a class="maruku-ref" href="#SetAsGroupoid"></a>.</p> </div> <div class="num_prop" id="FreeLoopSpaceOfGroupoid"> <h6 id="proposition_60">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>X</mi><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></munder><mi>X</mi></mrow><annotation encoding="application/x-tex"> [\Pi(S^1), X] \simeq X \underset{[\Pi(S^0), X]}{\times}X </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_86">Proof</h6> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\Pi_1(S^0) \simeq \ast \coprod \ast</annotation></semantics></math>. Therefore the <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><msubsup><mo stretchy="false">]</mo> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">[\Pi(S^0), X_\bullet]^I_\bullet</annotation></semantics></math> has</p> <ul> <li> <p>objects are pairs of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math>;</p> </li> <li> <p>morphisms are commuting squares of such.</p> </li> </ul> <p>Now the fiber product in def. <a class="maruku-ref" href="#HomotopyFiberProductOfGroupoids"></a> picks in there those pairs of morphisms for which both start at the same object, and both end at the same object. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><msub><mo stretchy="false">]</mo> <mo>•</mo></msub></mrow></munder><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><msubsup><mo stretchy="false">]</mo> <mo>•</mo> <mi>I</mi></msubsup><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><msub><mo stretchy="false">]</mo> <mo>•</mo></msub></mrow></munder><mi>X</mi></mrow><annotation encoding="application/x-tex">X_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} [\Pi(S^0), X_\bullet]^I_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} X</annotation></semantics></math> is the groupoid whose</p> <ul> <li> <p>objects are diagrams in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>x</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><mo>↗</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow \\ x_0 && x_1 \\ & \searrow \nearrow } </annotation></semantics></math></div></li> <li> <p>morphism are cylinder-diagrams over these.</p> </li> </ul> <p>One finds along the lines of example <a class="maruku-ref" href="#BZIsPiSOne"></a> that this is equivalent to maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(S^1)</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> and homotopies between these.</p> </div> <div class="num_remark"> <h6 id="remark_42">Remark</h6> <p>Even though all these models of the circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(S^1)</annotation></semantics></math> are equivalent, below the special appearance of the circle in the proof of prop. <a class="maruku-ref" href="#FreeLoopSpaceOfGroupoid"></a> as the combination of two semi-circles will be important for the following proofs. As we see in a moment, this is the natural way in which the circle appears as the composition of an <a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a> with a <a class="existingWikiWord" href="/nlab/show/coevaluation+map">coevaluation map</a>.</p> </div> <div class="num_example" id="AdjointActionGroupoidFromFreeLoopSpaceObject"> <h6 id="example_61">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, the <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a> of its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">G//_{ad} G</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a>, def. <a class="maruku-ref" href="#ActionGroupoid"></a>, of the <a class="existingWikiWord" href="/nlab/show/adjoint+action">adjoint action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on itself:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Pi(S^1), \mathbf{B}G] \simeq G//_{ad} G \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_62">Example</h6> <p>For an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\flat U(1)</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≃</mo><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Pi(S^1), \mathbf{B}\flat U(1)] \simeq \flat U(1)//_{ad} \flat U(1) \simeq (\flat U(1)) \times (\mathbf{B}\flat U(1)) \,. </annotation></semantics></math></div></div> <div class="num_example" id="GroupCharacterAsClassFunctionByFreeLoopSpace"> <h6 id="example_63">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \colon G \to \flat U(1)</annotation></semantics></math> be a group homomorphism, hence a <a class="existingWikiWord" href="/nlab/show/group+character">group character</a>. By example <a class="maruku-ref" href="#DeloopingGroupoid"></a> this has a <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> to a groupoid homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>c</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}c \;\colon\; \mathbf{B}G \to \mathbf{B}\flat U(1) \,. </annotation></semantics></math></div> <p>Under the <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a> construction this becomes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>c</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [\Pi(S^1), \mathbf{B}c] \;\colon\; [\Pi(S^1), \mathbf{B}G] \to [\Pi(S^1), \mathbf{B}\flat U(1)] </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>c</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mi>G</mi><mo>→</mo><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Pi(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to \flat U(1) \times \mathbf{B}U(1) \,. </annotation></semantics></math></div> <p>So by postcomposing with the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> on the first factor we recover from the general <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of groupoids the statement that a group character is a <a class="existingWikiWord" href="/nlab/show/class+function">class function</a> on <a class="existingWikiWord" href="/nlab/show/conjugacy+classes">conjugacy classes</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>c</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mi>G</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Pi(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to U(1) \,. </annotation></semantics></math></div></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_defn" id="NerveOfGroupoid"> <h6 id="definition_85">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, def. <a class="maruku-ref" href="#Groupoid"></a>, its <em><a class="existingWikiWord" href="/nlab/show/simplicial+nerve">simplicial nerve</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">N(\mathcal{G}_\bullet)_\bullet</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>≔</mo><msubsup><mi>𝒢</mi> <mn>1</mn> <mrow><msubsup><mo>×</mo> <mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow> <mi>n</mi></msubsup></mrow></msubsup></mrow><annotation encoding="application/x-tex"> N(\mathcal{G}_\bullet)_n \coloneqq \mathcal{G}_1^{\times_{\mathcal{G}_0}^n} </annotation></semantics></math></div> <p>the set of sequences of composable morphisms of length <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>;</p> <p>with face maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>k</mi></msub><mo lspace="verythinmathspace">:</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> d_k \colon N(\mathcal{G}_\bullet)_{n+1} \to N(\mathcal{G}_\bullet)_{n} </annotation></semantics></math></div> <p>being,</p> <ul> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> the functions that remembers the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th object;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math></p> <ul> <li> <p>the two outer face maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">d_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">d_n</annotation></semantics></math> are given by forgetting the first and the last morphism in such a sequence, respectively;</p> </li> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math> inner face maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">d_{0 \lt k \lt n}</annotation></semantics></math> are given by composing the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th morphism with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k+1</annotation></semantics></math>st in the sequence.</p> </li> </ul> </li> </ul> <p>The degeneracy maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>k</mi></msub><mo lspace="verythinmathspace">:</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mi>n</mi><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> s_k \colon N(\mathcal{G}_\bullet)n \to N(\mathcal{G}_\bullet)_{n+1} \,. </annotation></semantics></math></div> <p>are given by inserting an <a class="existingWikiWord" href="/nlab/show/identity">identity</a> morphism on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_43">Remark</h6> <p>Spelling this out in more detail: write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mi>n</mi></msub><mo>=</mo><mrow><mo>{</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>n</mi></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathcal{G}_n = \left\{ x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_2 \stackrel{f_{2,3}}{\to} \cdots \stackrel{f_{n-1,n}}{\to} x_n \right\} </annotation></semantics></math></div> <p>for the set of sequences of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> composable morphisms. Given any element of this set and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \lt k \lt n </annotation></semantics></math>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≔</mo><msub><mi>f</mi> <mrow><mi>i</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>f</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> f_{i-1,i+1} \coloneqq f_{i,i+1} \circ f_{i-1,i} </annotation></semantics></math></div> <p>for the comosition of the two morphism that share the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th vertex.