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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</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='#AbstractStatement'>Abstract statement</a></li> <li><a href='#realizations'>Realizations</a></li> <ul> <li><a href='#for_groupoids'>For groupoids</a></li> <li><a href='#For2Groupoids'>For 2-groupoids</a></li> <li><a href='#for_graygroupoids'>For Gray-groupoids</a></li> <li><a href='#for_grothendieck_groupoids'>For Grothendieck ∞-groupoids</a></li> <li><a href='#ForKanComplexes'>For Kan complexes</a></li> <li><a href='#ForAlgebraicKanComplexes'>For algebraic Kan complexes</a></li> <li><a href='#Cubical'>For cubical sets</a></li> <li><a href='#for__groups_and_fold_groupoids'>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>cat</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">cat^n</annotation></semantics></math> groups and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-fold groupoids</a></li> <li><a href='#for_segalgroupoids'>For Segal-groupoids</a></li> <li><a href='#ForStrictOmegaCategoriesWithWeakInverses'>For strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories with weak inverses</a></li> <li><a href='#ForDgmSet'>For diagrammatic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty, 0)</annotation></semantics></math>-categories.</a></li> </ul> <li><a href='#generalizations'>Generalizations</a></li> <ul> <li><a href='#for_stratified_spaces'>For stratified spaces</a></li> <li><a href='#for_spectra'>For spectra</a></li> </ul> <li><a href='#References'>References and a bit of history.</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>homotopy hypothesis</em> is the <a class="existingWikiWord" href="/nlab/show/hypothesis">hypothesis</a> or <a class="existingWikiWord" href="/nlab/show/assertion">assertion</a> that</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> are <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28infinity%2C1%29-categories">equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> at their <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>,</li> </ul> <p>or rather the stronger statement that</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-groupoids">n-groupoids</a> are equivalent to this localization of those topological spaces which are <a class="existingWikiWord" href="/nlab/show/homotopy+n-types">homotopy n-types</a> for all <a class="existingWikiWord" href="/nlab/show/extended+natural+numbers">extended natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mover><mi>ℕ</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">n \in \bar{\mathbb{N}}</annotation></semantics></math></li> </ul> <p>and moreover</p> <ul> <li>this equivalence is induced by the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> construction.</li> </ul> <p>What this actually means in detail depends on which definition of <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is being used and to which precise incarnation of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> it is being compared to.</p> <p>There are some definitions of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> for which the homotopy hypothesis is a proven <em>theorem</em>, notably for the usual definition via <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> (see at <em><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a></em>).</p> <p>Depending on where in the spectrum between <a class="existingWikiWord" href="/nlab/show/geometric+definitions+of+higher+categories">geometric definitions of higher categories</a> and <a class="existingWikiWord" href="/nlab/show/algebraic+definitions+of+higher+categories">algebraic definitions of higher categories</a> a given definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids is located, the statement may be more or less obvious.</p> <p>For instance there is some justification for <em>defining</em> 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 to <em>be</em> equivalently a topological space considered modulo weak homotopy equivalence (see at <em><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></em>). For this definition the homotopy hypothesis is of course a tautology.</p> <p>A definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid that is still very geometrical but much more combinatorial is that given by <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>. For these the homotopy hypothesis has a (non-trivial but fairly tractable) proof. The equivalence between Kan complexes and <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> obtained this way is at the heart of all traditional <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>A genuine algebraic definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids for which the homotopy theory has a (non-trivial but tractable) proof is given by <a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a>es.</p> <p>However for other algebraic definitions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids not much indication for how to prove the homotopy hypothesis is known. The definition of <a class="existingWikiWord" href="/nlab/show/Trimble+%E2%88%9E-category">Trimble ∞-category</a> stands out as an algebraic definition that has the notion of <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> built right into it, but also here it seems unclear at the moment how to make progress with proving the homotopy hypothesis.</p> <p>In fact, generally the homotopy hypothesis is regarded as a <em>consistency condition</em> for definitions in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>:</p> <ul> <li>Any definition of <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> should be such that there is a <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid">fundamental n-groupoid</a>-construction with values in the corresponding <a class="existingWikiWord" href="/nlab/show/%28n%2C0%29-categories">(n,0)-categories</a> which makes these equivalent to <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a>s.