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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"> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='homotopy_theory'>Homotopy theory</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%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+type+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'>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/%28infinity%2C1%29-category'>(∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%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/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/homotopy'>right homotopy</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/path+space+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/localization+at+geometric+homotopies'>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+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/fundamental+infinity-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+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/fundamental+%28infinity%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+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> <h4 id='model_category_theory'>Model category theory</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a>, <a class='existingWikiWord' href='/nlab/show/model+%28%E2%88%9E%2C1%29-category'>model $\infty$-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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_1' xmlns='http://www.w3.org/1998/Math/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+limit'>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/enriched+monoidal+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+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/model+structure+on+functors'>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/slice+model+structure'>on slice categories</a> </li> <li> Bousfield localization<a href='/nlab/new/Bousfield+localization+of+model+categories'>?</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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_2' xmlns='http://www.w3.org/1998/Math/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/%28infinity%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/%28infinity%2C1%29-categorical+hom-space'>(∞,1)-categorical hom-space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/locally+presentable+%28infinity%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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_3' xmlns='http://www.w3.org/1998/Math/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+infinity-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+left+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+omega-groupoids'>on strict ∞-groupoids</a>, <a class='existingWikiWord' href='/nlab/show/canonical+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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_4' xmlns='http://www.w3.org/1998/Math/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/Borel+model+structure'>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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_5' xmlns='http://www.w3.org/1998/Math/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+dg-algebras'>model structure on dgc-algebras</a></li> </ul> for rational equivariant <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_6' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_7' xmlns='http://www.w3.org/1998/Math/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+homotopy+n-types'>for n-groupoids</a>/<a class='existingWikiWord' href='/nlab/show/model+structure+for+homotopy+n-types'>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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_8' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-algebras general <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_10' xmlns='http://www.w3.org/1998/Math/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+in+a+monoidal+model+category'>on monoids</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_11' xmlns='http://www.w3.org/1998/Math/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-infinity+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+for+ring+spectra'>model structure on ring spectra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/model+structure+for+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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_12' xmlns='http://www.w3.org/1998/Math/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+relative+categories'>on categories with weak equivalences</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/model+structure+for+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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_13' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_14' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_15' xmlns='http://www.w3.org/1998/Math/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+omega-categories'>on strict ∞-categories</a> </li> </ul> for <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_16' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_17' xmlns='http://www.w3.org/1998/Math/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/%C4%8Cech+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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#Idea'>Idea</a></li><li><a href='#definition'>Definition</a><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><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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_18' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_19' xmlns='http://www.w3.org/1998/Math/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+equivalence'>weak equivalences</a> to become actual <a class='existingWikiWord' href='/nlab/show/isomorphism'>isomorphisms</a>, also called the <a class='existingWikiWord' href='/nlab/show/localization'>localization</a> at the weak equivalences <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_20' xmlns='http://www.w3.org/1998/Math/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+space'>topological spaces</a> with weak equivalences those <a class='existingWikiWord' href='/nlab/show/continuous+map'>continuous functions</a> which are <a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalences</a> or <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalence'>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/Ho%28Top%29'>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+complex'>CW-complexes</a> with <a class='existingWikiWord' href='/nlab/show/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/stuff%2C+structure%2C+property'>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/%28infinity%2C1%29-category'>(∞,1)-category</a> is associated a homotopy category, whose morphisms are literally the <a class='existingWikiWord' href='/nlab/show/homotopy+class'>homotopy classes</a> of the original morphisms. See at <a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%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/%28infinity%2C1%29-category'>(∞,1)-category</a> obtained by universally forcing the <a class='existingWikiWord' href='/nlab/show/weak+equivalence'>weak equivalences</a> to become actual <a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>homotopy equivalences</a>, also called the <a class='existingWikiWord' href='/nlab/show/simplicial+localization'>simplicial localization</a> <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_21' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_22' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_23' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_24' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_25' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, the set, <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_27' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_28' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_29' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_30' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_31' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_32' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, and then the notation <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_34' xmlns='http://www.w3.org/1998/Math/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/simplicial+localization'>hammock localisation</a> of Dwyer and Kan, as then <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_35' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_36' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_37' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_38' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_39' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_40' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_41' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> at the collection <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_43' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_44' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and functor <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_46' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> sends all <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_48' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, there exists a functor <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_50' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_51' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_52' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_53' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_54' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_55' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_56' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_57' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_58' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_59' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_60' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_61' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_62' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_63' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and whose morphisms are commutative squares in <math class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_65' xmlns='http://www.w3.org/1998/Math/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+equivalence'>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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_66' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_67' xmlns='http://www.w3.org/1998/Math/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-isomorphism'>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/spectrum'>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+category'>combinatorial model categories</a> see <a class='existingWikiWord' href='/nlab/show/CombModCat'>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> </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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_68' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2c70d271c54cae83a170d9447604d1b0838e832e_69' xmlns='http://www.w3.org/1998/Math/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/Ho%28Top%29'>classical homotopy category</a>. </div>  <div class="revisedby"> Revision on June 24, 2022 at 10:59:51 by <a href="/nlab/author/Urs+Schreiber" style="color: #005c19">Urs Schreiber</a> See the <a href="/nlab/history/homotopy+category" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><a href="/nlab/show/homotopy+category" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (to current)<a href="/nlab/revision/homotopy+category/33" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (33 more)<a href="/nlab/show/homotopy+category" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/homotopy+category/34" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/homotopy+category" accesskey="S" class="navlink" id="history" rel="nofollow">History (34 revisions)</a><a href="/nlab/rollback/homotopy+category?rev=34" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/homotopy+category/34/cite" style="color: black">Cite</a> <a href="/nlab/source/homotopy+category/34" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>