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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="cohesive_toposes">Cohesive <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>-Toposes</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></strong></p> <p><strong>Backround</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+cohesive+toposes">motivation for cohesive toposes</a></p> </li> </ul> <p><strong>Definition</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">strongly ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">totally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a> / <a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> / <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <p><strong>Presentation over a site</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+site">locally connected site</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+site">locally ∞-connected site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+site">connected site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+site">∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+site">strongly connected site</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+site">strongly ∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+site">totally connected site</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+site">totally ∞-connected site</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+site">local site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-local+site">∞-local site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-cohesive+site">∞-cohesive site</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, <a class="existingWikiWord" href="/nlab/show/discrete+groupoid">discrete groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-topological+%E2%88%9E-groupoid">D-topological ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological groupoid</a>, <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>, <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-groupoid">Lie 2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+super+%E2%88%9E-groupoid">smooth super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a>, <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a>, <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/synthetic+differential+super+%E2%88%9E-groupoid">synthetic differential super ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion+of+global-+over+G-equivariant+homotopy+theory">cohesion of global- over G-equivariant homotopy theory</a></strong></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> <ul> <li><a href='#observation'>Observation</a></li> </ul> <li><a href='#structures_in_'>Structures in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">Disc\infty Grpd</annotation></semantics></math></a></li> <ul> <li><a href='#TopAsCohesiveInfinTopos'>Geometric homotopy and Galois theory</a></li> <ul> <li><a href='#observation_2'>Observation</a></li> </ul> <li><a href='#cohomology_and_principal_bundles'>Cohomology and principal <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>-bundles</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>The term <em>discrete <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>-groupoid</em> or <em>bare <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>-groupoid</em> or <em>geometrically <a class="existingWikiWord" href="/nlab/show/discrete+object">discrete</a> <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+type">geometric homotopy type</a></em> is essentially synonymous to just <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></em> or just <em><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></em>. It is used for emphasis in contexts where one considers <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 with extra <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+type">geometric structure</a> (e.g. <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive</a> <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">structure</a>) to indicate that this extra structure is being disregarded, or rather that the special case of <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete</a> such structure is considered.</p> <h2 id="definition">Definition</h2> <div class="un_observation"> <h6 id="observation">Observation</h6> <p>The <a class="existingWikiWord" href="/nlab/show/terminal+object+in+an+%28%E2%88%9E%2C1%29-category">terminal</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-sheaf (∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is trivially a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a>, where each of the defining four <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mi>Disc</mi><mo>⊣</mo><mi>Γ</mi><mo>⊣</mo><mi>coDisc</mi><mo stretchy="false">)</mo><mo>:</mo><mn>∞</mn><mi>Grpd</mi><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \infty Grpd \to \infty Grpd</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a>.</p> </div> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>In the context of <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a>es we say that <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> defines <strong>discrete cohesion</strong> and refer to its objects as <strong>discrete <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</strong>.</p> <p>More generally, given any other <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</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>⊣</mo><mi>Disc</mi><mo>⊣</mo><mi>Γ</mi><mo>⊣</mo><mi>codisc</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>Disc</mi></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (\Pi \dashv Disc \dashv \Gamma \dashv codisc) : \mathbf{H} \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi></mrow><annotation encoding="application/x-tex">Disc</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> functor is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+%28%E2%88%9E%2C1%29-functor">full and faithful (∞,1)-functor</a> and hence embeds <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> as a full <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-category">sub-(∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. A general object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is a <em>cohesive <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>-groupoid</em> . We say <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>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> is a <strong>discrete <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>-groupoid</strong> if it is in 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><mi>Disc</mi></mrow><annotation encoding="application/x-tex">Disc</annotation></semantics></math>.