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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#remarks'>Remarks</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#waldhausen_category_of_a_small_abelian_category'>Waldhausen category of a small abelian category</a></li> <li><a href='#waldhausen_category_of_a_small_exact_category'>Waldhausen category of a small exact category</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>Waldhausen category</em> <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">C'</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> equipped with a bit of extra structure that allows us to consider it as a presentation (via <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a>) of an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-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> such that the extra structure allows us to conveniently compute the <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>K</mi></mstyle><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{K}(C)</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">C</annotation></semantics></math>.</p> <p>Notably a Waldhausen category provides the notion of co<a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a>s, which are crucial structures controlling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>K</mi></mstyle><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{K}(C)</annotation></semantics></math>. Dual to the discussion at <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>, ordinary <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>s in Waldhausen 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><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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\hookrightarrow& B \\ \downarrow && \downarrow \\ 0 &\to& B//A } </annotation></semantics></math></div> <p>with <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 \hookrightarrow B</annotation></semantics></math> a special morphism called a Waldhausen cofibration compute <em>homotopy pushout</em>s and hence coexact sequences in the corresponding <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a>.</p> <p>Using this, the <a class="existingWikiWord" href="/nlab/show/Waldhausen+S-construction">Waldhausen S-construction</a> on <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">C'</annotation></semantics></math> is an algorithm for computing the <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> spectrum 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>.</p> <h2 id="definition">Definition</h2> <p>Waldhausen in his work in <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> introduced the notion of a category with cofibrations and weak equivalences, nowadays known as <em>Waldhausen category</em>. As the original name suggests, this is a 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">C</annotation></semantics></math> with zero object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, equipped with a choice of two classes of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">cof</mi></mrow><annotation encoding="application/x-tex">\mathrm{cof}</annotation></semantics></math> of the cofibrations and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>.</mo><mi>e</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">w.e.</annotation></semantics></math> of weak equivalences such that</p> <ul> <li> <p>(C1) all isomorphisms are cofibrations</p> </li> <li> <p>(C2) there is a zero object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and for any object <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 unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">0\to a</annotation></semantics></math> is a cofibration</p> </li> <li> <p>(C3) 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\hookrightarrow b</annotation></semantics></math> is a cofibration and <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\to c</annotation></semantics></math> any morphism then the pushout <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>→</mo><mi>b</mi><msub><mo>∪</mo> <mi>a</mi></msub><mi>c</mi></mrow><annotation encoding="application/x-tex">c\to b\cup_a c</annotation></semantics></math> is a cofibration</p> </li> <li> <p>(W1) all isomorphisms are weak equivalences</p> </li> <li> <p>(W2) weak equivalences are closed under composition (make a subcategory)</p> </li> <li> <p>(W3) “glueing for weak equivalences”: Given any commutative 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>D</mi></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mo>∼</mo></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>∼</mo></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>∼</mo></msup></mtd></mtr> <mtr><mtd><mi>D</mi><mo>′</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>A</mi><mo>′</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>B</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ D &\leftarrow& A &\hookrightarrow &B\\ \downarrow^\sim&& \downarrow^\sim &&\downarrow^\sim\\ D' &\leftarrow &A' &\hookrightarrow &B' } </annotation></semantics></math></div> <p>in which the vertical arrows are weak equivalences and right horizontal maps cofibrations, the induced map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>∪</mo> <mi>A</mi></msub><mi>D</mi><mo>↪</mo><mi>B</mi><mo>′</mo><msub><mo>∪</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub><mi>D</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">B\cup_A D\hookrightarrow B'\cup_{A'} D'</annotation></semantics></math> is a weak equivalence.</p> </li> </ul> <p>The axioms imply that for any cofibration <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\hookrightarrow B</annotation></semantics></math> there is a cofibration sequence <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>B</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A\hookrightarrow B\to B/A</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">B/A</annotation></semantics></math> is the choice of the cokernel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>∪</mo> <mi>A</mi></msub><mn>0</mn></mrow><annotation encoding="application/x-tex">B\cup_A 0</annotation></semantics></math>.</p> <p>Given a Waldhausen 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">C</annotation></semantics></math> whose weak equivalence classes from a set, one defines <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K_0(C)</annotation></semantics></math> as an abelian group whose elements are the weak equivalence classes modulo the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>B</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A]+[B/A]=[B]</annotation></semantics></math> for any cofibration sequence <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>B</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A\hookrightarrow B\to B/A</annotation></semantics></math>.