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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> <h4 id="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></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='#properties'>Properties</a></li> <ul> <li><a href='#AsKanComplexes'>As Kan complexes</a></li> <li><a href='#FiberSequences'>Fiber sequences</a></li> <li><a href='#free_simplicial_groups'>Free simplicial groups</a></li> <li><a href='#looping_and_delooping'>Looping and delooping</a></li> <li><a href='#AsInfinityGroups'>As <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>-groups</a></li> <li><a href='#ClosedMonoidalStructure'>Closed monoidal structure</a></li> </ul> <li><a href='#DeloopingAndBundle'>Delooping and simplicial principal bundles</a></li> <ul> <li><a href='#Delooping'>Delooping</a></li> <ul> <li><a href='#delooping_modeled_by_'>Delooping modeled by <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></a></li> <li><a href='#delooping_modeled_by__2'>Delooping modeled by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">d B G</annotation></semantics></math></a></li> <li><a href='#examples'>Examples</a></li> </ul> <li><a href='#UniversalBundle'>Cocycles</a></li> <li><a href='#PrincipalBundles'>Simplicial Principal bundles</a></li> <ul> <li><a href='#UniversalSimplicialBundle'>Universal simplicial <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 bundle</a></li> <li><a href='#SimplicialBundleReferences'>References</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references_2'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Recall that a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is a combinatorial <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model</a> for a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. This relation is most immediate when the simplicial set is in fact a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> (an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>).</p> <p>A <em>simplicial group</em> is a simplicial set with the structure of a <a class="existingWikiWord" href="/nlab/show/group">group</a> on it. It turns out that this necessarily means that it is also a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>. Therefore a simplicial group is</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> with an extra group structure on it;</p> </li> <li> <p>a model for a topological space with a group structure.</p> </li> </ul> <p>Accordingly (as discussed at <a class="existingWikiWord" href="/nlab/show/group">group</a>) a simplicial group <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> gives rise to</p> <ul> <li> <p>a one-object <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 <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> whose explicit standard realization as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is denoted <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> </li> <li> <p>an <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{E} G</annotation></semantics></math> whose explicit standard realization as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> (even a simplicial group, again) is denoted <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> </li> <li> <p>such that there is a fibration</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd> <mtd><mo>:</mo><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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mo>:</mo><mo>=</mo></mtd> <mtd><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{E} G &:=& W G \\ \downarrow && \downarrow \\ \mathbf{B} G &:=& \bar W G } </annotation></semantics></math></div> <p>which is the <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal G-bundle</a>.</p> </li> </ul> <p>Simplicial abelian groups are models for <a class="existingWikiWord" href="/nlab/show/connective+spectrum">connective</a> <a class="existingWikiWord" href="/nlab/show/module+spectrum">modules</a> over the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Mac+Lane+spectrum">Eilenberg-Mac Lane spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mstyle mathvariant="bold"><mi>Z</mi></mstyle></mrow><annotation encoding="application/x-tex">H \mathbf{Z}</annotation></semantics></math>; see <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> and <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a>.</p> <h2 id="definition">Definition</h2> <p>A <strong>simplicial group</strong>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, is a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> of <a class="existingWikiWord" href="/nlab/show/group">group</a>s.</p> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> of simplicial groups is the category of functors from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^{op}</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a>. It will be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Simp</mo><mo lspace="0em" rspace="thinmathspace">Grp</mo></mrow><annotation encoding="application/x-tex">\Simp\Grp</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="AsKanComplexes">As Kan complexes</h3> <p> <div class='num_theorem' id='EverySimplicialGroupIsAKanComplex'> <h6>Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> any simplicial group (by <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetting</a> the group <a class="existingWikiWord" href="/nlab/show/structure">structure</a>) is a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>.</p> </div> </p> <p>This is due to <a href="#Moore54">Moore 1954, Theorem 3 on p. 18-04</a>, review in <a href="#Quillen67">Quillen 67, II 3.8</a>, <a href="#MaySimplicialObjects">May 67, Theorem 17.1</a>,<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math> <a href="#Curtis71">Curtis 1971, Sec. 3, Lem. 3.1</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math> <a href="#Weibel94">Weibel 1994, Lem. 8.2.8</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math> <a href="#JoyalTierney05">Joyal & Tierney 2005, p. 14</a>.</p> <p>In fact, not only are the <a class="existingWikiWord" href="/nlab/show/horn">horn</a> fillers guaranteed to exist, but there is an algorithm that provides explicit fillers. This implies that constructions on a simplicial group that use fillers of horns can often be adjusted to be functorial by using the algorithmically defined fillers. An argument that just uses ‘existence’ of fillers can be refined to give something more ‘coherent’.</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a simplicial group.