</p> <p>With this, face map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math> acts simply by “removing the index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>”:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>2</mn></msub><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mn>2</mn></msub><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d_0 \colon (x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto (x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n ) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mi>⋯</mi><mover><mo>→</mo><mrow></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mi>⋯</mi><mover><mo>→</mo><mrow></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d_{0\lt k \lt n} \colon ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n ) \mapsto ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n ) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_n \colon ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} ) \,. </annotation></semantics></math></div> <p>Similarly, writing</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>≔</mo><msub><mi>id</mi> <mrow><msub><mi>x</mi> <mi>k</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> f_{k,k} \coloneqq id_{x_k} </annotation></semantics></math></div> <p>for the identity morphism on the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math>, then the degenarcy map acts by “repeating the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th index”</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>k</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><mo>→</mo><msub><mi>x</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><mo>→</mo><msub><mi>x</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>,</mo><mi>k</mi></mrow></msub></mrow></mover><msub><mi>x</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> s_k \colon ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \mapsto ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \,. </annotation></semantics></math></div> <p>This makes it manifest that these functions organise into a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_61">Proposition</h6> <p>These collections of maps in def. <a class="maruku-ref" href="#NerveOfGroupoid"></a> satisfy the <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a>, hence make the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>. Moreover, this simplicial set is a Kan complex, where each <a class="existingWikiWord" href="/nlab/show/horn">horn</a> has a <em>unique</em> filler (extension to a <a class="existingWikiWord" href="/nlab/show/simplex">simplex</a>).</p> </div> <p>(A 2-<a class="existingWikiWord" href="/nlab/show/simplicial+coskeleton">coskeletal</a> Kan complex.)</p> <div class="num_prop"> <h6 id="proposition_62">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> operation constitutes a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo lspace="verythinmathspace">:</mo><mi>Grpd</mi><mo>→</mo><mi>KanCplx</mi><mo>↪</mo><mi>sSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N \colon Grpd \to KanCplx \hookrightarrow sSet \,. </annotation></semantics></math></div></div> <h4 id="DoldKanCorrespondence">Chain complexes as Kan complexes</h4> <p>In the familiar construction of <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> recalled <a href="#SimplicialHomology">above</a> one constructs the <em>alternating face map chain complex</em> of the <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> of singular simplices, def. <a class="maruku-ref" href="#ComplexOfChainsOnASimplicialSet"></a>. This construction is natural and straightforward, but the result chain complex tends to be very “large” even if its <a class="existingWikiWord" href="/nlab/show/chain+homology+groups">chain homology groups</a> end up being very “small”. But in the context of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> one is to consider all objects notup to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, but of to <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>, which for <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> means up to <em><a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a></em>. Hence one should look for the natural construction of “smaller” chain complexes that are still quasi-isomorphic to these alternating face map complexes. This is accomplished by the <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a> construction:</p> <div class="num_defn" id="AlternatingFaceMapComplex"> <h6 id="definition_86">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> its <strong><a class="existingWikiWord" href="/nlab/show/alternating+face+map+complex">alternating face map complex</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(C A)_\bullet</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> which</p> <ul> <li> <p>in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is given by the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_n</annotation></semantics></math> itself</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>:</mo><mo>=</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> (C A)_n := A_n </annotation></semantics></math></div></li> <li> <p>with <a class="existingWikiWord" href="/nlab/show/differential">differential</a> given by the alternating sum of face maps (using the abelian group structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>n</mi></msub><mo>≔</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>d</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial_n \coloneqq \sum_{i = 0}^n (-1)^i d_i \;\colon\; (C A)_n \to (C A)_{n-1} \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_lemma" id="AlternatingSumOfFacesInNilpotent"> <h6 id="lemma_30">Lemma</h6> <p>The differential in def. <a class="maruku-ref" href="#AlternatingFaceMapComplex"></a> is well-defined in that it indeed squares to 0.</p> </div> <div class="proof"> <h6 id="proof_87">Proof</h6> <p>Using the <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identity</a>, prop. <a class="maruku-ref" href="#SimplicialIdentities"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>d</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i \circ d_j = d_{j-1} \circ d_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \lt j</annotation></semantics></math> one finds:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mo>∂</mo> <mi>n</mi></msub><msub><mo>∂</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>≥</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>≥</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>≥</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><msub><mi>d</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>j</mi></msub><mo>−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></msup><msub><mi>d</mi> <mi>k</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \partial_n \partial_{n+1} & = \sum_{i, j} (-1)^{i+j} d_i \circ d_{j} \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_i \circ d_j \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_{j-1} \circ d_i \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j - \sum_{i \leq k} (-1)^{i+k} d_{k} \circ d_i \\ &= 0 \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="NormalizedChainComplexOnGeneralGroup"> <h6 id="definition_87">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, its <em><a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a></em> or <em>Moore complex</em> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>-graded <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>N</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>,</mo><mo>∂</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((N A)_\bullet,\partial )</annotation></semantics></math> which</p> <ul> <li> <p>is in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> the joint <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mi>ker</mi><mspace width="thinmathspace"></mspace><msubsup><mi>d</mi> <mi>i</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex"> (N A)_n=\bigcap_{i=1}^{n}ker\,d_i^n </annotation></semantics></math></div> <p>of all face maps except the 0-face;</p> </li> <li> <p>with differential given by the remaining 0-face map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>n</mi></msub><mo>:</mo><mo>=</mo><msubsup><mi>d</mi> <mn>0</mn> <mi>n</mi></msubsup><msub><mo stretchy="false">|</mo> <mrow><mo stretchy="false">(</mo><mi>N</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></msub><mo>:</mo><mo stretchy="false">(</mo><mi>N</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>N</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial_n := d_0^n|_{(N A)_n} : (N A)_n \rightarrow (N A)_{n-1} \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_remark"> <h6 id="remark_44">Remark</h6> <p>We may think of the elements of the complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">N A</annotation></semantics></math>, def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a>, in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> as being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/disks">disks</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> all whose <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> is captured by a single face:</p> <ul> <li> <p>an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>N</mi><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">g \in N G_1</annotation></semantics></math> in degree 1 is a 1-disk</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mover><mo>→</mo><mi>g</mi></mover><mo>∂</mo><mi>g</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> 1 \stackrel{g}{\to} \partial g \,, </annotation></semantics></math></div></li> <li> <p>an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>N</mi><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">h \in N G_2</annotation></semantics></math> is a 2-disk</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mn>1</mn></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>h</mi></msup></mtd> <mtd><msup><mo>↘</mo> <mrow><mo>∂</mo><mi>h</mi></mrow></msup></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mn>1</mn></mover></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h} \\ 1 &&\stackrel{1}{\to}&& 1 } \,, </annotation></semantics></math></div></li> <li> <p>a degree 2 element in the kernel of the boundary map is such a 2-disk that is closed to a 2-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mn>1</mn></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>h</mi></msup></mtd> <mtd><msup><mo>↘</mo> <mrow><mo>∂</mo><mi>h</mi><mo>=</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mn>1</mn></mover></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h = 1} \\ 1 &&\stackrel{1}{\to}&& 1 } \,, </annotation></semantics></math></div></li> </ul> <p>etc.