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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="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> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <ul> <li><a href='#observation'>Observation</a></li> </ul> <li><a href='#monoidal_structure'>Monoidal structure</a></li> <li><a href='#QuillenAdjunctions'>Quillen adjunctions</a></li> <li><a href='#simplicial_structure'>Simplicial structure</a></li> <li><a href='#geometric_realization_is_a_reedy_cofibrant_replacement'>Geometric realization is a Reedy cofibrant replacement</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <a class="existingWikiWord" href="/nlab/show/Arne+Strom">Arne Strøm</a> proved that the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of all <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> has a structure of a Quillen <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> where <ul> <li> fibrations are <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibrations">Hurewicz fibrations</a>, </li> <li> cofibrations are closed <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibrations">Hurewicz cofibrations</a> </li> <li> and the role of weak equivalences is played by (strong) <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> (as opposed to the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> of the “standard” <a class="existingWikiWord" href="/nlab/show/Quillen+model+structure+on+topological+spaces">Quillen model structure on topological spaces</a>). </li> </ul> The theorem might have been a <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a> at the time, but the paper (<a href="#Strom72">Strøm 1972</a>) has a number of subtleties. Strøm’s proofs are not that well-known today and use techniques better known to the topologists of that time, and there is consequently a slight controversy among topologists now. One of these is that there are modern reproofs, but these modern techniques essentially use <a class="existingWikiWord" href="/nlab/show/compactly+generated+spaces">compactly generated spaces</a>, while Strøm’s proofs succeeded in avoiding that assumption. However, for many applications nowadays, one is mainly interested in the analogous model structure on the category of <a class="existingWikiWord" href="/nlab/show/k-spaces">k-spaces</a>, or of <a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a>s (<a class="existingWikiWord" href="/nlab/show/weak+Hausdorff+space">weak Hausdorff</a> <a class="existingWikiWord" href="/nlab/show/k-spaces">k-spaces</a>). Note that any cofibration in the latter category is closed. <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <div class="num_prop"> <h6 id="observation">Observation</h6> In the Strøm model structure, every object is both a <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a> and a <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant object</a>. </div> This is a most rare property for a non-trivial model structure. <h3 id="monoidal_structure">Monoidal structure</h3> The Strøm model structure on the category of <a class="existingWikiWord" href="/nlab/show/compactly+generated+spaces">compactly generated spaces</a> is a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a>. This is proven in section 6.4 of <a class="existingWikiWord" href="/nlab/show/A+Concise+Course+in+Algebraic+Topology">A Concise Course in Algebraic Topology</a> (without that language) using the fact that a subspace inclusion is a Hurewicz cofibration if and only if it is an <a class="existingWikiWord" href="/nlab/show/NDR-pair">NDR-pair</a>. <h3 id="QuillenAdjunctions">Quillen adjunctions</h3> The <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo lspace="verythinmathspace">:</mo><mi>Top</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">id \colon Top \to Top</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">left Quillen</a> from the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> (or the mixed model structure) to the Strøm model structure, and of course right Quillen in the other direction. <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Strom</mi></msub><mover><munder><mo>⟶</mo><mi>id</mi></munder><mover><mo>⟵</mo><mi>id</mi></mover></mover><msub><mi>Top</mi> <mi>Quillen</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top_{Strom} \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} Top_{Quillen} \,. </annotation></semantics></math></div> This is just the observation that any <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, and any <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>—or dually, that any <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> inclusion is a Hurewicz cofibration. It follows, by composition, that the (<a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a>)-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo lspace="verythinmathspace">:</mo><mi>sSet</mi><mo>⇆</mo><mi>Top</mi><mo lspace="verythinmathspace">:</mo><mi>Sing</mi></mrow><annotation encoding="application/x-tex"> {\vert-\vert} \colon sSet \leftrightarrows Top \colon Sing</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> between the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a> and the Strøm model structure. <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Strom</mi></msub><mover><munder><mo>⟶</mo><mi>id</mi></munder><mover><mo>⟵</mo><mi>id</mi></mover></mover><msub><mi>Top</mi> <mi>Quillen</mi></msub><mover><munder><mo>⟶</mo><mi>Sing</mi></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top_{Strom} \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} Top_{Quillen} \stackrel{\overset{{\vert-\vert}}{\longleftarrow}}{\underset{Sing}{\longrightarrow}} sSet_{Quillen} \,. </annotation></semantics></math></div> <h3 id="simplicial_structure">Simplicial structure</h3> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> denotes