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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </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> <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> 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>… 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>, … see also <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> Introductions <ul> <li> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a> </li> </ul> Definitions <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </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/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> Paths and cylinders <ul> <li> <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/path+object">path object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a> </li> </ul> </li> </ul> Homotopy groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a> </li> <li> <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> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <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> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> Basic facts <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> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem </li> </ul> </div></div> <h4 id="model_category_theory">Model category theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, <a class="existingWikiWord" href="/nlab/show/model+%28infinity%2C1%29-category">model <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>-category</a> Definitions <ul> <li> <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> (<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">of a model category</a> </li> </ul> Morphisms <ul> <li> <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> </li> </ul> </li> </ul> Universal constructions <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> <a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a> <a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a> </li> </ul> Refinements <ul> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+category">locally cartesian closed model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a> </li> </ul> Producing new model structures <ul> <li> <a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a> </li> </ul> Presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories <ul> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a> </li> </ul> Model structures <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> for <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 <a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+semi-simplicial+sets">on semi-simplicial sets</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-groupoids">on strict ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/natural+model+structure+on+groupoids">on groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">on chain complexes</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a> related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a> </li> </ul> for equivariant <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 <ul> <li> <a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a> (<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>) </li> </ul> for rational <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 <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> for rational equivariant <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 <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a> </li> </ul> for <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>-groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-groupoids</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-types</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a> </li> </ul> for <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>-algebras general <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>-algebras <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">on simplicial T-algebras</a>, on <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a> </li> </ul> specific <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>-algebras <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">on dg-algebras</a> and on <a class="existingWikiWord" href="/nlab/show/simplicial+ring">on simplicial rings</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">on cosimplicial rings</a> related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a> </li> </ul> for stable/spectrum objects <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a> </li> </ul> 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 <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+quasi-categories">Joyal model for quasi-categories</a> (and its <a class="existingWikiWord" href="/nlab/show/model+structure+for+cubical+quasicategories">cubical version</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a> </li> </ul> for stable <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 <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> 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>-operads <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">on operads</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">for Segal operads</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">for dendroidal complete Segal spaces</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">for dendroidal Cartesian fibrations</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-categories <ul> <li> <a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a> </li> </ul> 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>-sheaves / <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>-stacks <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a> <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a>/<a class="existingWikiWord" href="/nlab/show/Cech+model+structure+on+simplicial+presheaves">Cech model structure</a>/<a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#for_simplicially_enriched_categories'>For simplicially enriched categories</a></li> <li><a href='#for_categories_with_weak_equivalences'>For categories with weak equivalences</a></li> </ul> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> Given a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},W)</annotation></semantics></math>, then its homotopy category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\mathcal{C})</annotation></semantics></math> is, if it exists, the result of universally forcing the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> to become actual <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, also called the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> at the weak equivalences <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>⟶</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>=</mo><mi>𝒞</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C} \longrightarrow Ho(\mathcal{C})= \mathcal{C}[W^{-1}] \,. </annotation></semantics></math></div> The classical example is the category of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> with weak equivalences those <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> which are <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> or <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>. The corresponding homotopy category is often referred to as “the homotopy category”, by default, or the “<a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a>” for emphasis. This turns out to be equivalent to the category of topological spaces or (for weak homotopy equivalences) of just those <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> with <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>-<a class="existingWikiWord" href="/nlab/show/homotopy+class">classes</a> of continuous functions between them, whence the name “homotopy category”. The existence of a homotopy category, as well as tractable presentations of it typically require extra <a class="existingWikiWord" href="/nlab/show/properties">properties</a> of the class of weak equivalences (such as that they admit a <a class="existingWikiWord" href="/nlab/show/calculus+of+fractions">calculus of fractions</a>) or even extra <a class="existingWikiWord" href="/nlab/show/structure">structure</a> (such as <a class="existingWikiWord" href="/nlab/show/fibration+category">fibration category</a>/<a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a> structure, or full <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure, or further enhancements of that to <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> structures, etc). See at <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a> for more on this. More generally, to every <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> is associated a homotopy category, whose morphisms are literally the <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of the original morphisms. See at <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a> for more on this. These two concepts of “homotopy category” are compatible: to a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> is associated, if it exists, an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> obtained by universally forcing the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> to become actual <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a>, also called the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>W</mi></msub><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">L_W \mathcal{C}</annotation></semantics></math> at the weak