</p> </div> <div class="num_defn" id="DegenerateElement"> <h6 id="definition_88">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (or in fact any <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>), then an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a \in A_{n+1}</annotation></semantics></math> is called <em>degenerate</em> (or <em><a class="existingWikiWord" href="/nlab/show/thin+element">thin</a></em>) if it is in the <a class="existingWikiWord" href="/nlab/show/image">image</a> of one of the simplicial degeneracy maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">s_i \colon A_n \to A_{n+1}</annotation></semantics></math>. All elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">A_0</annotation></semantics></math> are regarded a non-degenerate. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">⟨</mo><msub><mo>∪</mo> <mi>i</mi></msub><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> D (A_{n+1}) \coloneqq \langle \cup_i s_i(A_{n}) \rangle \hookrightarrow A_{n+1} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">A_{n+1}</annotation></semantics></math> which is generated by the degenerate elements (i.e. the smallest subgroup containing all the degenerate elements).</p> </div> <div class="num_defn" id="ComplexModuloDegeneracies"> <h6 id="definition_89">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> its <strong>alternating face maps chain complex modulo degeneracies</strong>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C A)/(D A)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> <ul> <li> <p>which in degree 0 equals is just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>≔</mo><msub><mi>A</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">((C A)/D(A))_0 \coloneqq A_0</annotation></semantics></math>;</p> </li> <li> <p>which in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> group obtained by dividing out the group of degenerate elements, def. <a class="maruku-ref" href="#DegenerateElement"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>:</mo><mo>=</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ((C A)/D(A))_{n+1} := A_{n+1} / D(A_{n+1}) </annotation></semantics></math></div></li> <li> <p>whose <a class="existingWikiWord" href="/nlab/show/differential">differential</a> is the induced action of the alternating sum of faces on the quotient (which is well-defined by lemma <a class="maruku-ref" href="#LeftCosetsDisjoint"></a>).</p> </li> </ul> </div> <div class="num_lemma" id="LeftCosetsDisjoint"> <h6 id="lemma_31">Lemma</h6> <p>Def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a> is indeed well defined in that the alternating face map differential respects the degenerate subcomplex.</p> </div> <div class="proof"> <h6 id="proof_88">Proof</h6> <p>Using the mixed <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a> we find that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">s_j(a) \in A_n</annotation></semantics></math> a degenerate element, its boundary is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>s</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>s</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mi>j</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><mi>a</mi><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>></mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>s</mi> <mi>j</mi></msub><msub><mi>d</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>s</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>></mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>s</mi> <mi>j</mi></msub><msub><mi>d</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \sum_i (-1)^i d_i s_j(a) &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i = j, j+1} (-1)^i a + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \\ &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \end{aligned} </annotation></semantics></math></div> <p>which is again a combination of elements in the image of the degeneracy maps.</p> </div> <div class="num_prop" id="NormalizedIntoModuloDegeneraciesIsIsomorpism"> <h6 id="proposition_63">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, the evident composite of natural morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mi>A</mi><mover><mo>↪</mo><mi>i</mi></mover><mi>A</mi><mover><mo>→</mo><mi>p</mi></mover><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N A \stackrel{i}{\hookrightarrow} A \stackrel{p}{\to} (C A)/(D A) </annotation></semantics></math></div> <p>from the normalized chain complex, def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a>, into the alternating face map complex modulo degeneracies, def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a>, (inclusion followed by projection to the quotient) is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of chain complexes.</p> </div> <p>e.g. (<a href="Moore+complex#GoerssJardine">Goerss-Jardine, theorem III 2.1</a>).</p> <div class="num_cor" id="SplittingOffDegenerateCells"> <h6 id="corollary_8">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a>, there is a splitting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>N</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊕</mo><msub><mi>D</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A) </annotation></semantics></math></div> <p>of the alternating face map complex, def. <a class="maruku-ref" href="#AlternatingFaceMapComplex"></a> as a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>, where the first direct summand is <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a> of def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a> and the second is the degenerate cells from def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a>.</p> </div> <div class="proof"> <h6 id="proof_89">Proof</h6> <p>By prop. <a class="maruku-ref" href="#NormalizedIntoModuloDegeneraciesIsIsomorpism"></a> there is an <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> to the diagonal composite in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>p</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>N</mi><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A } \,. </annotation></semantics></math></div> <p>This hence exhibits a <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">splitting</a> of the <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> given by the quotient by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">D A</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>D</mi><mi>A</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>C</mi><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>p</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↙</mo> <mrow><msub><mpadded width="0"><mo>≃</mo></mpadded> <mi>iso</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>N</mi><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& D A &\hookrightarrow & C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) &\to & 0 \\ && && {}^{\mathllap{i}}\uparrow & \swarrow_{\mathrlap{\simeq}_{iso}} \\ && && N A } \,. </annotation></semantics></math></div></div> <div class="num_theorem" id="EMTheorem"> <h6 id="theorem_eilenbergmaclane">Theorem (Eilenberg-MacLane)</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, then the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mi>A</mi><mo>↪</mo><mi>C</mi><mi>A</mi></mrow><annotation encoding="application/x-tex"> N A \hookrightarrow C A </annotation></semantics></math></div> <p>of the normalized chain complex, def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a> into the full alternating face map complex, def. <a class="maruku-ref" href="#AlternatingFaceMapComplex"></a>, is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a> <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> and in fact a natural chain <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, i.e. the complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_\bullet(X)</annotation></semantics></math> is null-homotopic.</p> </div> <p>(<a href="Moore+complex#GoerssJardine">Goerss-Jardine, theorem III 2.4</a>)</p> <div class="num_cor"> <h6 id="corollary_9">Corollary</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, then the projection <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (C A) \longrightarrow (C A)/(D A) </annotation></semantics></math></div> <p>from its alternating face maps complex, def. <a class="maruku-ref" href="#AlternatingFaceMapComplex"></a>, to the alternating face map complex modulo degeneracies, def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a>, is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>.