</p> <p>Here is the explicit algorithm that computes the horn fillers:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>y</mi> <mn>0</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>y</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>y</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>y</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(y_0,\ldots, y_{k-1}, -,y_{k+1}, \ldots, y_n)</annotation></semantics></math> give a <a class="existingWikiWord" href="/nlab/show/horn">horn</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">G_{n-1}</annotation></semantics></math>, so the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">y_i</annotation></semantics></math>s are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math> simplices that fit together as if they were all but one, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">k^{th}</annotation></semantics></math> one, of the faces of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex. There are three cases:</p> <ol> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k = 0</annotation></semantics></math>:</p> <ul> <li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>n</mi></msub><mo>=</mo><msub><mi>s</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">w_n = s_{n-1}y_n</annotation></semantics></math> and then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i = w_{i+1}(s_{i-1}d_i w_{i+1})^{-1}s_{i-1}y_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i = n-1, \ldots, 1</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">w_1 </annotation></semantics></math> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i w_1 = y_i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">i\neq 0</annotation></semantics></math>;</li> </ul> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0\lt k \lt n</annotation></semantics></math>:</p> <ul> <li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>0</mn></msub><mo>=</mo><msub><mi>s</mi> <mn>0</mn></msub><msub><mi>y</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">w_0 = s_0 y_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mi>i</mi></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i = 1, \ldots, k-1</annotation></semantics></math>, then take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>n</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>n</mi></msub><msub><mi>w</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">w_n = w_{k-1}(s_{n-1}d_n w_{k-1})^{-1}s_{n-1}y_n</annotation></semantics></math>, and finally a downwards induction given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i = w_{i+1}(s_{i-1}d_{i}w_{i+1})^{-1}s_{i-1}y_i</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i = n-1, \ldots, k+1</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">w_{k+1}</annotation></semantics></math> gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_{i}w_{k+1} = y_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≠</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">i \neq k</annotation></semantics></math>;</li> </ul> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k=n</annotation></semantics></math>:</p> <ul> <li>use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>0</mn></msub><mo>=</mo><msub><mi>s</mi> <mn>0</mn></msub><msub><mi>y</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">w_0 = s_0 y_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>w</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msub><mi>s</mi> <mi>i</mi></msub><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i = 1, \ldots, n-1</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">w_{n-1}</annotation></semantics></math> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>w</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i w_{n-1} = y_i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≠</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">i\neq n</annotation></semantics></math>.</li> </ul> </li> </ol> </div> <div class="num_remark"> <h6 id="remarks">Remarks</h6> <ul> <li> <p>The filler for any horn can be chosen to be a <em>product of degenerate elements</em>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a> of a simplicial group, <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>, can be calculated as the homology groups of the <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. This is, in general, a non-Abelian chain complex.</p> </li> <li> <p>A simplicial group can be considered as a <a class="existingWikiWord" href="/nlab/show/simplicial+groupoid">simplicial groupoid</a> having exactly one object. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a simplicial group, the suggested notation for the corresponding simplicially enriched groupoid would be <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> according to notational conventions suggested elsewhere in the nLab.</p> </li> <li> <p>There is a functor due to Dwyer and Kan, called the <a class="existingWikiWord" href="/nlab/show/Dwyer-Kan+loop+groupoid">Dwyer-Kan loop groupoid</a> that takes a simplicial set to a simplicial groupoid. This has a left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{W}</annotation></semantics></math> (see below), called the <em><a class="existingWikiWord" href="/nlab/show/simplicial+classifying+space">simplicial classifying space</a></em> functor, and together they give an equivalence of categories between the homotopy category of simplical sets and that of simplicial groupoids. We thus have that all homotopy types are modelled by simplicial groupoids … and for connected homotopy types by simplicial groups. One <em>important fact</em> to note in this equivalence is that it shifts dimension by 1, so if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(K)</annotation></semantics></math> is the simplicial group corresponding to the connected simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_k(K)</annotation></semantics></math> is the same as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_{k-1}(G(K))</annotation></semantics></math>. This is important when considering algebraic models for a <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a>.</p> </li> </ul> </div> <h3 id="FiberSequences">Fiber sequences</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a simplicial group and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">G_0</annotation></semantics></math> its group of 0-cells, regarded as a simplicially constant simplicial group. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">G/G_0</annotation></semantics></math> for the evident <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of simplicial groups.