</p> </div> <div class="un_remark"> <h6 id="remark">Remark</h6> <p>This generalizes the traditional use of the terms <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a> and <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>:</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a> is equivalently a 0-<a class="existingWikiWord" href="/nlab/show/truncated">truncated</a> discrete <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>-groupoid;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> is equivalently a 0-<a class="existingWikiWord" href="/nlab/show/truncated">truncated</a> <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object</a> in discrete <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.</p> </li> </ul> </div> <h2 id="structures_in_">Structures in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">Disc\infty Grpd</annotation></semantics></math></h2> <p>We discuss now some of the general abstract <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos+--+structures">structures in a cohesive (∞,1)-topos</a> realized in discrete <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.</p> <h3 id="TopAsCohesiveInfinTopos">Geometric homotopy and Galois theory</h3> <p>We discuss the general absatract notion of geometric homotopy in cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes (see <a href="http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#Homotopy">here</a>) in the context of discrete cohesion.</p> <p>By the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es <a class="existingWikiWord" href="/nlab/show/Top">Top</a> and <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> are <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalent</a>, hence indistinguishable by general abstract constructions in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a>. However, in practice it can be useful to distinguish them as two different <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentations</a> for an equivalence class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes.</p> <p>For that purposes consider the following</p> <div class="un_defn"> <h6 id="definition_3">Definition</h6> <p>Define the <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo>:</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex"> Top := N(Top_{Quillen})^\circ </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><mn>∞</mn><mi>Grpd</mi><mo>:</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \infty Grpd := N(sSet_{Quillen})^\circ \,, </annotation></semantics></math></div> <p>where on the right we have the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">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> and the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N((-)^\circ)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a> of the <a class="existingWikiWord" href="/nlab/show/simplicial+category">simplicial category</a> given by the full <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-subcategory of these <a class="existingWikiWord" href="/nlab/show/simplicial+model+categories">simplicial model categories</a> on fibrant-cofibrant objects.</p> </div> <p>For</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><mo>⊣</mo><mi>Sing</mi></mrow><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mover><munder><mo>→</mo><mi>Sing</mi></munder><mover><mo>←</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> ({|-| \dashv Sing}) : Top_{Quillen} \stackrel{\overset{{|-|}}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen} </annotation></semantics></math></div> <p>the standard <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem given by the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a>-functor and <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>, 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>𝕃</mi><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>⊣</mo><mi>ℝ</mi><mi>Sing</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mover><munder><mo>→</mo><mrow><mi>ℝ</mi><mi>Sing</mi></mrow></munder><mover><mo>←</mo><mrow><mi>𝕃</mi><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mrow></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (\mathbb{L} {|-|} \dashv \mathbb{R}Sing) : Top \stackrel{\overset{\mathbb{L}{|-|}}{\leftarrow}}{\underset{\mathbb{R}Sing}{\to}} \infty Grpd </annotation></semantics></math></div> <p>for the corresponding <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>s (the image under the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a> of the restriction of <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">{|-|}</annotation></semantics></math> and <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> to fibrant-cofibrant objects followed by functorial fibrant-cofibrant replacement) that constitute a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a>s modeled as morphisms of <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>.</p> <p>Since this is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a> either functor serves as the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> and <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> and so we have</p> <div class="un_observation"> <h6 id="observation_2">Observation</h6> <p><a class="existingWikiWord" href="/nlab/show/Top">Top</a> is exhibited a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> by setting</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>⊣</mo><mi>Disc</mi><mo>⊣</mo><mi>Γ</mi><mo>⊣</mo><mi>coDisc</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mover><mover><mover><munder><mo>←</mo><mrow><mi>𝕃</mi><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mrow></munder><mover><mo>→</mo><mrow><mi>ℝ</mi><mi>Sing</mi></mrow></mover></mover><mover><mo>←</mo><mrow><mi>𝕃</mi><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mrow></mover></mover><mover><mo>→</mo><mrow><mi>ℝ</mi><mi>Sing</mi></mrow></mover></mover><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : Top \stackrel{\overset{\mathbb{R}Sing}{\to}}{\stackrel{\overset{\mathbb{L}{|-|}}{\leftarrow}}{\stackrel{\overset{\mathbb{R}Sing}{\to}}{\underset{\mathbb{L}{|-|}}{\leftarrow}}}} \infty Grpd \,. \,. </annotation></semantics></math></div> <p>In particular a presentation of the intrinsic <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> is given by the familiar <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> construction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℝ</mi><mi>Sing</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi(X) \simeq \mathbb{R} Sing X \,. </annotation></semantics></math></div></div> <div class="un_remark"> <h6 id="remark_2">Remark</h6> <p>While degenerate, it is sometimes useful to make this example of a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> explicit. For instance it allows to think of simplicial models for topological fibrations in terms of topological <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>. Some remarks on this are in <a href="http://ncatlab.org/nlab/show/higher+parallel+transport#FlatInTop">Flat higher parallel transport in Top</a>.