</p> </div> <div class="proof"> <h6 id="proof_90">Proof</h6> <p>Consider the pre-composition of the map with the inclusion of the normalized chain complex, def. <a class="maruku-ref" href="#NormalizedChainComplexOnGeneralGroup"></a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>p</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>C</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>D</mi><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>N</mi><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A } </annotation></semantics></math></div> <p>By theorem <a class="maruku-ref" href="#EMTheorem"></a> the vertical map is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> and by prop. <a class="maruku-ref" href="#NormalizedIntoModuloDegeneraciesIsIsomorpism"></a> the composite diagonal map is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, hence in particular also a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>. Since quasi-isomorphisms satisfy the <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> property, it follows that also the map in question is a quasi-isomorphism.</p> </div> <div class="num_example" id="ChainsOnThe1Simplex"> <h6 id="example_64">Example</h6> <p>Consider the 1-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[\Delta[1]]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> which in each degree is the <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> on the simplices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math>.</p> <p>This simplicial abelian group starts out as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><mover><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover><mo>⟶</mo></mover><msup><mi>ℤ</mi> <mn>4</mn></msup><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover><msup><mi>ℤ</mi> <mn>3</mn></msup><mover><munder><mo>⟶</mo><mrow><msub><mo>∂</mo> <mn>1</mn></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><mo>∂</mo> <mn>0</mn></msub></mrow></mover></mover><msup><mi>ℤ</mi> <mn>2</mn></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbb{Z}[\Delta[1]] = \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} \mathbb{Z}^4 \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} \mathbb{Z}^3 \stackrel{\overset{\partial_0}{\longrightarrow}}{\underset{\partial_1}{\longrightarrow}} \mathbb{Z}^2 \right) </annotation></semantics></math></div> <p>(where we are indicating only the face maps for notational simplicity).</p> <p>Here the first <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℤ</mi> <mn>2</mn></msup><mo>=</mo><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}^2 = \mathbb{Z}\oplus \mathbb{Z}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of two copies of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, is the group of 0-chains generated from the two endpoints <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math>, i.e. the abelian group of formal linear combinations of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℤ</mi> <mn>2</mn></msup><mo>≃</mo><mrow><mo>{</mo><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>ℤ</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}^2 \simeq \left\{ a \cdot (0) + b \cdot (1) | a,b \in \mathbb{Z}\right\} \,. </annotation></semantics></math></div> <p>The second <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℤ</mi> <mn>3</mn></msup><mo>≃</mo><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}^3 \simeq \mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z}</annotation></semantics></math> is the abelian group generated from the three (!) 1-simplicies in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math>, namely the non-degenerate edge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0\to 1)</annotation></semantics></math> and the two degenerate cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0 \to 0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1 \to 1)</annotation></semantics></math>, hence the abelian group of formal linear combinations of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℤ</mi> <mn>3</mn></msup><mo>≃</mo><mrow><mo>{</mo><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>ℤ</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}^3 \simeq \left\{ a \cdot (0\to 0) + b \cdot (0 \to 1) + c \cdot (1 \to 1) | a,b,c \in \mathbb{Z}\right\} \,. </annotation></semantics></math></div> <p>The two face maps act on the basis 1-cells as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>i</mi><mo>→</mo><mi>j</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \partial_1 \colon (i \to j) \mapsto (i) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>i</mi><mo>→</mo><mi>j</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial_0 \colon (i \to j) \mapsto (j) \,. </annotation></semantics></math></div> <p>Now of course most of the (infinitely!) many simplices inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math> are degenerate. In fact the only non-degenerate simplices are the two 0-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1)</annotation></semantics></math> and the 1-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0 \to 1)</annotation></semantics></math>. Hence the alternating face maps complex modulo degeneracies, def. <a class="maruku-ref" href="#ComplexModuloDegeneracies"></a>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[\Delta[1]]</annotation></semantics></math> is simply this:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>ℤ</mi><mover><mo>⟶</mo><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mover><msup><mi>ℤ</mi> <mn>2</mn></msup><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (C (\mathbb{Z}[\Delta[1]])) / D (\mathbb{Z}[\Delta[1]])) = \left( \cdots \to 0 \to 0 \to \mathbb{Z} \stackrel{\left(1 \atop -1\right)}{\longrightarrow} \mathbb{Z}^2 \right) \,. </annotation></semantics></math></div> <p>Notice that alternatively we could consider the topological 1-simplex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></msup><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta^1 = [0,1]</annotation></semantics></math> and its <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(\Delta^1)</annotation></semantics></math> in place of the smaller <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math>, then the free simplicial abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">(</mo><mi>Sing</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}(Sing(\Delta^1))</annotation></semantics></math> of that. The corresponding alternating face map chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">(</mo><mi>Sing</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\mathbb{Z}(Sing(\Delta^1)))</annotation></semantics></math> is “huge” in that in each positive degree it has a free abelian group on uncountably many generators. Quotienting out the degenerate cells still leaves uncountably many generators in each positive degree (while every singular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> is “thin”, only those whose parameterization is as induced by a degeneracy map are actually regarded as degenerate cells here). Hence even after normalization the singular simplicial chain complex is “huge”. Nevertheless it is quasi-isomorphic to the tiny chain complex found above.</p> </div> <p>The statement of the <em><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></em> now is the following.</p> <div class="num_theorem"> <h6 id="theorem_6">Theorem</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>A</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mover><mo>→</mo><mo>←</mo></mover><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Γ</mi></mrow><annotation encoding="application/x-tex"> N \;\colon\; A^{\Delta^{op}} \stackrel{\leftarrow}{\to} Ch_\bullet^+(A) \;\colon\; \Gamma </annotation></semantics></math></div> <p>between</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/category+of+simplicial+objects">category of simplicial objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of connective chain complexes</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>;</p> </li> </ul> <p>where</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/normalized+chains+complex">normalized chains complex</a>/normalized <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a> functor.</li> </ul> </div> <p>(<a href="Dold-Kan+correspondence#Dold58">Dold 58</a>, <a href="Dold-Kan+correspondence#Kan58">Kan 58</a>, <a href="Dold-Kan+correspondence#DoldPuppe61">Dold-Puppe 61</a>).</p> <div class="num_theorem"> <h6 id="theorem_kan">Theorem (Kan)</h6> <p>For the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the category <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s, the functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a> with respect to the cosimplicial chain complex</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mi>Δ</mi><mo>→</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}[-]: \Delta \to Ch_+(Ab) </annotation></semantics></math></div> <p>that sends the standard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> to the normalized <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a> of the free simplicial abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>ℤ</mi></msub><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_{\mathbb{Z}}(\Delta^n)</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^n</annotation></semantics></math>, i.e.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma(V) : [k] \mapsto Hom_{Ch_\bullet^+(Ab)}(N(\mathbb{Z}(\Delta[k])), V) \,. </annotation></semantics></math></div></div> <p>This is due to (<a href="Dold-Kan+correspondence#Kan58">Kan 58</a>).