</p> <p>The evident morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub><mo>→</mo><mi>G</mi><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Ω</mi><mi>G</mi><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> G_0 \to G \to G/G_0 \simeq \mathbf{B} \Omega G/G_0 </annotation></semantics></math></div> <p>form a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>One checks that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any simplicial set and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a simplicial group acting freely on it, the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> X \to X/G </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>. This is for instance (<a href="#Weibel94">Weibel, exercise 8.2.6</a>). By the disucssion at <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> it is therefore sufficient to observe that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>G</mi> <mn>0</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G_0 &\to& * \\ \downarrow && \downarrow \\ G &\to& G/G_0 } </annotation></semantics></math></div> <p>is an ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of simplicial sets. This is clear since the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">G_0</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is degreewise free (being the action of a subgroup).</p> </div> <div class="num_example"> <h6 id="example">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>δ</mi></mover><msub><mi>G</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G_1 \stackrel{\delta}{\to} G_0)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a> of groups, 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><msub><mi>G</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>δ</mi></mover><msub><mi>G</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>1</mn></msub><mover><munder><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munder><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>δ</mi><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mover><msub><mi>G</mi> <mn>0</mn></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> [G_1 \stackrel{\delta}{\to} G_0] = \left( G_0 \times G_1 \stackrel{\overset{(\delta p_2)\cdot p_1}{\to}}{\underset{p_1}{\to}} G_0 \right) </annotation></semantics></math></div> <p>for <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> which is the corresponding <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">[</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">N[G_1 \to G_0]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> being the corresponding simplicial group. Then the above says that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub><mo>→</mo><mo stretchy="false">[</mo><msub><mi>G</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>δ</mi></mover><msub><mi>G</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> G_0 \to [G_1 \stackrel{\delta}{\to} G_0] \to \mathbf{B}G_1 </annotation></semantics></math></div> <p>is a fiber sequence of <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>s.</p> </div> <h3 id="free_simplicial_groups">Free simplicial groups</h3> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>The <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">forgetful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><mi>AbSGrpg</mi><mo>→</mo><mi>SSet</mi></mrow><annotation encoding="application/x-tex"> U : AbSGrpg \to SSet </annotation></semantics></math></div> <p>from simplicial <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s to the underlying <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>:</mo><mi>SSet</mi><mo>→</mo><mi>AbSimpGrp</mi></mrow><annotation encoding="application/x-tex"> \mathbb{Z} : SSet \to AbSimpGrp </annotation></semantics></math></div> <p>from <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> to abelian simplicial groups, the <strong>free simplicial abelian group</strong> functor that sends the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices to the free abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℤ</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><mi>ℤ</mi><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">(\mathbb{Z}X)_n = \mathbb{Z} X_n</annotation></semantics></math> over it.</p> <p>This functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> has the following properties:</p> <ul> <li> <p>it preserves weak equivalences</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} X</annotation></semantics></math> is a cofibrant simplicial group</p> </li> </ul> </div> <p>(…)</p> <h3 id="looping_and_delooping">Looping and delooping</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mn>0</mn></msub><mo>↪</mo></mrow><annotation encoding="application/x-tex">sSet_0 \hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> be the category of <em>reduced simplicial sets</em> (simplicial sets with a single 0-cell).</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mi>sSet</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X \in sSet_0</annotation></semantics></math> define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>∈</mo><mi>sGrpd</mi></mrow><annotation encoding="application/x-tex">\Omega X \in sGrpd</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>F</mi><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>s</mi> <mn>0</mn></msub><mi>F</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega X : [n] \mapsto (F X_{n+1})/ s_0 F(X_n) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>:</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><mo>.</mo><mo>.</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega X : ([n] \to [k]) \mapsto ... </annotation></semantics></math></div></div> <h3 id="AsInfinityGroups">As <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>-groups</h3> <p>Simplicial groups are models for <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>s. This is exhibited by the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a>. See also <a href="http://ncatlab.org/nlab/show/groupoid+object+in+an+(infinity%2C1)-category#ModelsInInfGrpd">models for group objects in ∞Grpd</a>.</p> <p>Another equivalent model is that of <a class="existingWikiWord" href="/nlab/show/connected">connected</a> <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es.</p> <p>At the abstract level of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> this equivalence is induced by forming <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>s and <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo>:</mo><mn>∞</mn><msub><mi>Grpd</mi> <mi>con</mi></msub><mover><mo>→</mo><mo>←</mo></mover><mn>∞</mn><mi>Grp</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega : \infty Grpd_{con} \stackrel{\leftarrow}{\to} \infty Grp : \mathbf{B} \,. </annotation></semantics></math></div> <p>This <a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">(∞,1)-equivalence</a> is modeled by a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> whose <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> Quillen functor is the operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{W}</annotation></semantics></math> discussed above.