</li> </ul> <p>One way to justify this condition is by recourse to the proven cases of the homotopy hypothesis: experience shows that the collections of all three models – topological spaces, Kan complexes, algebraic Kan complexes – provide a model for <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> that supports the general abstract <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> – specifically <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> – that one expects in analogy to how <a class="existingWikiWord" href="/nlab/show/Set">Set</a> supports ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a> theory. Any other definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids is hoped/required to reproduce this, and hence is hoped/required to satisfy the homotopy hypothesis.</p> <p>Apart from having different models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids that lend themselves more or less to a comparison with topological spaces, there is also the issue as to how to conceive of the notion of <em>equivalence</em> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids.</p> <p>The usual, unstated, implication is that the notion of <em>equivalence</em> of <a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a>s used to model <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a>s is the appropriate <em><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a>-theoretic</em> notion of <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a>. It is in this way that, for instance, it is known that 1-<a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>s model homotopy 1-types (see below).</p> <p>The reason this is important to specify is that there are other notions of equivalence on categorical structures which model homotopy types in other ways. For example, if we declare a functor between categories to be a weak equivalence iff its <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> is a weak equivalence of <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s, then <em>all</em> homotopy types can be modeled by 1-categories in this way; see the <a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a> for 1-categories.</p> <p>Finally, in analogy to the homotopy hypothesis, there are also attempts to relate general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-categories">(∞,n)-categories</a> (not necessarily groupoidal) to <a class="existingWikiWord" href="/nlab/show/directed+topological+space">directed topological space</a>s by a <a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,n)-category</a>-construction. There have been claims that a <em>directed homotopy hypothesis</em> can be proven, but at the moment there does not seem to be a published statement.</p> <h2 id="AbstractStatement">Abstract statement</h2> <p>The following general abstract statement of the homotopy hypothesis is often useful to make explicit.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>There is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo stretchy="false">)</mo><mo>:</mo></mrow><annotation encoding="application/x-tex">(\Pi \dashv |-|) : </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> </div> <p>This statement can be formulated, holds true and is proven below at least for the standard definitions of these two <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> (see <a href="#ForKanComplexes">section on Kan complexes</a>, <a href="#ForAlgebraicKanComplexes">section on algebraic Kan complexes</a>).</p> <h2 id="realizations">Realizations</h2> <p>We discuss various different definitions of <a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a>s and <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s and the formulation and proof of the homotopy hypothesis for them, to the extent that it is available.</p> <h3 id="for_groupoids">For groupoids</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> of <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>s, <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s, and <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s is equivalent to the 2-category of <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy 1-types</a>, <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a>s, and <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> classes of homotopies.</p> </div> <p>See <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis+for+1-types">homotopy hypothesis for 1-types</a> for more.</p> <h3 id="For2Groupoids">For 2-groupoids</h3> <p>(…) <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>s model all <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy 2-type</a>s (…)</p> <p><a class="existingWikiWord" href="/nlab/show/strict+2-groupoid">strict 2-groupoid</a>s suffice (…) (but note that strict 2-functors are not sufficient to model all maps between 2-types)</p> <h3 id="for_graygroupoids">For Gray-groupoids</h3> <p>It is known that not all homotopy 3-types can be modeled by strict 3-groupoids, but that <a class="existingWikiWord" href="/nlab/show/Gray-groupoid">Gray-groupoid</a>s (<a class="existingWikiWord" href="/nlab/show/semi-strict+infinity-category">semi-strict 3-groupoids</a>) suffice; the obstruction is the <a class="existingWikiWord" href="/nlab/show/Whitehead+product">Whitehead product</a> which arises from a nontrivial <a class="existingWikiWord" href="/nlab/show/interchanger">interchanger</a>.</p> <h3 id="for_grothendieck_groupoids">For Grothendieck ∞-groupoids</h3> <p>For a review of progress on proving the homotopy hypothesis for Grothendieck’s original conception of ∞-groupoid in <em>Pursuing Stacks</em>, as later developed by <a class="existingWikiWord" href="/nlab/show/Georges+Maltsiniotis">Georges Maltsiniotis</a>, see <a href="#HenryLanari">Henry and Lanari</a>‘s 2019 paper.</p> <h3 id="ForKanComplexes">For Kan complexes</h3> <p>We write <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> for <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> equipped with the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>. The cofibrant-fibrant objects 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> are precisely the <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>.