equivalences. The homotopy categories of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},W)</annotation></semantics></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>W</mi></msub><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">L_W \mathcal{C}</annotation></semantics></math> coincide, which justifies the terminology “homotopy category” generally. <h2 id="definition">Definition</h2> <h3 id="for_simplicially_enriched_categories">For simplicially enriched categories</h3> Given a simplicially enriched category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, we can form for each pair of objects, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math>, of objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, the set, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0C(x,y)</annotation></semantics></math>, of connected components of the ‘function space’ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(x,y)</annotation></semantics></math>. As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math> preserves finite limits, this gives a category, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(C)</annotation></semantics></math>. As 1-simplices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(x,y)</annotation></semantics></math> can be often interpreted as being homotopies, this category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(C)</annotation></semantics></math> is often called the homotopy category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, and then the notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> may be used. This notions is closely related to the next, by using, say the <a class="existingWikiWord" href="/nlab/show/hammock+localisation">hammock localisation</a> of Dwyer and Kan, as then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math> of that simplicially enriched category, coincides with the following. <h3 id="for_categories_with_weak_equivalences">For categories with weak equivalences</h3> Given a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> (such as a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>), its homotopy category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> is – if it exists – the <a class="existingWikiWord" href="/nlab/show/category">category</a> which is universal with the property that there is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Q : C \to Ho(C) </annotation></semantics></math></div> that sends every weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math>. One also writes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>C</mi></mrow><annotation encoding="application/x-tex">Ho(C) := W^{-1}C</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C[W^{-1}]</annotation></semantics></math> and calls it the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> at the collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> of weak equivalences. More in detail, the universality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> means the following: <ul> <li>for any (possibly <a class="existingWikiWord" href="/nlab/show/large+category">large</a>) category <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 functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">F : C \to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> sends all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">w \in W</annotation></semantics></math> to isomorphisms 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>, there exists a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>Q</mi></msub><mo>:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">F_Q : Ho(C) \to A</annotation></semantics></math> and a natural isomorphism</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>F</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>Q</mi></msup></mtd> <mtd><msup><mo>⇓</mo> <mo>≃</mo></msup></mtd> <mtd><msub><mo>↗</mo> <mrow><msub><mi>F</mi> <mi>Q</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C &&\stackrel{F}{\to}& A \\ \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q} \\ Ho(C) } </annotation></semantics></math></div> <ul> <li>the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Q</mi> <mo>*</mo></msup><mo>:</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q^* : Func(Ho(C),A) \to Func(C,A)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>.</li> </ul> The second condition implies that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>Q</mi></msub></mrow><annotation encoding="application/x-tex">F_Q</annotation></semantics></math> in the first condition is unique up to unique isomorphism. <h2 id="properties">Properties</h2> <ul> <li> If it exists, the homotopy category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> is unique up to <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>. </li> <li> As described at <a class="existingWikiWord" href="/nlab/show/localization">localization</a>, in general, the morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> must be constructed using zigzags of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in which the backwards-pointing arrows are weak equivalences. This means that in general, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> need not be <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a> even if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is. However, in many cases (such as any <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>) there is a more direct description of the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> classes of maps in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> between suitably “good” (fibrant and cofibrant) objects. </li> <li> In <a class="existingWikiWord" href="/nlab/show/2-limit">2-categorical terms</a>, the homotopy category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)</annotation></semantics></math> is the coinverter of the canonical 2-cell <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>→</mo></mtd></mtr> <mtr><mtd><mi>W</mi></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mi>C</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>→</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{& \to \\ W & \Downarrow & C\\ & \to}</annotation></semantics></math></div> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is the category whose objects are morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and whose morphisms are commutative squares in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. </li> </ul> <h2 id="examples">Examples</h2> <ul> <li> In classical <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, the homotopy category refers to the homotopy category <a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> with weak equivalences taken to be <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>. </li> <li> <a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a> is often restricted to the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of spaces of the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> (the full subcategory of CW-complexes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top)</annotation></semantics></math>). This is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(sSet_{Quillen})</annotation></semantics></math>, the homotopy category of the standard Quillen-<a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>. This equivalence is one aspect of the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>. </li> <li> In <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> the localization of the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> at the <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> is called the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a>. But see also at <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+chain+complexes">homotopy category of chain complexes</a>. </li> <li> In <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> one considers the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a> of <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a>. </li> <li> In <a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a> one considers the <a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+category">equivariant stable homotopy category</a> of spectra. </li> <li> For the homotopy category of <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, see <a class="existingWikiWord" href="/nlab/show/Ho%28Cat%29">Ho(Cat)</a>. </li> <li> For the homotopy category of that of <a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a> see <a class="existingWikiWord" href="/nlab/show/Ho%28CombModCat%29">Ho(CombModCat)</a>. </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category of an (infinity,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+2-category">homotopy 2-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+category">simplicial homotopy category</a> </li> </ul> <h2 id="references">References</h2> Early discussion: <ul> <li id="Brown65"><a class="existingWikiWord" href="/nlab/show/Edgar+Brown">Edgar Brown</a>, Abstract homotopy theory, Trans. AMS 119 no. 1 (1965) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1090/S0002-9947-1965-0182970-6">doi:10.1090/S0002-9947-1965-0182970-6</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> For more see the references at <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> or <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a>. </body></html> </div> <div class="revisedby"> Last revised on December 26, 2023 at 00:35:21. 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