</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><msub><mi>sSet</mi> <mn>0</mn></msub><mover><mo>⟶</mo><mover><mo>⟵</mo><mrow><msub><mo>≃</mo> <mi>Quillen</mi></msub></mrow></mover></mover><mi>sGrp</mi><mo>:</mo><mover><mi>W</mi><mo>¯</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{G} : sSet_0 \stackrel{\stackrel{\simeq_{Quillen}}{\longleftarrow}}{\longrightarrow} sGrp : \overline{W} \,. </annotation></semantics></math></div> <p>This is for instance in <a href="#GoerssJardine">GoerssJardine, chapter 5</a>.</p> <p>See also <a href="http://ncatlab.org/nlab/show/groupoid+object+in+an+(infinity%2C1)-category#ModelsInInfGrpd">group object in an (∞,1)-category – models for groups in ∞Grpd</a>.</p> <h3 id="ClosedMonoidalStructure">Closed monoidal structure</h3> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sAb</mi></mrow><annotation encoding="application/x-tex">sAb</annotation></semantics></math> of simplicial abelian groups is naturally a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, with the tensor product being degreewise that of abelian groups. This is indeed a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A, B</annotation></semantics></math> The <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A,B]</annotation></semantics></math> is the simplicial abelian group whose underlying simplicial set is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>sAb</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [A,B] : [n] \mapsto Hom_{sAb}(A \otimes \mathbb{Z}[\Delta[n]], B) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mi>sSet</mi><mo>→</mo><mi>sAb</mi></mrow><annotation encoding="application/x-tex">\mathbf{Z}[-] : sSet \to sAb</annotation></semantics></math> is degreewise the <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> abelian group functor.</p> <h2 id="DeloopingAndBundle">Delooping and simplicial principal bundles</h2> <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 simplicial group, we describe its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \in sSet</annotation></semantics></math> and the corresponding <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}G \to \mathbf{B}G</annotation></semantics></math> such that 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><msub><mi>P</mi> <mo>•</mo></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mo>•</mo></msub></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P_\bullet &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ X_\bullet &\stackrel{g}{\to}& \mathbf{B}G } </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> models the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>P</mi> <mo>•</mo></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mo>•</mo></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P_\bullet &\to& * \\ \downarrow &\swArrow& \downarrow \\ X_\bullet &\to& \mathbf{B}G } \,. </annotation></semantics></math></div> <p>in the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a> and hence produces the <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">P_\bullet \to X_\bullet</annotation></semantics></math> classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X_\bullet \to \mathbf{B}G</annotation></semantics></math>.</p> <p>For all these constructions exist very explicit combinatorial formulas that go by the symbols</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">\overline{W}G</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></p> </li> <li> <p><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> for the <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}G</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>:</mo><msub><mi>X</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\tau : X_\bullet \to G_\bullet</annotation></semantics></math> (called the <em><a class="existingWikiWord" href="/nlab/show/twisting+function">twisting function</a></em>) for the <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X_\bullet \to \mathbf{B}G</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><msub><mo>×</mo> <mi>g</mi></msub><mi>W</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">X_\bullet \times_g W G</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">P_\bullet</annotation></semantics></math> (called <em><a class="existingWikiWord" href="/nlab/show/twisted+Cartesian+product">twisted Cartesian product</a></em> ).</p> </li> </ul> <p>All of these constructions are functorial and hence lift from the context of <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s to that of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> over some <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><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. There they provide models for strict <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28infinity%2C1%29-category">group objects</a>, <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> and <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s in the corresponding <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. In particular in the projective model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]</annotation></semantics></math> the pullback of the objectwise <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>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">W G \to \overline{W}G</annotation></semantics></math> is still a homotopy pullback and models the corresponding 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.</p> <h3 id="Delooping">Delooping</h3> <p>A simplicial group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28infinity%2C1%29-category">group object</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to the category of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es. Accordingly, there should be a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> which is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, i.e. a Kan complex with an essentially unique object, such that the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> of that Kan complex reproduces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>An explicit construction of <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> from <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> goes traditionally by the symbol <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>KanCplx</mi></mrow><annotation encoding="application/x-tex">\bar W G \in KanCplx</annotation></semantics></math>. Another one by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">d B G</annotation></semantics></math>.