</p> <p>Also we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> for <a class="existingWikiWord" href="/nlab/show/Top">Top</a> equipped with the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>.</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>⊣</mo><mi>Sing</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mover><munder><mo>→</mo><mi>Sing</mi></munder><mover><mo>←</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (|-| \dashv Sing) : Top \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen} \,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|-|</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> and the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(-)</annotation></semantics></math> is forming the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(-)</annotation></semantics></math>.</p> <p>This induces an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presented</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Top">Top</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></li> </ul> </div> <p>(<a href="#Quillen67">Quillen 67</a>), see e.g. (<a href="#GoerssJardine96">Goerss-Jardine 96, section I.11</a>, <a href="#JoyalTierney05">Joyal-Tierney 05, chapter I</a>)</p> <h3 id="ForAlgebraicKanComplexes">For algebraic Kan complexes</h3> <p>An <a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a> is an <a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+categories">algebraic definition of higher groupoids</a> obtained by taking the ordinary definition of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> and equipping these with <em>choices</em> of <a class="existingWikiWord" href="/nlab/show/horn">horn</a>-fillers. These choices encode specified composition operations, specified <a class="existingWikiWord" href="/nlab/show/associator">associator</a>s for these, specified <em>pentagonators</em> and so on.</p> <p>Algebraic Kan complexes constitute a genuine algebraic model in that they are precisely the <a class="existingWikiWord" href="/nlab/show/algebras+over+a+monad">algebras over a monad</a> on <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>.</p> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">model structure on algebraic Kan complexes</a> is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Alg</mi><mi>Kan</mi><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (F \dashv U) : Alg Kan \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet_{Quillen} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The proof is spelled out at <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">model structure on algebraic fibrant objects</a>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>With the above <a href="ForKanComplexes">homotopy hypothesis-theorem for Kan complexes</a> this gives a zig-zag of Quillen equivalences between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alg</mi><mi>Kan</mi></mrow><annotation encoding="application/x-tex">Alg Kan</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Alg</mi><mi>Kan</mi><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub><mover><munder><mo>←</mo><mi>Sing</mi></munder><mover><mo>→</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></mover><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Alg Kan \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet_{Quillen} \stackrel{\overset{|-|}{\to}}{\underset{Sing}{\leftarrow}} Top \,. </annotation></semantics></math></div> <p>This already yields the homotopy hypothesis for algebraic Kan complexes at the level of the corresponding <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented (∞,1)-categories</a> (as discussed there)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Alg</mi><msup><mi>C</mi> <mo>∘</mo></msup><mo>≃</mo><msup><mi>Top</mi> <mo>∘</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Alg C^\circ \simeq Top^\circ \,. </annotation></semantics></math></div></div> <p>But there is also a direct Quillen equivalence:</p> <div class="num_def"> <h6 id="definition">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\Lambda^n_k</annotation></semantics></math> for the topological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> and its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/horn">horn</a>.</p> <p>Fix any choice of <a class="existingWikiWord" href="/nlab/show/retract">retract</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex"> R(n,k) : \Delta^n \to \Lambda^n_k </annotation></semantics></math></div> <p>for all topological <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mi>i</mi> <mi>n</mi></msubsup><mo>↪</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Lambda^n_i \hookrightarrow \Delta^n</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> equip the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math> with the stucture of an <a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a> by taking the filler of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Λ</mi> <mi>k</mi></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Lambda_k[n] \to Sing X</annotation></semantics></math> to be given by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>⊣</mo><mi>Sing</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(|-| \dashv Sing)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mover><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Delta^n \stackrel{R(n,k)}{\to} \Lambda^n_k \to X</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Alg</mi><mi>Kan</mi></mrow><annotation encoding="application/x-tex">\Pi_\infty(X) \in Alg Kan</annotation></semantics></math> for the resulting algebraic Kan complex.</p> <p>This construction constitutes a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mo>→</mo><mi>Alg</mi><mi>C</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Pi_\infty(-) : Top \to Alg C \,, </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><msub><mi>Π</mi> <mn>∞</mn></msub><mo>=</mo><mi>Sing</mi></mrow><annotation encoding="application/x-tex">U \Pi_\infty = Sing</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The choices of fillers in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_\infty(X)</annotation></semantics></math> may be thought of as explicit choice of reparameterizations of paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. These choices are arbitrary, but by the general statement at <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">model structure on algebraic fibrant objects</a>, any two chocies yield equivalent objects.