</p> </div> <div class="un_remark"> <h6 id="remark_3">Remark</h6> <p>Notice that the <a class="existingWikiWord" href="/nlab/show/topology">topology</a> that enters the explicit construction of the objects in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> here does <em>not</em> show up as <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive structure</a>. A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> here is a model for a <em>discrete</em> <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>-groupoid, the <a class="existingWikiWord" href="/nlab/show/topology">topology</a> only serves to allow the construction 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>. For discussion of <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 equipped with genuine <em>topological cohesion</em> see <a class="existingWikiWord" href="/nlab/show/Euclidean-topological+%E2%88%9E-groupoid">Euclidean-topological ∞-groupoid</a>.</p> </div> <h3 id="cohomology_and_principal_bundles">Cohomology and principal <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>-bundles</h3> <p>We discuss the general abstract notion of cohomology and principal <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>-bundles a in cohesive <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>-toposes (see <a href="http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#Cohomology">here</a>) in the context of discrete cohesion.</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">sGrp</mi><mo>=</mo><mi mathvariant="normal">Grp</mi><mo stretchy="false">(</mo><mi mathvariant="normal">sSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{sGrp} = \mathrm{Grp}(\mathrm{sSet})</annotation></semantics></math> for the category of <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a>s.</p> </div> <p>A classical reference is section 17 of <a href="#May">May</a>.</p> <div class="num_prop"> <h6 id="DiscreteLoopingDelooping">Proposition</h6> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">sGrpd</mi></mrow><annotation encoding="application/x-tex">\mathrm{sGrpd}</annotation></semantics></math> inherits a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred</a> along the forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi mathvariant="normal">sGrp</mi><mo>→</mo><mi mathvariant="normal">sSet</mi></mrow><annotation encoding="application/x-tex">F : \mathrm{sGrp} \to \mathrm{sSet}</annotation></semantics></math>.</p> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">sSet</mi> <mn>0</mn></msub><mo>↪</mo><mi mathvariant="normal">sSet</mi></mrow><annotation encoding="application/x-tex">\mathrm{sSet}_0 \hookrightarrow \mathrm{sSet}</annotation></semantics></math> of reduced simplicial sets (simplicial sets with a single vertex) carries a model category structure whose weak equivalences and cofibrations are those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">sSet</mi> <mi mathvariant="normal">Quillen</mi></msub></mrow><annotation encoding="application/x-tex">\mathrm{sSet}_{\mathrm{Quillen}}</annotation></semantics></math>.</p> <p>There is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</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>G</mi><mo>⊣</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo><mo>:</mo><mi>sGrp</mi><mover><munder><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover></munder><mover><mo>←</mo><mi>G</mi></mover></mover><msub><mi>sSet</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> (G \dashv \bar W) : sGrp \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} sSet_{0} </annotation></semantics></math></div> <p>which <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presents</a> the abstract <a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a> equivalence of <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>-categories</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>Ω</mi><mo>⊣</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">)</mo><mo>:</mo><mn>∞</mn><mi>Grpd</mi><mover><munder><mo>→</mo><mi>B</mi></munder><mover><mo>←</mo><mi>Ω</mi></mover></mover><mn>∞</mn><msub><mi>Grpd</mi> <mi>connected</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (\Omega \dashv \mathbf{B}) : \infty Grpd \stackrel{\overset{\Omega}{\leftarrow}}{\underset{B}{\to}} \infty Grpd_{connected} \,, </annotation></semantics></math></div></div> <p>The model structures and the Quillen equivalence are classical, discussed in (<a href="#GoerssJardine">GoerssJardine, section V</a>)</p> <p>This means on abstract grounds that 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/simplicial+group">simplicial group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi><mo>∈</mo><mi mathvariant="normal">sSet</mi></mrow><annotation encoding="application/x-tex">\bar W G \in \mathrm{sSet}</annotation></semantics></math> is a model of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying</a> <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> object <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> for discrte <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/principal+%E2%88%9E-bundles">principal ∞-bundles</a>. The following statements assert that these principal <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>-bundles themselves can be modeled as ordinary <a class="existingWikiWord" href="/nlab/show/simplicial+principal+bundle">simplicial principal bundle</a>s</p> <div class="num_defn"> <h6 id="definition_5">Definition</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/simplicial+group">simplicial group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">\bar W G</annotation></semantics></math> the model for <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> given by the above proposition, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex"> W G \to \bar W G </annotation></semantics></math></div> <p>for the simplicial <a class="existingWikiWord" href="/nlab/show/decalage">decalage</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">\bar W G</annotation></semantics></math>.</p> </div> <p>This characterization of the object going by the classical name <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">W G</annotation></semantics></math> is made fairly explicit in (<a href="#Duskin">Duskin, p. 85</a>).</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">W G \to \bar W G</annotation></semantics></math> is a Kan fibration resolution of the point inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">{*} \to \bar W G</annotation></semantics></math>.</p> </div> <p>This follows directly from the characterization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">W G \to \bar W G</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/decalage">decalage</a>. Pieces of this statement appear in (<a href="#May">May</a>): lemma 18.2 there gives the fibration property, prop. 21.5 the contractibility of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">W G</annotation></semantics></math>.