</p> <p>More explicitly we have the following</p> <div class="num_prop" id="ExplicitUnitAndCounit"> <h6 id="proposition_64">Proposition</h6> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup></mrow><annotation encoding="application/x-tex">V \in Ch_\bullet^+</annotation></semantics></math> the simplicial abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(V)</annotation></semantics></math> is in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><munder><mo>→</mo><mi>surj</mi></munder><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></munder><msub><mi>V</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> \Gamma(V)_n = \bigoplus_{[n] \underset{surj}{\to} [k]} V_k </annotation></semantics></math></div> <p>and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\theta : [m] \to [n]</annotation></semantics></math> a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> the corresponding map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><msub><mo stretchy="false">)</mo> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\Gamma(V)_n \to \Gamma(V)_m</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>θ</mi> <mo>*</mo></msup><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><munder><mo>→</mo><mi>surj</mi></munder><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></munder><msub><mi>V</mi> <mi>k</mi></msub><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><munder><mo>→</mo><mi>surj</mi></munder><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo></mrow></munder><msub><mi>V</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex"> \theta^* : \bigoplus_{[n] \underset{surj}{\to} [k]} V_k \to \bigoplus_{[m] \underset{surj}{\to} [r]} V_r </annotation></semantics></math></div> <p>is given on the summand indexed by some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\sigma : [n] \to [k]</annotation></semantics></math> by the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msup><mi>d</mi> <mo>*</mo></msup></mrow></mover><msub><mi>V</mi> <mi>s</mi></msub><mo>↪</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><munder><mo>→</mo><mi>surj</mi></munder><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo></mrow></munder><msub><mi>V</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex"> V_k \stackrel{d^*}{\to} V_s \hookrightarrow \bigoplus_{[m] \underset{surj}{\to} [r]} V_r </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mi>t</mi></mover><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mi>d</mi></mover><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [m] \stackrel{t}{\to} [s] \stackrel{d}{\to} [k] </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/weak+factorization+system">epi-mono factorization</a> of the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mi>θ</mi></mover><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mi>σ</mi></mover><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[m] \stackrel{\theta}{\to} [n] \stackrel{\sigma}{\to} [k]</annotation></semantics></math>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mi>N</mi><mo>→</mo><mi>Id</mi></mrow><annotation encoding="application/x-tex">\Gamma N \to Id</annotation></semantics></math> is given on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>sAb</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">A \in sAb^{\Delta^{op}}</annotation></semantics></math> by the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><munder><mo>→</mo><mi>surj</mi></munder><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></munder><mo stretchy="false">(</mo><mi>N</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><mo>→</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> \bigoplus_{[n] \underset{surj}{\to} [k]} (N A)_k \to A_n </annotation></semantics></math></div> <p>which on the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>mand indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\sigma : [n] \to [k]</annotation></semantics></math> is the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><msub><mi>A</mi> <mi>k</mi></msub><mo>↪</mo><msub><mi>A</mi> <mi>k</mi></msub><mover><mo>→</mo><mrow><msup><mi>σ</mi> <mo>*</mo></msup></mrow></mover><msub><mi>A</mi> <mi>n</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N A_k \hookrightarrow A_k \stackrel{\sigma^*}{\to} A_n \,. </annotation></semantics></math></div></li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Id</mi><mo>→</mo><mi>N</mi><mi>Γ</mi></mrow><annotation encoding="application/x-tex">Id \to N \Gamma</annotation></semantics></math> is on a chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> given by the composite of the projection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> V \to C(\Gamma(V)) \to C(\Gamma(C))/D(\Gamma(V)) </annotation></semantics></math></div> <p>with the inverse</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>N</mi><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(\Gamma(V))/D(\Gamma(V)) \to N \Gamma(V) </annotation></semantics></math></div> <p>of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>D</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N \Gamma(V) \hookrightarrow C(\Gamma(V)) \to C(\Gamma(V))/D(\Gamma(V)) </annotation></semantics></math></div> <p>(which is indeed an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, as discussed at <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>).</p> </li> </ul> </div> <p>This is spelled out in (<a href="Moore+complex#Goerssjardine">Goerss-Jardine, prop. 2.2 in section III.2</a>).</p> <div class="num_prop"> <h6 id="proposition_65">Proposition</h6> <p>With the explicit choice for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mi>N</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>Id</mi></mrow><annotation encoding="application/x-tex">\Gamma N \stackrel{\simeq}{\to} Id</annotation></semantics></math> as <a href="#ExplicitUnitAndCounit">above</a> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> form an <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Γ</mi><mo>⊣</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Gamma \dashv N)</annotation></semantics></math></p> </div> <p>This is for instance (<a href="Dold-Kan+correspondence#Weilbel">Weibel, exercise 8.4.2</a>).</p> <div class="num_remark"> <h6 id="remark_45">Remark</h6> <p>It follows that with the inverse structure maps, we also have an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> the other way round: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N \dashv \Gamma)</annotation></semantics></math>.</p> </div> <p>Hence in conclusion the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> allows us to regard <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> (in non-negative degree) as, in particular, special <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>. In fact as simplicial sets they are <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> and hence <a class="existingWikiWord" href="/nlab/show/infinity-groupoids">infinity-groupoids</a>:</p> <div class="num_theorem" id="MooreTheorem"> <h6 id="theorem_j_c_moore">Theorem (J. C. Moore)</h6> <p>The <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> underlying any <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> (by forgetting the group structure) is a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>.</p> </div> <p>This is due to (<a href="simplicial+group#Moore54">Moore, 1954</a>)</p> <p>In fact, not only are the <a class="existingWikiWord" href="/nlab/show/horn">horn</a> fillers guaranteed to exist, but there is an algorithm that provides explicit fillers. This implies that constructions on a simplicial group that use fillers of horns can often be adjusted to be functorial by using the algorithmically defined fillers. An argument that just uses ‘existence’ of fillers can be refined to give something more ‘coherent’.</p> <div class="proof"> <h6 id="proof_91">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a simplicial group.</p> <p>Here is the explicit algorithm that computes the horn fillers:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>y</mi> <mn>0</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>y</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>y</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>y</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(y_0,\ldots, y_{k-1}, -,y_{k+1}, \ldots, y_n)</annotation></semantics></math> give a <a class="existingWikiWord" href="/nlab/show/horn">horn</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">G_{n-1}</annotation></semantics></math>, so the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">y_i</annotation></semantics></math>s are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math> simplices that fit together as if they were all but one, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">k^{th}</annotation></semantics></math> one, of the faces of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex. There are three cases:</p> <ol> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k = 0</annotation></semantics></math>:</p> <ul> <li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>n</mi></msub><mo>=</mo><msub><mi>s</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">w_n = s_{n-1}y_n</annotation></semantics></math> and then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i = w_{i+1}(s_{i-1}d_i w_{i+1})^{-1}s_{i-1}y_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mi>n</mi><mo>,</mo><mi>…</mi><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i = n, \ldots, 1</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">w_1 </annotation></semantics></math> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i w_1 = y_i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">i\neq 0</annotation></semantics></math>;</li> </ul> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0\lt k \lt n</annotation></semantics></math>:</p> <ul> <li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>0</mn></msub><mo>=</mo><msub><mi>s</mi> <mn>0</mn></msub><msub><mi>y</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">w_0 = s_0 y_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mi>i</mi></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i = 0, \ldots, k-1</annotation></semantics></math>, then