</p> </div> <div class="num_def"> <h6 id="definition_2">Definition</h6> <p>Given choices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(-,-)</annotation></semantics></math> of horn retracts as above, define a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><msub><mo stretchy="false">|</mo> <mi>r</mi></msub><mo>:</mo><mi>Alg</mi><mi>Kan</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> |-|_r : Alg Kan \to Top </annotation></semantics></math></div> <p>called <strong>reduced geometric realization</strong> by taking it on an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Alg</mi><mi>Kan</mi></mrow><annotation encoding="application/x-tex">A \in Alg Kan</annotation></semantics></math> to be given by the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>A</mi><msub><mo stretchy="false">|</mo> <mi>r</mi></msub><mo>:</mo><mo>=</mo><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>k</mi></msub><mo>→</mo><mi>A</mi></mrow></munder><msup><mi>Δ</mi> <mi>n</mi></msup><mover><mo>→</mo><mo>→</mo></mover><mo stretchy="false">|</mo><mi>U</mi><mi>A</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> |A|_r := \lim_{\to}(\coprod_{\Lambda[n]_k \to A} \Delta^n \stackrel{\to}{\to} |U A|) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>U</mi><mi>A</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|U A|</annotation></semantics></math> is the ordinary <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of the underlying simplicial set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and where the two maps are</p> <ol> <li> <p>the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|-|</annotation></semantics></math> of the distinguished fillers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>U</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\Delta[n] \to U A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>;</p> </li> <li> <p>the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mover><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo>→</mo><mo stretchy="false">|</mo><mi>U</mi><mi>X</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">\Delta^n \stackrel{R(n,k)}{\to} \Lambda^n_k \to |U X|</annotation></semantics></math> .</p> </li> </ol> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><msub><mo stretchy="false">|</mo> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">|-|_r</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_\infty</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><msub><mo stretchy="false">|</mo> <mi>r</mi></msub><mo>⊣</mo><msub><mi>Π</mi> <mn>∞</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (|-|_r \dashv \Pi_\infty) \,. </annotation></semantics></math></div></div> <p>This is (<a href="#Nikolaus">Nikolaus, prop. 3.4</a>).</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We check the hom-isomorphism. A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo stretchy="false">|</mo><mi>A</mi><msub><mo stretchy="false">|</mo> <mi>r</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : |A|_r \to X</annotation></semantics></math> is by definition of the coequalizer the same as a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo>:</mo><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\tilde f : |A| \to X</annotation></semantics></math> such that for each horn <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>k</mi></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h : \Lambda[n]_k \to A</annotation></semantics></math> with distinguished filler <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>h</mi><mo stretchy="false">^</mo></mover><mo>:</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\hat h : \Delta[n] \to A</annotation></semantics></math> the composites</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mover><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mover><mo>→</mo><mrow><mo stretchy="false">|</mo><mi>h</mi><mo stretchy="false">|</mo></mrow></mover><mo stretchy="false">|</mo><mi>U</mi><mi>A</mi><mo stretchy="false">|</mo><mover><mo>→</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> \Delta^n \stackrel{R(n,k)}{\to} \Lambda^n_k \stackrel{|h|}{\to} |U A| \stackrel{\tilde f}{\to} X </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><mo stretchy="false">|</mo><mover><mi>h</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">|</mo></mrow></mover><mo stretchy="false">|</mo><mi>U</mi><mi>A</mi><mo stretchy="false">|</mo><mover><mo>→</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> \Delta^n \stackrel{|\hat h|}{\to} |U A| \stackrel{\tilde f}{\to} X </annotation></semantics></math></div> <p>are equal. This means equivalently that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>⊣</mo><mi>Sing</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(|-| \dashv Sing)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>U</mi><mi>A</mi><mo>→</mo><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\tilde \tilde f : U A \to Sing X</annotation></semantics></math> sends distinguished fillers in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to distinguished fillers in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_\infty(X)</annotation></semantics></math> and is hence a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alg</mi><mi>Kan</mi></mrow><annotation encoding="application/x-tex">Alg Kan</annotation></semantics></math>.</p> <p>This construction shows that the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>:</mo><mo stretchy="false">|</mo><mi>A</mi><msub><mo stretchy="false">|</mo> <mi>r</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mover><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>A</mi><mo>→</mo><msub><mi>Π</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f : |A|_r \to X) \mapsto (\tilde \tilde f : A \to \Pi_\infty(X))</annotation></semantics></math> thus obtained is a bijection.