</p> <div class="num_cor"> <h6 id="corollary">Corollary</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/simplicial+group">simplicial group</a>, the sequence of simplicial sets</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>W</mi><mi>G</mi><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex"> G \to W G \to \bar W G </annotation></semantics></math></div> <p>is a presentation of the <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</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><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G \to * \to \mathbf{B}G \,. </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">W G \to \bar W G</annotation></semantics></math> is a model for the universal <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>-principal discrete <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>-bundle (see <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>):</p> <p>every <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>-principal discrete <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>-bundle <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi mathvariant="normal">Grpd</mi></mrow><annotation encoding="application/x-tex">\infty \mathrm{Grpd}</annotation></semantics></math>, which by definition is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</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>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\to& \mathbf{B}G } </annotation></semantics></math></div> <p>in <span class="newWikiWord">?Gpd<a href="/nlab/new/%3FGpd">?</a></span>, is presented in the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a> by the ordinary <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>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>W</mi><mi>G</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><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P &\to& W G \\ \downarrow && \downarrow \\ X &\to& \bar W G } \,. </annotation></semantics></math></div></div> <p>The explicit statement that the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>W</mi><mi>G</mi><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">G \to W G \to \bar W G</annotation></semantics></math> is a model for the looping fiber sequence appears on p. 239 of <em><a class="existingWikiWord" href="/nlab/show/Crossed+Menagerie">Crossed Menagerie</a></em> . The universality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">W G \to \bar W G</annotation></semantics></math> 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>-principal simplicial bundles is the topic of section 21 in (<a href="#May">May</a>), where however it is not made explicit that the “<a class="existingWikiWord" href="/nlab/show/twisted+cartesian+product">twisted cartesian product</a>s” considered there are precisely the models for the pullbacks as above. This is made explicit on page 148 of <em><a class="existingWikiWord" href="/nlab/show/Crossed+Menagerie">Crossed Menagerie</a></em>.</p> <p>In <em><a class="existingWikiWord" href="/nlab/show/Euclidean-topological+%E2%88%9E-groupoid">Euclidean-topological ∞-groupoid</a></em> we discuss how this model of discrete principal <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>-bundles by simplicial principal bundles lifts to a model of topological principal <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>-bundles by simplicial topological bundles principal over <a class="existingWikiWord" href="/nlab/show/simplicial+topological+group">simplicial topological group</a>s.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+groupoid">discrete groupoid</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometries of physics</a></strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>(<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher</a>) <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/site">site</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">∞-sheaf ∞-topos</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/discrete+geometry">discrete geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/terminal+category">Point</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Discrete%E2%88%9EGroupoid">Discrete∞Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SmoothSet">SmoothSet</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGroupoid">Smooth∞Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">formal geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FormalCartSp">FormalCartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FormalSmoothSet">FormalSmoothSet</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FormalSmooth%E2%88%9EGroupoid">FormalSmooth∞Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SuperFormalCartSp">SuperFormalCartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SuperFormalSmooth%E2%88%9EGroupoid">SuperFormalSmooth∞Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">Simplicial groups</a> and <a class="existingWikiWord" href="/nlab/show/simplicial+principal+bundles">simplicial principal bundles</a> are discussed in</p> <ul id="May"> <li><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>Simplicial Objects in Algebraic Topology</em> (<a href="http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu">djvu</a>).</li> </ul> <p>and section V of</p> <ul id="GoerssJardine"> <li><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a> and <a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>, 1999, <em>Simplicial Homotopy Theory</em>, number 174 in Progress in Mathematics, Birkhauser. (<a href="http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html">ps</a>)</li> </ul> <p>The relation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">W G \to \bar W G</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/decalage">decalage</a> is mentioned on p. 85 of</p> <ul id="Duskin"> <li><a class="existingWikiWord" href="/nlab/show/John+Duskin">John Duskin</a>, <em>Simplicial methods and the interpretation of “triple” cohomology</em>, number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc. (1975)</li> </ul> <p>Discrete cohesion is the topic of section 3.1 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em> ,</li> </ul> <p>where much of the above material is taken from.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on June 25, 2018 at 13:00:32. See the <a href="/nlab/history/geometrically+discrete+infinity-groupoid" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/geometrically+discrete+infinity-groupoid" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/geometrically+discrete+infinity-groupoid/15" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/geometrically+discrete+infinity-groupoid" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/geometrically+discrete+infinity-groupoid" accesskey="S" class="navlink" id="history" rel="nofollow">History (15 revisions)</a> <a href="/nlab/show/geometrically+discrete+infinity-groupoid/cite" style="color: black">Cite</a> <a href="/nlab/print/geometrically+discrete+infinity-groupoid" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/geometrically+discrete+infinity-groupoid" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>