take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>n</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>nw</mi></msub><msub><mo></mo><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">w_n = w_{k-1}(s_{n-1}d_nw_{k-1})^{-1}s_{n-1}y_n</annotation></semantics></math>, and finally a downwards induction given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i = w_{i+1}(s_{i-1}d_{i}w_{i+1})^{-1}s_{i-1}y_i</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mi>n</mi><mo>,</mo><mi>…</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i = n, \ldots, k+1</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">w_{k+1}</annotation></semantics></math> gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_{i}w_{k+1} = y_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≠</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">i \neq k</annotation></semantics></math>;</li> </ul> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k=n</annotation></semantics></math>:</p> <ul> <li>use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>0</mn></msub><mo>=</mo><msub><mi>s</mi> <mn>0</mn></msub><msub><mi>y</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">w_0 = s_0 y_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mi>i</mi></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i = 0, \ldots, n-1</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">w_{n-1}</annotation></semantics></math> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i w_{n-1} = y_i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≠</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">i\neq n</annotation></semantics></math>.</li> </ul> </li> </ol> </div> <h3 id="geometric_realization">Geometric realization</h3> <p>So far we we have considered passing from <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> to <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> by applying the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> functor of def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a>. Now we discuss a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> of this functor, called <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, which turns a simplicial set into a topological space by identifying each of its abstract <a class="existingWikiWord" href="/nlab/show/n-simplices">n-simplices</a> with the standard topological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex.</p> <p>This is an example of a general abstract phenomenon:</p> <div class="num_prop" id="NerveAndRealizationAdjunction"> <h6 id="proposition_66">Proposition</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>D</mi><mo>⟶</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \delta \;\colon\; D \longrightarrow \mathcal{C} </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> with all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. Then the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>-functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> N \;\colon\; \mathcal{C} \longrightarrow [D^{op}, Set] </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N(X) \coloneqq \mathcal{C}(\delta(-),X) </annotation></semantics></math></div> <p>has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert-\vert}</annotation></semantics></math>, called <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>⊣</mo><mi>N</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mover><munder><mo>⟶</mo><mi>N</mi></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></mover><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> ({\vert-\vert} \dashv N) \;\colon\; \mathcal{C} \stackrel{\overset{{\vert-\vert}}{\longleftarrow}}{\underset{N}{\longrightarrow}} [D^{op}, Set] </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msup><mi>δ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {\vert S\vert} = \int^{d \in D} \delta(d) \cdot S(d) \,. </annotation></semantics></math></div></div> <p>(<a href="nerve+and+realization#Kan58">Kan 58</a>)</p> <div class="proof"> <h6 id="proof_92">Proof</h6> <p>By basic propeties of <a class="existingWikiWord" href="/nlab/show/ends">ends</a> and <a class="existingWikiWord" href="/nlab/show/coends">coends</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>Set</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>Set</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>𝒞</mi><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>𝒞</mi><mo stretchy="false">(</mo><msup><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msup><mi>δ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>S</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [D^{op}, Set](S,N(X)) & = \int_{d \in D} Set(S(d), N(X)(d)) \\ & = \int_{d\in D} Set(S(d), \mathcal{C}(\delta(d),X)) \\ & \simeq \int_{d \in D} \mathcal{C}(\delta(d) \cdot S(d), X) \\ & \simeq \mathcal{C}(\int^{d \in D} \delta(d) \cdot S(d), X) \\ & = \mathcal{C}({\vert S\vert}, X) \,. \end{aligned} </annotation></semantics></math></div></div> <div class="num_example" id="TopologicalRealizationOfSimplicialSets"> <h6 id="example_65">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi></mrow><annotation encoding="application/x-tex">Sing</annotation></semantics></math> of def. <a class="maruku-ref" href="#SingularSimplicialComplex"></a> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo lspace="verythinmathspace">:</mo><mi>sSet</mi><mo>⟶</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> {\vert-\vert} \colon sSet \longrightarrow Top </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><msup><mi>Δ</mi> <mi>n</mi></msup><mo>⋅</mo><msub><mi>S</mi> <mi>n</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {\vert S \vert} = \int^{[n]\in \Delta} \Delta^n \cdot S_n \,. </annotation></semantics></math></div></div> <p>Topological geometric realization takes values in particularly nice topological spaces.</p> <div class="num_defn" id="TopologicalRealizationOfsSetLandsInCWComplexes"> <h6 id="proposition_67">Proposition</h6> <p>The topological <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> in example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a> takes values in <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>.</p> </div> <p>(e.g. <a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, chapter I, prop. 2.3</a>)</p> <p>Thus for a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mrow><mo stretchy="false">|</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">|</mo></mrow><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\epsilon_X \colon {\vert Sing X\vert} \longrightarrow X</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>-adjunction is a candidate for a replacement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by a CW-complex. For this, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϵ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon_X</annotation></semantics></math> should be at least a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, i.e. induce <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on all <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>. Since homotopy groups are built from maps into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> out of <a class="existingWikiWord" href="/nlab/show/compact+topological+spaces">compact topological spaces</a> it is plausible that this works if the topology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is entirely detected by maps out of compact topological spaces into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Topological spaces with this property are called <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated</a>.</p> <p>We take <em><a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a></em> to imply <em><a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff topological space</a></em>.</p> <div class="num_defn" id="kTop"> <h6 id="definition_90">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is called <strong>compactly open</strong> or <strong>compactly closed</strong>, respectively, if for every <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon K \longrightarrow X</annotation></semantics></math> out of a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a> the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">f^{-1}(U) \subset K</annotation></semantics></math> is open or closed, respectively.</p> <p>A topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a></strong> if each of its compactly closed subspaces is already closed.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> Top_{cg} \hookrightarrow Top </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> on the compactly generated topological spaces.</p> </div> <p>Often the condition is added that a compactly closed topological space be also a <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+topological+space">weakly Hausdorff topological space</a>.</p> <div class="num_example" id="ExamplesOfCompactlyGeneratedTopologiclSpaces"> <h6 id="example_66">Example</h6> <p>Examples of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>, def. <a class="maruku-ref" href="#kTop"></a>, include</p> <ul> <li> <p>every <a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/locally+compact+space">locally compact space</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/first+countable+space">first countable space</a></p> </li> </ul> </div> <div class="num_cor" id="TopologicalRealizationOfSSetLandsInkTop"> <h6 id="corollary_10">Corollary</h6> <p>The topological <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> functor of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> in example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a> takes values in <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>sSet</mi><mo>⟶</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex"> {\vert - \vert} \;\colon\; sSet \longrightarrow Top_{cg} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_93">Proof</h6> <p>By example <a class="maruku-ref" href="#ExamplesOfCompactlyGeneratedTopologiclSpaces"></a> and prop. <a class="maruku-ref" href="#TopologicalRealizationOfsSetLandsInCWComplexes"></a>.