</p> </div> <div class="num_theorem"> <h6 id="theorem_4">Theorem</h6> <p>We have an identity:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Alg</mi><mi>Kan</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>U</mi></mpadded></msup><mo>↙</mo></mtd> <mtd><msup><mo>⇓</mo> <mo>=</mo></msup></mtd> <mtd><msup><mo>↖</mo> <mpadded width="0"><mrow><msub><mi>Π</mi> <mn>∞</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>sSet</mi></mtd> <mtd></mtd> <mtd><munder><mo>←</mo><mi>Sing</mi></munder></mtd> <mtd></mtd> <mtd><mi>Top</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && Alg Kan \\ & {}^{\mathllap{U}}\swarrow & \Downarrow^{=}& \nwarrow^{\mathrlap{\Pi_\infty}} \\ sSet &&\underset{Sing}{\leftarrow}&& Top } </annotation></semantics></math></div> <p>and a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Alg</mi><mi>Kan</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>F</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mo>≃</mo></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo stretchy="false">|</mo><mo>−</mo><msub><mo stretchy="false">|</mo> <mi>r</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>sSet</mi></mtd> <mtd></mtd> <mtd><munder><mo>→</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></munder></mtd> <mtd></mtd> <mtd><mi>Top</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Alg Kan \\ & {}^{\mathllap{F}}\nearrow & \Downarrow^{\simeq}& \searrow^{\mathrlap{|-|_r}} \\ sSet &&\underset{|-|}{\to}&& Top } \,. </annotation></semantics></math></div></div> <p>This is (<a href="#Nikolaus">Nikolaus, corollary 3.5</a>)</p> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>The identity is evident by definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_\infty</annotation></semantics></math>.</p> <p>Using this, we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><msub><mo stretchy="false">|</mo> <mi>r</mi></msub><mo>∘</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo>∘</mo><msub><mi>Π</mi> <mn>∞</mn></msub><mo>=</mo><mi>Sing</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (|-|_r \circ F \dashv U \circ \Pi_\infty = Sing) \,. </annotation></semantics></math></div> <p>So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><msub><mo stretchy="false">|</mo> <mi>r</mi></msub><mo>∘</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">|-|_r \circ F</annotation></semantics></math> is another <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi></mrow><annotation encoding="application/x-tex">Sing</annotation></semantics></math> and hence naturally isomorphism to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|-|</annotation></semantics></math>.</p> </div> <div class="num_corollary"> <h6 id="corollary">Corollary</h6> <p>The adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><msub><mo stretchy="false">|</mo> <mi>r</mi></msub><mo>⊣</mo><msub><mi>Π</mi> <mn>∞</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mi>Alg</mi><mi>C</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> (|-|_r \dashv \Pi_\infty) : Alg C \to Top </annotation></semantics></math></div> <p>constitutes a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </div> <p>This is <a href="#Nilkolaus">Nikolaus, corollary 3.6</a></p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By the above theorem and the <a href="http://ncatlab.org/nlab/show/Quillen+equivalence#TwoOutOfThree">2-out-of-3-property</a> of Quillen equivalences.</p> </div> <h3 id="Cubical">For cubical sets</h3> <p>Also <a class="existingWikiWord" href="/nlab/show/cubical+set">cubical set</a>s may serve as a model for homotopy theory.</p> <p>There is an evident <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>-valued <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>□</mo><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> \Box \to sSet </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/cube+category">cube category</a> to <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>, which sends the cubical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cube to the simplicial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cube</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mi>n</mi></msup><mo>↦</mo><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo> <mrow><mo>×</mo><mi>n</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{1}^n \mapsto (\Delta[1])^{\times n} \,. </annotation></semantics></math></div> <p>Similarly there is a canonical <a class="existingWikiWord" href="/nlab/show/Top">Top</a>-valued functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>□</mo><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> \Box \to Top </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mi>n</mi></msup><mo>↦</mo><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>Top</mi> <mn>1</mn></msubsup><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{1}^n \mapsto (\Delta^1_{Top})^n \,. </annotation></semantics></math></div> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>⊣</mo><msub><mi>Sing</mi> <mo>□</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mover><munder><mo>→</mo><mrow><msub><mi>Sing</mi> <mo>□</mo></msub></mrow></munder><mover><mo>←</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></mover><msup><mi>Set</mi> <mrow><msup><mo>□</mo> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> (|-| \dashv Sing_\Box) : Top \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing_\Box}{\to}} Set^{\Box^{op}} </annotation></semantics></math></div> <p>is the cubical analogue of the simplicial nerve and realization discussed <a href="#ForKanComplexes">above</a>.</p> <div class="num_theorem"> <h6 id="theorem_5">Theorem</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">model structure on cubical sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mo>□</mo> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{\Box^{op}}</annotation></semantics></math> whose</p> <ul> <li> <p>weak equivalences are the morphisms that become weak equivalences under geometric realization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|-|</annotation></semantics></math>;</p> </li> <li> <p>cofibrations are the <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>s.