</p> <h4 id="delooping_modeled_by_">Delooping modeled by <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></h4> <p>It is immediate to deloop the simplicial group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/simplicial+groupoid">simplicial groupoid</a> that in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is the 1-<a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> with a single object and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">G_k</annotation></semantics></math> as its collection of morphisms.</p> <div class="num_defn"> <h6 id="definition_3">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">\mathcal{G}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+groupoid">simplicial groupoid</a> that on objects is a constant simplicial set, define a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\bar W \mathcal{G}</annotation></semantics></math> as follows.</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>:</mo><mo>=</mo><mi>ob</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\overline{W}\mathcal{G})_0 := ob(\mathcal{G}_0)</annotation></semantics></math>, the set of objects of the groupoid of 0-simplices (and hence of the groupoid at each level);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>=</mo><mi>Mor</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\overline{W}\mathcal{G})_1 = Mor(\mathcal{G}_0)</annotation></semantics></math>, the collection of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s of the groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_0</annotation></semantics></math>:</p> </li> </ul> <p>and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math>,</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mi>h</mi> <mi>i</mi></msub><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\overline{W}\mathcal{G})_n = \{(h_{n-1}, \ldots ,h_0)| h_i \in Mor(\mathcal{G}_i)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mn>0</mn><mo><</mo><mi>i</mi><mo><</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">s(h_{i-1}) = t(h_i), 0\lt i\lt n\}</annotation></semantics></math>.</li> </ul> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> are generic symbols for the domain and codomain mappings of all the groupoids involved. The face and degeneracy mappings between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>𝒢</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\overline{W}(\mathcal{G})_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>𝒢</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\overline{W}(\mathcal{G})_0</annotation></semantics></math> are the source and target maps and the identity maps of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_0</annotation></semantics></math>, respectively; whilst the face and degeneracy maps at higher levels are given as follows:</p> <p>The face and degeneracy maps are given by</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_0(h_{n-1}, \ldots, h_0) = (h_{n-2}, \ldots, h_0)</annotation></semantics></math>;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>i</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \lt i\lt n</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>d</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>d</mi> <mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>d</mi> <mn>0</mn></msub><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msub><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_i(h_{n-1}, \ldots, h_0) = (d_{i-1}h_{n-1}, d_{i-2}h_{n-2}, \ldots, d_0h_{n-i}h_{n-i-1},h_{n-i-2}, \ldots , h_0)</annotation></semantics></math>;</p> </li> </ul> <p>and</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><msub><mi>h</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_n(h_{n-1}, \ldots, h_0) = (d_{n-1}h_{n-1}, d_{n-2}h_{n-2}, \ldots, d_1h_{1})</annotation></semantics></math>;</li> </ul> <p>whilst</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mrow><mi>dom</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s_0(h_{n-1}, \ldots, h_0) = (id_{dom(h_{n-1})},h_{n-1}, \ldots, h_0) </annotation></semantics></math>;</li> </ul> <p>and,</p> <ul> <li>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0\lt i \leq n</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>s</mi> <mn>0</mn></msub><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mi>id</mi> <mrow><mi>cod</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>h</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s_i(h_{n-1}, \ldots, h_0) = (s_{i-1}h_{n-1}, \ldots, s_0h_{n-i}, id_{cod(h_{n-i})},h_{n-i-1}, \ldots, h_0) </annotation></semantics></math>.</li> </ul> </div> <div class="num_defn"> <h6 id="definition_4">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 simplicial group and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math> the corresponding one-object <a class="existingWikiWord" href="/nlab/show/simplicial+groupoid">simplicial groupoid</a>, one writes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi><mo>:</mo><mo>=</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>𝒢</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \overline{W}G := \overline{W}\mathcal{G} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The above construction has a straightforward <a class="existingWikiWord" href="/nlab/show/internalization">internalization</a> to contexts other than <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. For instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> in <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>s or in <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>s, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">\overline{W}G</annotation></semantics></math> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>:</mo><mo>=</mo><msub><mi>G</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>×</mo><msub><mi>G</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>×</mo><mi>⋯</mi><mo>×</mo><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> (\overline{W}G)_n := G_{n-1} \times G_{n-2} \times \cdots \times G_0 </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> in this context (<a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s, etc.)