</p> </div> <div class="num_prop" id="kTopIsCoreflectiveInTop"> <h6 id="proposition_68">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top_{cg} \hookrightarrow Top</annotation></semantics></math> of def. <a class="maruku-ref" href="#kTop"></a> has the following properties</p> <ol> <li> <p>It is a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub><mover><munder><mo>⟵</mo><mi>k</mi></munder><mo>↪</mo></mover><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top_{cg} \stackrel{\hookrightarrow}{\underset{k}{\longleftarrow}} Top \,. </annotation></semantics></math></div> <p>The coreflection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k(X)</annotation></semantics></math> of a topological space is given by adding to the open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> all compactly open subsets, def. <a class="maruku-ref" href="#kTop"></a>.</p> </li> <li> <p>It has all small <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>.</p> <p>The colimits are computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>, the limits are the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of the limits as computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>.</p> </li> <li> <p>It is a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> is the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of the Cartesian product formed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>.</p> </li> </ol> </div> <p>This is due to (<a href="compactly+generated+topological+space#Steenrod67">Steenrod 67</a>), expanded on in (<a href="compactly+generated+topological+space#Lewis78">Lewis 78, appendix A</a>). One says that prop. <a class="maruku-ref" href="#kTopIsCoreflectiveInTop"></a> with example <a class="maruku-ref" href="#ExamplesOfCompactlyGeneratedTopologiclSpaces"></a> makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> a “<a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>”.</p> <div class="num_prop" id="Timesk"> <h6 id="proposition_69">Proposition</h6> <p>Regarded, via corollary <a class="maruku-ref" href="#TopologicalRealizationOfSSetLandsInkTop"></a> as a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo lspace="verythinmathspace">:</mo><mi>sSet</mi><mo>→</mo><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">{\vert - \vert} \colon sSet \to Top_{cg}</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> preserves <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>.</p> </div> <p>See at <em><a href="geometric+realization#GeometricRealizationIsLeftExact">Geometric realization is left exact</a></em>.</p> <div class="proof"> <h6 id="proof_idea_3">Proof idea</h6> <p>The key step in the proof is to use the <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>cg</mi></msub></mrow><annotation encoding="application/x-tex">Top_{cg}</annotation></semantics></math> (prop. <a class="maruku-ref" href="#kTopIsCoreflectiveInTop"></a>). This gives that the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> and hence preserves colimits in each variable, so that the <a class="existingWikiWord" href="/nlab/show/coend">coend</a> in the definition of the geometric realization may be taken out of Cartesian products.</p> </div> <div class="num_lemma"> <h6 id="lemma_32">Lemma</h6> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>, of a <a class="existingWikiWord" href="/nlab/show/minimal+Kan+fibration">minimal Kan fibration</a>, def. <a class="maruku-ref" href="#MinimalKanFibration"></a> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a>.</p> </div> <p>This is due to (<a class="existingWikiWord" href="/nlab/show/Calculus+of+fractions+and+homotopy+theory">Gabriel-Zisman 67</a>). See for instance (<a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, chapter I, corollary 10.8, theorem 10.9</a>).</p> <div class="proof"> <h6 id="proof_idea_4">Proof idea</h6> <p>By prop. <a class="maruku-ref" href="#MinimalKanFibrationAreFiberBundles"></a> minimal Kan fibrations are simplicial <a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a>, locally trivial over each simplex in the base. By prop. <a class="maruku-ref" href="#Timesk"></a> this property translates to their <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> also being a locally trivial <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, hence in particular a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>.</p> </div> <div class="num_prop" id="GeometricRealizationOfKanFibrationIsSerreFibration"> <h6 id="proposition_70">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>, of any <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, def. <a class="maruku-ref" href="#KanFibration"></a> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, def. <a class="maruku-ref" href="#SerreFibration"></a>.</p> </div> <p>This is due to (<a href="Kan+fibration#Quillen68">Quillen 68</a>). See for instance (<a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, chapter I, theorem 10.10</a>).</p> <div class="num_prop" id="UnitOfSingularNerveAndRealizationIsWEOnKanComplexes"> <h6 id="proposition_71">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, then the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">unit</a> of the <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> (prop. <a class="maruku-ref" href="#NerveAndRealizationAdjunction"></a>, example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⟶</mo><mi>Sing</mi><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex"> S \longrightarrow Sing {\vert S \vert} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, def. <a class="maruku-ref" href="#WeakHomotopyEquivalence"></a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, then the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">|</mo></mrow><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> {\vert Sing X\vert} \longrightarrow X </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </div> <p>e.g. (<a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, chapter I, prop. 11.1 and p. 63</a>).</p> <div class="proof"> <h6 id="proof_idea_5">Proof idea</h6> <p>Use prop. <a class="maruku-ref" href="#SingDetextsAndReflectsFibrations"></a> and prop. <a class="maruku-ref" href="#GeometricRealizationOfKanFibrationIsSerreFibration"></a> applied to the <a class="existingWikiWord" href="/nlab/show/path+fibration">path fibration</a> to proceed by <a class="existingWikiWord" href="/nlab/show/induction">induction</a>.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="the_classical_model_structure_on_simplicial_sets">The classical model structure on simplicial sets</h3> <div class="num_defn" id="ClassesOfMorphismsOnsSetQuillen"> <h6 id="definition_91">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>)</strong></p> <p>The classical model structure on <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math>, has the following distinguished classes of morphisms:</p> <ul> <li> <p>The classical <strong>weak equivalences</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> are the morphisms whose <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a>, is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>;</p> </li> <li> <p>The classical <strong>fibrations</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> are the <strong><a class="existingWikiWord" href="/nlab/show/Kan+fibrations">Kan fibrations</a></strong>, def. <a class="maruku-ref" href="#KanFibration"></a>;</p> </li> <li> <p>The classical <strong>cofibrations</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> of simplicial sets, i.e. the degreewise <a class="existingWikiWord" href="/nlab/show/injections">injections</a>.</p> </li> </ul> </div> <div class="num_prop"> <h6 id="proposition_72">Proposition</h6> <p>In model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math>, def. <a class="maruku-ref" href="#ClassesOfMorphismsOnsSetQuillen"></a>, the following holds.</p> <ul> <li> <p>The fibrant objects are precisely the <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>.</p> </li> <li> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> of fibrant simplicial sets / <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> is a weak equivalence precisely if it induces an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on all <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a>, def. <a class="maruku-ref" href="#UnderlyingSetsOfSimplicialHomotopyGroups"></a>.</p> </li> <li> <p>All simplicial sets are cofibrant with respect to this model structure.</p> </li> </ul> </div> <div class="num_prop"> <h6 id="proposition_73">Proposition</h6> <p>The <strong>acyclic fibrations</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math>(i.e. the maps that are both fibrations as well as weak equivalences) between <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> are precisely the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> that have the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> with respect to all inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\partial \Delta[n] \hookrightarrow \Delta[n]</annotation></semantics></math> of boundaries of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices into their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mo>∃</mo></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \partial \Delta[n] &\to& X \\ \downarrow &{}^\exists\nearrow& \downarrow^f \\ \Delta[n] &\to& Y } \,. </annotation></semantics></math></div></div> <p>This appears spelled out for instance as (<a href="classical+model+structure+on+simplicial+sets#GoerssJardine99">Goerss-Jardine 99, theorem 11.2</a>).