</p> </li> </ul> </div> <p>This is <a href="#Jardine">Jardine, sections 3.</a></p> <div class="num_theorem"> <h6 id="theorem_6">Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit of the adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><msub><mi>Sing</mi> <mo>□</mo></msub><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \to Sing_\Box(|A|) </annotation></semantics></math></div> <p>is a weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mo>□</mo></msup></mrow><annotation encoding="application/x-tex">Set^{\Box}</annotation></semantics></math> for every cubical set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>The counit of the adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msub><mi>Sing</mi> <mo>□</mo></msub><mi>X</mi><mo stretchy="false">|</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> |Sing_\Box X| \to X </annotation></semantics></math></div> <p>is a weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> for every topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>It follows that we have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> induced on the <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>Set</mi> <mrow><msup><mo>□</mo> <mi>op</mi></msup></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(Top) \simeq Ho(Set^{\Box^{op}}) \,. </annotation></semantics></math></div></div> <p>This is <a href="#Jardine">Jardine, theorem 29, corollary 30</a>.</p> <h3 id="for__groups_and_fold_groupoids">For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>cat</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">cat^n</annotation></semantics></math> groups and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-fold groupoids</h3> <p>Loday’s notion of a <a class="existingWikiWord" href="/nlab/show/cat-n-group">cat-n-group</a> corresponds to the connected version of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-fold groupoid. We will restrict our discussion to that connected case.</p> <div class="num_theorem"> <h6 id="theorem_7">Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of <a class="existingWikiWord" href="/nlab/show/cat-n-group">cat-n-group</a>s is equivalent to that of <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> <a class="existingWikiWord" href="/nlab/show/connected">connected</a> <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n+1-type</a>s.</p> </div> <p>This is proven in (<a href="#Loday">Loday</a>). (There are some glitches in his proof and these were fixed by various authors (Steiner, Gilbert, ..) and then detailed proofs were given by Bullejos, Cegarra, Duskin and separately, using the equivalent formulation of crossed n-cubes, by Porter. Detailed references and some more commentary is at <a class="existingWikiWord" href="/nlab/show/cat-n-group">cat-n-group</a>.)</p> <h3 id="for_segalgroupoids">For Segal-groupoids</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>There is realization/singular complex <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>⊣</mo><mi>sSing</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mo>→</mo><mi>SegalGrpd</mi></mrow><annotation encoding="application/x-tex"> (|-| \dashv sSing) : Top \to SegalGrpd </annotation></semantics></math></div> <p>for <a class="existingWikiWord" href="/nlab/show/Segal+groupoid">Segal groupoid</a>s,</p> <p>Its unit is an equivalence of <a class="existingWikiWord" href="/nlab/show/Segal+categories">Segal categories</a> and its counit a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> of topological spaces.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>This is lemma 6.3.21 and corollary 6.3.24 in (<a href="#Pellissier">Pellissier</a>)</p> </div> <h3 id="ForStrictOmegaCategoriesWithWeakInverses">For strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories with weak inverses</h3> <p>While <a class="existingWikiWord" href="/nlab/show/strict+omega-groupoids">strict omega-groupoids</a> in the sense of <a class="existingWikiWord" href="/nlab/show/strict+omega-categories">strict omega-categories</a> with <em>strict</em> inverses are far from modelling all homotopy types, strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories with all <em>weak</em> inverses come closer. In (<a href="#KapranovVoevodsky">Kapranov-Voevodsky</a>) it was argued that these are in fact sufficient, but a mistake in the argument was found in (<a href="#Simpson">Simpson, Cor. 5.2</a>) (see also <a href="Simpson%27s+conjecture#History">here</a>).</p> <p>The issue however is somewhat subtle, as highlighted by Voevodsky <a href="http://ncatlab.org/nlab/show/homotopy+type+theory#VoevodskyIASTalk2014">here</a>. For more on this see at <em><a class="existingWikiWord" href="/nlab/show/Simpson%27s+conjecture">Simpson's conjecture</a></em>.</p> <h3 id="ForDgmSet">For diagrammatic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty, 0)</annotation></semantics></math>-categories.</h3> <p><a class="existingWikiWord" href="/nlab/show/diagrammatic+set">Diagrammatic sets</a> are a topologically sound alternative to <a class="existingWikiWord" href="/nlab/show/computad">polygraphs</a>: a diagrammatic set is a presheaf over the <a class="existingWikiWord" href="/nlab/show/atom+category">atom category</a>, whose objects include many of the <a class="existingWikiWord" href="/nlab/show/geometric+shape+for+higher+structures">geometric shape for higher structures</a>.</p> <p>There, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty, \infty)</annotation></semantics></math>-category is a diagrammatic set in which any suitable assemblage 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>-cells can be composed, via a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n + 1)</annotation></semantics></math>-cell, into a single <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cell. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">0 \le n \le \infty</annotation></semantics></math>, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty, n)</annotation></semantics></math>-category is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty, \infty)</annotation></semantics></math>-category in which all cells of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\gt n</annotation></semantics></math> are “weakly invertible”.