</p> <p>In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">G : C^{op} \to sSet</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a> that is objectwise a simplicial group, then we have the simplicial presheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi><mo>:</mo><mi>c</mi><mo>↦</mo><mover><mi>W</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \overline{W}G : c \mapsto \overline{W}(G(c)) \,. </annotation></semantics></math></div></div> <h4 id="delooping_modeled_by__2">Delooping modeled by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">d B G</annotation></semantics></math></h4> <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>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/bisimplicial+set">bisimplicial set</a> obtained by taking degreewise the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>B</mi><mi>G</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">d B G \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> for its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi><mo>≃</mo><mi>d</mi><mi>B</mi><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \bar W G \simeq d B G \,. </annotation></semantics></math></div></div> <p>This is shown for instance in (<a href="#JardineLuo">JardineLuo</a>) and in (<a href="#CegarraRemedios">CegarraRemedios</a>).</p> <h4 id="examples">Examples</h4> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is an ordinary <a class="existingWikiWord" href="/nlab/show/group">group</a>, regarded as a simplicially constant simplicial group, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">\overline{W}G</annotation></semantics></math> is the usual bar complex of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><mi>G</mi><mo>×</mo><mi>G</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>G</mi><mover><mo>→</mo><mo>→</mo></mover><mo>*</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \overline{W}G = \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} * \right) \,. </annotation></semantics></math></div> <h3 id="UniversalBundle">Cocycles</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> a morphism</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><msub><mi>X</mi> <mo>•</mo></msub><mo>→</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex"> g : X_\bullet \to \overline{W}G </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> corresponds precisely to what is called a <a class="existingWikiWord" href="/nlab/show/twisting+function">twisting function</a>, a family of maps</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>ϕ</mi><mo stretchy="false">(</mo><mi>g</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>:</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>G</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \{\phi(g)_n : X_n \to G_{n-1}\} </annotation></semantics></math></div> <p>satisfying the relations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>d</mi> <mn>0</mn></msub><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>1</mn></msub><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>0</mn></msub><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mi>i</mi></msub><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>i</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr> <mtr><mtd><msub><mi>s</mi> <mi>i</mi></msub><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>i</mi><mo>≥</mo><mn>0</mn><mo>,</mo></mtd></mtr> <mtr><mtd><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>0</mn></msub><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>G</mi></msub><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ d_0 \phi(x) = \phi(d_1 x)(\phi(d_0 x))^{-1} \\ d_i \phi(x) = \phi(d_{i+1}x), i\gt 0, \\ s_i\phi(x) = \phi(s_{i+1}x), i\geq 0, \\ \phi(s_0 x) = 1_G. } </annotation></semantics></math></div> <h3 id="PrincipalBundles">Simplicial Principal bundles</h3> <p>Simplicial groups model all <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>s in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. Accordingly all <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> should be modeled by <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><strong>(principal action)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a simplicial group. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, an <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>E</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> \rho : E \times G \to E </annotation></semantics></math></div> <p>is called <strong>principal</strong> if it is degreewise principal, i.e. if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the only elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><msub><mi>G</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">g \in G_n</annotation></semantics></math> that have any fixed point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><msub><mi>E</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">e \in E_n</annotation></semantics></math> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\rho(e,g) = e</annotation></semantics></math> are the neutral elements.</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>The canonical action</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>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> G \times G \to G </annotation></semantics></math></div> <p>of any simplicial group on itself is principal.</p> </div> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p><strong>(simplicial principal bundle)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a simplicial group, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is called a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<strong>simplicial principal bundle</strong> if</p> <ul> <li> <p>the action is principal;</p> </li> <li> <p>the base is isomorphic to the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">/</mo><mi>G</mi><mo>:</mo><mo>=</mo><msub><mi>lim</mi> <mo>→</mo></msub><mo stretchy="false">(</mo><mi>E</mi><mo>×</mo><mi>G</mi><mover><mrow><munder><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munder><mi>E</mi></mrow><mover><mo>→</mo><mi>ρ</mi></mover></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E/G := \lim_{\to}(E \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to} E})</annotation></semantics></math> by the action:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">/</mo><mi>G</mi><mo>≃</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E/G \simeq X \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>A simplicial <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 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> is necessarly a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This appears as Lemma 18.2 in <a href="#MaySimplicialObjects">MaySimpOb</a>.