</p> <p>Waldhausen then devises the so called S-construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↦</mo><msub><mi>S</mi> <mo>•</mo></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">C\mapsto S_\bullet C</annotation></semantics></math> from Waldhausen categories to simplicial categories with cofibrations and weak equivalences (hence one can iterate the construction producing multisimplicial categories).</p> <p>The <span class="newWikiWord">K-theory space<a href="/nlab/new/K-theory+space">?</a></span> of a Waldhausen construction is given by formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><msub><mi mathvariant="normal">hocolim</mi> <mrow><msup><mi>Δ</mi> <mi mathvariant="normal">op</mi></msup></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>N</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>w</mi><mo>.</mo><mi>e</mi><mo>.</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>n</mi></msub><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega\mathrm{hocolim}_{\Delta^{\mathrm{op}}}([n]\mapsto N_\bullet(w.e.(S_n C)))</annotation></semantics></math>, where <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 the loop space functor, <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 simplicial <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>, w.e. takes the (simplicial) subcategory of weak equivalence and <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><mo>∈</mo><mi>Δ</mi></mrow><annotation encoding="application/x-tex">[n]\in\Delta</annotation></semantics></math>. This construction can be improved (using iterated <a class="existingWikiWord" href="/nlab/show/Waldhausen+S-construction">Waldhausen S-construction</a>) to the <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> <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/spectrum">spectrum</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">C</annotation></semantics></math>; the K-theory space will be just the one-space of the K-theory spectrum.</p> <p>Then the K-groups are the <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s of the K-theory space.</p> <h2 id="remarks">Remarks</h2> <ul> <li>The axioms of a Waldhausen 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">C</annotation></semantics></math> are very similar to the axioms of a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> on the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> in which the initial object is also terminal. One difference is that the weak equivalences in a Waldhausen category are not required to satisfy <a class="existingWikiWord" href="/nlab/show/2-out-of-3">2-out-of-3</a>. For example, Waldhausen gives an example of a Waldhausen category where the weak equivalences are <span class="newWikiWord">simple homotopy equivalences<a href="/nlab/new/simple+homotopy+equivalences">?</a></span>. Another difference is in axiom W3, whose analog in a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> is the axioms that every object has a <a class="existingWikiWord" href="/nlab/show/path+object">path object</a>. It still follows that one has fibration sequences in a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>.</li> </ul> <h2 id="examples">Examples</h2> <h3 id="waldhausen_category_of_a_small_abelian_category">Waldhausen category of a small abelian category</h3> <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">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of bounded chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^b(C)</annotation></semantics></math> becomes a Waldhausen category by taking</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> of chain complexes;</p> </li> <li> <p>a cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>A</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">f : A_\bullet \to X_\bullet</annotation></semantics></math> is a chain morphism that is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</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">C</annotation></semantics></math> in each degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_n : A_n \to X_n</annotation></semantics></math>.</p> </li> </ul> <h3 id="waldhausen_category_of_a_small_exact_category">Waldhausen category of a small exact category</h3> <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">C</annotation></semantics></math> just a <a class="existingWikiWord" href="/nlab/show/Quillen+exact+category">Quillen exact category</a> with ambient <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat C</annotation></semantics></math> there is an analogous, slightly more sophisticated construction of a Waldhausen category structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^b(C)</annotation></semantics></math>:</p> <ul> <li> <p>weak equivalences are the morphisms that are <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>s when regarded as morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat C</annotation></semantics></math>;</p> </li> <li> <p>the cofibrations are the degreewise <em>admissible morphisms</em>, i.e. those morphisms <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 \to X</annotation></semantics></math> such that the pushout <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>X</mi><mo>→</mo><mi>A</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \to X \to A/X</annotation></semantics></math> computed in the ambient <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat C</annotation></semantics></math> is 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>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/category+of+cofibrant+objects">category of cofibrant objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> </ul> <h2 id="references">References</h2> <p>Waldhausen categories are discussed with an eye towards their application in the computation of <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>s in <a href="http://www.math.rutgers.edu/~weibel/Kbook/Kbook.II.pdf">chapter 2</a> of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, <em>The K-book: An introduction to algebraic K-theory</em> (<a href="http://www.math.rutgers.edu/~weibel/Kbook.html">web</a>)</li> </ul> <p>Section 1 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/R.+W.+Thomason">R. W. Thomason</a>, Thomas Trobaugh, <em>Higher algebraic K-theory of schemes and of derived categories</em>, <em>The Grothendieck Festschrift</em>, 1990, 247-435.(<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/tt.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 26, 2022 at 18:54:55. 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