</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>There is a model structure on diagrammatic sets whose fibrant objects are exactly the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty, 0)</annotation></semantics></math>-categories and which is Quillen equivalent to the standard model structure on simplicial sets.</p> </div> <br /> <div class='proof'> <h6>Proof</h6> <p>This is <a href="#ChanavatHadzihasanovic2024Model">Corollary 5.8</a>.</p> </div> </p> <h2 id="generalizations">Generalizations</h2> <h3 id="for_stratified_spaces">For stratified spaces</h3> <p>For a <a class="existingWikiWord" href="/nlab/show/categorified">categorified</a> version which finds an equivalence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty, 1)</annotation></semantics></math>-categories, see <a class="existingWikiWord" href="/nlab/show/stratified+homotopy+hypothesis">stratified homotopy hypothesis</a> (<a href="#AFR15">Ayala-Francis-Rozenblyum</a>).</p> <h3 id="for_spectra">For spectra</h3> <p>For a stabilized version which finds an equivalence between stable homotopy types (i.e. <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a>) and symmetric monoidal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids, see <a class="existingWikiWord" href="/nlab/show/stable+homotopy+hypothesis">stable homotopy hypothesis</a> (<a href="#JO12">Johnson-Osorno</a>, <a href="#GJO17">Gurski-Johnson-Osorno</a>)</p> <h2 id="References">References and a bit of history.</h2> <p>The equivalence between the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical homotopy theory of topological spaces</a> and the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">homotopy theory of Kan complexes</a> is due to</p> <ul> <li id="Quillen67"><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, <em>Homotopical Algebra</em>, LNM 43, Springer, (1967)</li> </ul> <p>Later in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, <em><a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">Pursuing Stacks</a></em>, 1983 (including and especially the ‘letter to <a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>’ given on the first few pages),</li> </ul> <p>it was argued that there were <a class="existingWikiWord" href="/nlab/show/algebraic+definitions+of+higher+categories">algebraic definitions of higher groupoids</a> that could be put forward, so that the resulting objects ought to have a homotopy theory equivalent to the classical homotopy theory of topological spaces, and, moreover, would provide useful tools in the interpretation of <a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian cohomology</a> classes. This extended ideas that Grothendieck had explored in letters to <a class="existingWikiWord" href="/nlab/show/Larry+Breen">Larry Breen</a> in the mid 1970s in which he had given a sketch of a theory 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>-stacks and their relation with the homotopy <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>-type of a space or more generally a topos.</p> <p>At this stage, (in the 1970s and early 1980s) more <em>geometric</em> or <em>combinatorial</em> definitions of infinity categories were not yet available, or, perhaps more accurately, had been discovered, but were not <em>recognised</em> as having such an infinity categoric interpretation; see <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>‘s Letter to Grothendieck (16 June 1983) and the discussion <a class="existingWikiWord" href="/nlab/files/Spaces+as+infinity-groupoids.pdf" title="">here</a> in <a class="existingWikiWord" href="/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a>. These models included Kan complexes which now are interpreted as being one model for infinity groupoids, and <a class="existingWikiWord" href="/nlab/show/quasicategory">weak Kan complex</a>es, as put forward by Boardman and Vogt, which give one of the main models for (weak) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories. From this perspective, Quillen’s result can be seen as being a precursor of one form of the Homotopy Hypothesis.</p> <p>The name “Homotopy Hypothesis” for this statement is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em>The Homotopy Hypothesis</em>, 2007 (<a href="http://math.ucr.edu/home/baez/homotopy/">web</a>, <a href="http://math.ucr.edu/home/baez/homotopy/homotopy.pdf">pdf</a>)</li> </ul> <p>More recently <a class="existingWikiWord" href="/nlab/show/Georges+Maltsiniotis">Georges Maltsiniotis</a> has revived the approach to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids that Grothendieck initiated in <em>Pursuing Stacks</em> - see <a class="existingWikiWord" href="/nlab/show/Grothendieck+%E2%88%9E-groupoids">Grothendieck ∞-groupoids</a> - and work has begun on proving the homotopy hypothesis for Grothendieck ∞-groupoids. For a review of progress so far, see:</p> <ul> <li id="HenryLanari"><a class="existingWikiWord" href="/nlab/show/Simon+Henry">Simon Henry</a>, <a class="existingWikiWord" href="/nlab/show/Edoardo+Lanari">Edoardo Lanari</a>, <em>On the homotopy hypothesis in dimension 3</em>, Theory and Applications of Categories <strong>39</strong> 26 (2023) 735-768 [<a href="https://arxiv.org/abs/1905.05625">arxiv/1905.05625</a>, <a href="http://www.tac.mta.ca/tac/volumes/39/26/39-26abs.html">tac:39-26</a>, <a href="http://www.tac.mta.ca/tac/volumes/39/26/39-26.pdf">pdf</a>].</li> </ul> <p>These authors have reduced the hypothesis to a conjecture which has so far been proved only up to dimension 3, hence the title of their paper.