</p> </div> <h4 id="UniversalSimplicialBundle">Universal simplicial <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 bundle</h4> <div class="num_defn"> <h6 id="definition_7">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 simplicial group, define the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">W G</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/decalage">decalage</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">\overline{W}G</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mi>G</mi><mo>:</mo><mo>=</mo><mi>Dec</mi><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> W G := Dec \overline{W}G \,. </annotation></semantics></math></div></div> <p>By the discussion at <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> this means that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, an ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>P</mi> <mo>•</mo></msub></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><msub><mi>X</mi> <mo>•</mo></msub></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P_\bullet &\to& W G \\ \downarrow && \downarrow \\ X_\bullet &\stackrel{g}{\to}& \overline{W}G } </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">P_\bullet</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>P</mi> <mo>•</mo></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mo>•</mo></msub></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ P_\bullet &\to& * \\ \downarrow &\swArrow& \downarrow \\ X_\bullet &\stackrel{g}{\to}& \overline{W}G } \,, </annotation></semantics></math></div> <p>i.e. as the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>We call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mo>•</mo></msub><mo>:</mo><mo>=</mo><msub><mi>X</mi> <mo>•</mo></msub><msup><mo>×</mo> <mi>g</mi></msup><mi>W</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">P_\bullet := X_\bullet \times^g W G</annotation></semantics></math> the simplicial <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+bundle">principal bundle</a> corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>ϕ</mi><mo>:</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>G</mi> <mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\phi : X_n \to G_{(n-1)}\}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/twisting+function">twisting function</a> corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mi>X</mi> <mo>•</mo></msub><mo>→</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">g : X_\bullet \to \overline{W}G</annotation></semantics></math> by the above discussion.</p> <p>Then the simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mo>•</mo></msub><mo>:</mo><mo>=</mo><msub><mi>X</mi> <mo>•</mo></msub><msub><mo>×</mo> <mi>g</mi></msub><mi>W</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">P_\bullet := X_\bullet \times_{g} W G</annotation></semantics></math> is explicitly given by the formula called the <a class="existingWikiWord" href="/nlab/show/twisted+Cartesian+product">twisted Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><msup><mo>×</mo> <mi>ϕ</mi></msup><msub><mi>G</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet \times^\phi G_\bullet</annotation></semantics></math>:</p> <p>its cells are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>n</mi></msub><mo>=</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>×</mo><msub><mi>G</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> P_n = X_n \times G_n </annotation></semantics></math></div> <p>with face and degeneracy maps</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>i</mi></msub><mi>x</mi><mo>,</mo><msub><mi>d</mi> <mi>i</mi></msub><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_i (x,g) = (d_i x , d_i g)</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">i \gt 0</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>0</mn></msub><mi>x</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>d</mi> <mn>0</mn></msub><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_0 (x,g) = (d_0 x, \phi(x) d_0 g)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><mi>x</mi><mo>,</mo><msub><mi>s</mi> <mi>i</mi></msub><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s_i (x,g) = (s_i x, s_i g)</annotation></semantics></math>.</p> </li> </ul> </div> <h4 id="SimplicialBundleReferences">References</h4> <p>Here are some pointers on where precisely in the literature the above statements can be found.</p> <p>One useful reference is</p> <ul> <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>There the abbreviation PCTP ( <em>principal twisted cartesian product</em> ) is used for what above we called just <a class="existingWikiWord" href="/nlab/show/twisted+Cartesian+product">twisted Cartesian product</a>s.</p> <p>The fact that every PTCP <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>ϕ</mi></msub><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times_\phi G \to X</annotation></semantics></math> defined by a <a class="existingWikiWord" href="/nlab/show/twisting+function">twisting function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> arises as the pullback 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>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">W G \to \overline{W}G</annotation></semantics></math> along a morphism of simplicial sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \overline{W}G</annotation></semantics></math> can be found there by combining</p> <ol> <li> <p>the last sentence on p. 81 which asserts that pullbacks of PTCPs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>ϕ</mi></msub><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times_\phi G \to X</annotation></semantics></math> along morphisms of simplicial sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : Y \to X</annotation></semantics></math> yield PTCPs corresponding to the composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>;</p> </li> <li> <p>section 21 which establishes that <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 the PTCP for some universal twisting function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(G)</annotation></semantics></math>.</p> </li> <li> <p>lemma 21.9 states in the language of composites of twisting functions that every PTCP comes from composing a cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mover><mi>W</mi><mo stretchy="false">¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">Y \to \bar W G</annotation></semantics></math> with the universal twisting function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(G)</annotation></semantics></math>. In view of the relation to pullbacks in item 1, this yields the statement in the form we stated it above.