</p> <p>In fact:</p> <div class="num_prop"> <h6 id="proposition_74">Proposition</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> with</p> <ul> <li> <p>generating cofibrations the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\partial \Delta[n] \to \Delta[n]</annotation></semantics></math>;</p> </li> <li> <p>generating acyclic cofibrations the <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^i[n] \to \Delta[n]</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_theorem"> <h6 id="theorem_7">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> be the smallest class of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> satisfying the following conditions:</p> <ol> <li>The class of monomorphisms that are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is closed under <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a>, and <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> property in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> and contains all the <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>.</li> <li>For all natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, the unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta [n] \to \Delta [0]</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> </ol> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is the class of weak homotopy equivalences.</p> </div> <div class="proof"> <h6 id="proof_94">Proof</h6> <ul> <li>First, notice that the horn inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mn>0</mn></msup><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^0 [1] \hookrightarrow \Delta [1]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mn>1</mn></msup><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^1 [1] \hookrightarrow \Delta [1]</annotation></semantics></math> are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>Suppose that the horn inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^k [m] \hookrightarrow \Delta [m]</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \lt n</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">0 \le k \le m</annotation></semantics></math>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \le l \le n</annotation></semantics></math>, the horn inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>l</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^l [n] \hookrightarrow \Delta [n]</annotation></semantics></math> is also in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>Quillen’s <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> then implies all the trivial cofibrations are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> is a trivial Kan fibration, then its right lifting property implies there is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s : Y \to X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>s</mi><mo>=</mo><msub><mi>id</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">p \circ s = id_Y</annotation></semantics></math>, and the two-out-of-three property implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s : Y \to X</annotation></semantics></math> is a trivial cofibration. Thus every trivial Kan fibration is also in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>Every weak homotopy equivalence factors as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">p \circ i</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a trivial Kan fibration and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is a trivial cofibration, so every weak homotopy equivalence is indeed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</li> <li>Finally, noting that the class of weak homotopy equivalences satisfies the conditions in the theorem, we deduce that it is the <em>smallest</em> such class.</li> </ul> </div> <p>As a corollary, we deduce that the classical model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> is the smallest (in terms of weak equivalences) model structure for which the cofibrations are the monomorphisms and the weak equivalences include the (combinatorial) homotopy equivalences.</p> <div class="num_prop"> <h6 id="proposition_75">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo>:</mo><mi>sSet</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\pi_0 : sSet \to Set</annotation></semantics></math> be the connected components functor, i.e. the left adjoint of the constant functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cst</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">cst : Set \to sSet</annotation></semantics></math>. A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f : Z \to W</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> is a weak homotopy equivalence if and only if the induced map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>f</mi></msup><mo>:</mo><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>W</mi></msup><mo>→</mo><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>Z</mi></msup></mrow><annotation encoding="application/x-tex">\pi_0 K^f : \pi_0 K^W \to \pi_0 K^Z</annotation></semantics></math></div> <p>is a bijection for all <em>Kan complexes</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_95">Proof</h6> <p>One direction is easy: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, then axiomS FOR <a class="existingWikiWord" href="/nlab/show/simplicial+model+categories">simplicial model categories</a> (Def. <a class="maruku-ref" href="#SimplicialModelCategory"></a>) implies the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo>:</mo><msup><mi>sSet</mi> <mi>op</mi></msup><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">K^{(-)} : sSet^{op} \to sSet</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/right+Quillen+functor">right Quillen functor</a>, so <a class="existingWikiWord" href="/nlab/show/Ken+Brown%27s+lemma">Ken Brown's lemma</a> (Prop. <a class="maruku-ref" href="#KenBrownLemma"></a>) implies that it preserves all weak homotopy equivalences; in particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo>:</mo><msup><mi>sSet</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\pi_0 K^{(-)} : sSet^{op} \to Set</annotation></semantics></math> sends weak homotopy equivalences to bijections.</p> <p>Conversely, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a Kan complex, there is a natural bijection between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">\pi_0 K^X</annotation></semantics></math> and the hom-set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>sSet</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho (sSet) (X, K)</annotation></semantics></math>, and thus by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f : Z \to W</annotation></semantics></math> such that the induced morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>W</mi></msup><mo>→</mo><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>K</mi> <mi>Z</mi></msup></mrow><annotation encoding="application/x-tex">\pi_0 K^W \to \pi_0 K^Z</annotation></semantics></math> is a bijection for all Kan complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is precisely a morphism that becomes an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>sSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho (sSet)</annotation></semantics></math>, i.e. a weak homotopy equivalence.</p> </div> <div class="num_theorem" id="TopsSetQuillenEquivalence"> <h6 id="theorem_8">Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> and <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a>/<a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>-<a class="existingWikiWord" href="/nlab/show/nerve+and+realization">adjunction</a> of example <a class="maruku-ref" href="#TopologicalRealizationOfSimplicialSets"></a> constitutes a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> of def. <a class="maruku-ref" href="#ClassesOfMorphismsOnsSetQuillen"></a> and the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>⊣</mo><mi>Sing</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Top</mi> <mi>Quillen</mi></msub><munderover><mrow><mphantom><mrow><msub><mrow></mrow> <mi>Q</mi></msub></mrow></mphantom><msub><mo>≃</mo> <mi>Q</mi></msub></mrow><munder><mo>⟶</mo><mi>Sing</mi></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></munderover><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> ({\vert -\vert}\dashv Sing) \;\colon\; Top_{Quillen} \underoverset {\underset{Sing}{\longrightarrow}} {\overset{{\vert -\vert}}{\longleftarrow}} {\phantom{{}_{Q}}\simeq_{Q}} sSet_{Quillen} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_96">Proof</h6> <p>First of all, the adjunction is indeed a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>: prop. <a class="maruku-ref" href="#SingDetextsAndReflectsFibrations"></a> says in particular that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(-)</annotation></semantics></math> takes <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a> to <a class="existingWikiWord" href="/nlab/show/Kan+fibrations">Kan fibrations</a> and prop. <a class="maruku-ref" href="#TopologicalRealizationOfsSetLandsInCWComplexes"></a> gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert-\vert}</annotation></semantics></math> sends monomorphisms of simplicial sets to <a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a>.</p> <p>Now prop. <a class="maruku-ref" href="#UnitOfSingularNerveAndRealizationIsWEOnKanComplexes"></a> says that the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> and <a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a> are weak equivalences, and hence the Quillen adjunction is a Quillen equivalence.</p> </div> </body></html> </div> <div class="revisedby"> <p> Last revised on October 23, 2022 at 15:17:17. 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