</p> <p>Technical reviews of Quillen’s proof of a version of the homotopy hypothesis include:</p> <ul> <li id="GoerssJardine96"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>, section I.11 of <em><a class="existingWikiWord" href="/nlab/show/Simplicial+homotopy+theory">Simplicial homotopy theory</a></em>, 1996</p> </li> <li id="JoyalTierney05"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a> <em>An introduction to simplicial homotopy theory</em>, 2005 (<a href="http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01">chapter I</a>, more notes <a href="http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern47.pdf">pdf</a>)</p> </li> </ul> <p>The homotopy hypothesis for strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories with weak inverses is discussed in</p> <ul> <li id="KapranovVoevodsky"><a class="existingWikiWord" href="/nlab/show/Mikhail+Kapranov">Mikhail Kapranov</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Voevodsky">Vladimir Voevodsky</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids and homotopy types</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 32 no. 1 (1991), p. 29-46</li> </ul> <p>but a mistake in the argument was pointed out in Corollary 5.2 of</p> <ul> <li id="Simpson"><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>Homotopy types of strict 3-groupoids</em> (<a href="http://arxiv.org/abs/math/9810059">arXiv:math/9810059</a>)</li> </ul> <p>The homotopy hypothesis for <a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complexes">algebraic Kan complexes</a> is established and discussed in</p> <ul id="Nikolaus"> <li><a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <em>Algebraic models for higher categories</em> (<a href="http://arxiv.org/abs/1003.1342">arXiv</a>)</li> </ul> <p>For progress on</p> <p>The homotopy hypothesis for <a class="existingWikiWord" href="/nlab/show/Segal+groupoids">Segal groupoids</a> is formulated in section 6.3.4 of</p> <ul> <li>Regis Pellissier, <em>Weak enriched categories - Categories enrichies faibles</em> PhD Thesis (2002) (<a href="http://arxiv.org/abs/math/0308246">arXiv:math/0308246</a>)</li> </ul> <p>Models of homotopy <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>-types by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Cat</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Cat^n</annotation></semantics></math>-groups are discussed in</p> <ul id="Loday"> <li><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <em>Spaces with finitely many non-trivial homotopy groups</em>, J. Pure Appl. Algebra 24 (1982), 179-202.</li> </ul> <p>More literature on models of homotopy types by strict higher groupoids is at</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>Computing Homotopy Types Using Crossed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>-Cubes of Groups</em> (<a href="http://arxiv.org/abs/math/0109091">arXiv</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Simona+Paoli">Simona Paoli</a>, <em>Internal categorical structures in homotopical algebra</em>, to appear in <em><a class="existingWikiWord" href="/johnbaez/show/Towards+Higher+Categories">Towards Higher Categories</a></em>, eds. <a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a> and <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a> (<a href="http://www.maths.mq.edu.au/~simonap/paoli_IMA.pdf">pdf</a>)</p> </li> </ul> <p>The first paper, as its title suggests, has an emphasis on using higher groupoids for computation of homotopical invariants, in fact by applying higher homotopy van Kampen Theorems. These theorems lead to algebraic colimit arguments in algebraic topology, implying results, often nonabelian, not obtainable by other methods. It is also remarkable that the precision of these results requires the use of strict structures, whereas the current emphasis in higher category theory is on non strict structures.</p> <p>The homotopy theory of <a class="existingWikiWord" href="/nlab/show/cubical+set">cubical set</a>s is discussed in</p> <ul id="Jardine"> <li>Jardine, <em>Model structure on cubical sets</em> (<a href="http://hopf.math.purdue.edu/Jardine/cubical2.pdf">pdf</a>)</li> </ul> <ul> <li>Maltsiniotis, G. La catégorie cubique avec connexions est une catégorie test stricte. Homology, Homotopy Appl. 11, (2) (2009) 309–326.</li> </ul> <p>Cubical methods are also essential in</p> <ul> <li>R. Brown, P.J. Higgins, R. Sivera, <em>Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids</em>, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011).</li> </ul> <p>Diagrammatic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty, 0)</annotation></semantics></math>-category are discussed in</p> <ul> <li id="ChanavatHadzihasanovic2024Homotopy"> <p>Chanavat, Hadzihasanovic, <em>Diagrammatic sets as a model of homotopy types</em>, 2024 (<a href="https://arxiv.org/abs/2407.06285v1">arXiv:2407.06285</a>)</p> </li> <li id="ChanavatHadzihasanovic2024Model"> <p>Chanavat, Hadzihasanovic, <em>Model structures for diagrammatic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,n)</annotation></semantics></math>-categories</em>, 2024 (<a href="https://www.arxiv.org/abs/2410.19053">arXiv:2410.19053</a>)</p> </li> </ul> <p>A version for <a class="existingWikiWord" href="/nlab/show/stratified+spaces">stratified spaces</a> is discussed in</p> <ul> <li id="AFR15"><a class="existingWikiWord" href="/nlab/show/David+Ayala">David Ayala</a>, <a class="existingWikiWord" href="/nlab/show/John+Francis">John Francis</a>, <a class="existingWikiWord" href="/nlab/show/Nick+Rozenblyum">Nick Rozenblyum</a>, <em>A stratified homotopy hypothesis</em>, <a href="http://arxiv.org/abs/1502.01713">arXiv</a></li> </ul> <p>A version for <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> is discussed in</p> <ul> <li id="JO12"> <p><a class="existingWikiWord" href="/nlab/show/Niles+Johnson">Niles Johnson</a> and <a class="existingWikiWord" href="/nlab/show/Angelica+Osorno">Angelica Osorno</a>, <em>Modeling stable one-types</em>, <a href="http://tac.mta.ca/tac/volumes/26/20/26-20abs.html">tac</a></p> </li> <li id="GJO17"> <p><a class="existingWikiWord" href="/nlab/show/Nick+Gurski">Nick Gurski</a>, <a class="existingWikiWord" href="/nlab/show/Niles+Johnson">Niles Johnson</a>, and <a class="existingWikiWord" href="/nlab/show/Angelica+Osorno">Angelica Osorno</a>, <em>The 2-dimensional stable homotopy hypothesis</em>, <a href="https://arxiv.org/abs/1712.07218">arxiv</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 7, 2024 at 17:40:19. 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