</p> </li> </ol> <p>An explicit version of the statement that <a class="existingWikiWord" href="/nlab/show/twisted+Cartesian+product">twisted Cartesian product</a>s are nothing but pullbacks of a <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a> is on <a href="http://ncatlab.org/timporter/files/menagerie10.pdf#page=148">page 148</a> of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em><a class="existingWikiWord" href="/timporter/show/crossed+menagerie">The Crossed Menagerie</a>.</em></li> </ul> <p>On <a href="http://ncatlab.org/timporter/files/menagerie10.pdf#page=239">page 239</a> there it is mentioned that</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>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex"> G \to W G \to \overline{W}G </annotation></semantics></math></div> <p>is a model for the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> <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>One place in the literature where the observation that <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 the <a class="existingWikiWord" href="/nlab/show/decalage">decalage</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>W</mi><mo>¯</mo></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">\overline{W}G</annotation></semantics></math> is mentioned fairly explicitly is page 85 of</p> <ul> <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> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group+action">simplicial group action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+topological+group">simplicial topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-group">3-group</a>, <a class="existingWikiWord" href="/nlab/show/2-crossed+module">2-crossed module</a> / <a class="existingWikiWord" href="/nlab/show/crossed+square">crossed square</a>, <a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>, <strong>simplicial group</strong>, <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a>, <a class="existingWikiWord" href="/nlab/show/hypercrossed+complex">hypercrossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a></p> </li> </ul> <h2 id="references_2">References</h2> <ul> <li id="Quillen67"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, II §3.8 of: <em><a class="existingWikiWord" href="/nlab/show/Homotopical+Algebra">Homotopical Algebra</a></em>, Lecture Notes in Mathematics <strong>43</strong>, Springer (1967) [<a href="https://doi.org/10.1007/BFb0097438">doi:10.1007/BFb0097438</a>]</p> </li> <li id="MaySimplicialObjects"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, Chapter IV, Section 17 of: <em>Simplicial Objects in Algebraic Topology</em> University of Chicago Press 1967 (<a href="https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html">ISBN:9780226511818</a>, <a href="http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu">djvu</a>, <a class="existingWikiWord" href="/nlab/files/May_SimplicialObjectsInAlgebraicTopology.pdf" title="pdf">pdf</a>)</p> </li> <li id="Curtis71"> <p><a class="existingWikiWord" href="/nlab/show/Edward+B.+Curtis">Edward B. Curtis</a>, Section 3 of: <em>Simplicial homotopy theory</em>, Advances in Mathematics 6 (1971) 107–209 (<a href="https://doi.org/10.1016/0001-8708(71)90015-6">doi:10.1016/0001-8708(71)90015-6</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=279808">MR279808</a>)</p> </li> <li id="GoerssJardine"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/J.+F.+Jardine">J. F. Jardine</a>, Chapter V of: <em><a class="existingWikiWord" href="/nlab/show/Simplicial+homotopy+theory">Simplicial homotopy theory</a></em> Progress in Mathematics, Birkhäuser (1999), Modern Birkhäuser Classics (2009) (<a href="https://link.springer.com/book/10.1007/978-3-0346-0189-4">doi:10.1007/978-3-0346-0189-4</a>, <a href="http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/">webpage</a>)</p> </li> <li id="Weibel94"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, Chapter 8 of: <em>An introduction to homological algebra</em>, Cambridge (1994)</p> </li> </ul> <p>The algorithm for finding the horn fillers in a simplicial group is given in the proof of theorem 17.1, page 67 there.</p> <p>This proof that simplicial groups are Kan complexes is originally due to Moore, see theorem 3.4 in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Moore">John Moore</a>, <em>Semi-Simplicial Complexes And Postnikov Systems</em>, in <em>Symposium International De Topologia Algebraica</em> , 1956 conference, book published in 1958</li> </ul> <p>which earlier appeared in more detail as Theorem 3 on p. 18-04 of</p> <ul> <li id="Moore54"><a class="existingWikiWord" href="/nlab/show/John+Moore">John Moore</a>, <em>Homotopie des complexes monoideaux, I</em>, Seminaire Henri Cartan (1954-55) [<a href="http://www.numdam.org/item?id=SHC_1954-1955__7_2_A8_0">numdam:SHC_1954-1955__7_2_A8_0</a>]</li> </ul> <p>and is often referenced to</p> <ul> <li id="Moore"><a class="existingWikiWord" href="/nlab/show/John+Moore">John Moore</a>, <em>Algebraic homotopy theory</em>, lecture notes, Princeton University, 1955–1956</li> </ul> <p>In fact, it seems that this is the origin of the very notion of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>.</p> <p>A proof is also on p. 14 of</p> <ul> <li id="JoyalTierney05"><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a> chapter I of <em>An introduction to simplicial homotopy theory</em>, 2005 (<a href="http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01">web</a>, <a class="existingWikiWord" href="/nlab/files/JoyalTierneySimplicialHomotopyTheory.pdf" title="pdf">pdf</a>)</li> </ul> <p>Section 1.3.3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em><a class="existingWikiWord" href="/timporter/show/crossed+menagerie">The Crossed Menagerie</a></em></li> </ul> <p>discusses simplicial groups in the context of <a class="existingWikiWord" href="/nlab/show/nonabelian+algebraic+topology">nonabelian algebraic topology</a>.</p> <p>Additional useful references include</p> <ul> <li id="JardineLuo"> <p><a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>, Luo, <em>Higher order principal bundles</em> (<a href="http://www.math.uiuc.edu/K-theory/0681/">web</a>)</p> </li> <li id="CegarraRemedios"> <p><a class="existingWikiWord" href="/nlab/show/Antonio+Cegarra">Antonio Cegarra</a>, <a class="existingWikiWord" href="/nlab/show/Josu%C3%A9+Remedios">Josué Remedios</a>, <em>The relationship between the diagonal and the bar constructions on a bisimplicial set</em> (<a href="http://www.ugr.es/~acegarra/Paperspdfs/TRBDWC.pdf">pdf</a>)</p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/simplicial+object">simplicial object</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on July 24, 2023 at 12:00:07. 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