the category of <a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a>s, then <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo lspace="verythinmathspace">:</mo><mi>sSet</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> {|-|} \colon sSet \to Top</annotation></semantics></math> preserves finite <a class="existingWikiWord" href="/nlab/show/products">products</a>, and hence is a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a>. Therefore, in this case the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>⊣</mo><mi>Sing</mi></mrow><annotation encoding="application/x-tex">{|-|} \dashv Sing</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+Quillen+adjunction">strong monoidal Quillen adjunction</a>, and hence makes the Strøm model structure into a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>. <h3 id="geometric_realization_is_a_reedy_cofibrant_replacement">Geometric realization is a Reedy cofibrant replacement</h3> Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo>⊣</mo><mi>Sing</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mover><munder><mo>→</mo><mi>Sing</mi></munder><mover><mo>←</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow></mover></mover></mrow><annotation encoding="application/x-tex">({\vert- \vert} \dashv Sing) : Top\stackrel{\overset{{|-|}}{\leftarrow}}{\underset{Sing}{\to}} </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> for the ordinary <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>/<a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> (see <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>). For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msub><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>×</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S_{\bullet,\bullet} : \Delta^{op} \times \Delta^{op} \to Set</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/bisimplicial+set">bisimplicial set</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">d S</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>X</mi><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>×</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mover><mo>→</mo><mi>S</mi></mover><mi>Set</mi></mrow><annotation encoding="application/x-tex">d X : \Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{S}{\to} Set</annotation></semantics></math>. <div class="num_prop"> <h6 id="proposition">Proposition</h6> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/simplicial+topological+space">simplicial topological space</a>, there is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> between the <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">geometric realization</a> of the <a class="existingWikiWord" href="/nlab/show/simplicial+topological+space">simplicial topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><mo stretchy="false">|</mo><mi>Sing</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">[n] \mapsto |Sing(X_n)|</annotation></semantics></math> and the ordinary <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> that is the diagonal of the <a class="existingWikiWord" href="/nlab/show/bisimplicial+set">bisimplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">Sing(X_\bullet)_\bullet</annotation></semantics></math> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>|</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><mo stretchy="false">|</mo><mi>Sing</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>|</mo></mrow><msub><mo>≃</mo> <mi>iso</mi></msub><mo stretchy="false">|</mo><mi>d</mi><mi>Sing</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left|[n] \mapsto |Sing(X_n)|\right| \simeq_{iso} | d Sing(X_\bullet)_\bullet | \,. </annotation></semantics></math></div> Moreover, the degreewise <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>⊣</mo><mi>Sing</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(|-| \dashv Sing)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">counit</a> yields a morphism <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><mo stretchy="false">|</mo><mi>Sing</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> ([n] \mapsto |Sing(X_n)|) \to X_\bullet </annotation></semantics></math></div> and this is a cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> in the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>Top</mi> <mi>Strom</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, Top_{Strom}]_{Reedy}</annotation></semantics></math> relative to the Strøm model structure. </div> See <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">geometric realization of simplicial topological spaces</a> for more details. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">model structure on topological spaces</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+pointed+topological+spaces">classical model structure on pointed topological spaces</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">model structure on Delta-generated topological spaces</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">model structure on topological spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hurewicz+model+structure+on+chain+complexes">Hurewicz model structure on chain complexes</a> </li> </ul> <h2 id="references">References</h2> The model structure was originally established in <ul> <li id="Strom72"><a class="existingWikiWord" href="/nlab/show/Arne+Str%C3%B8m">Arne Strøm</a>, The homotopy category is a homotopy category, Archiv der Mathematik 23 (1972) (<a href="https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/strom-the-homotopy-category-is-a-homotopy-category-1972.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Strom_HomotopyCategory.pdf" title="pdf">pdf</a>)</li> </ul> using results on <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibrations">Hurewicz cofibrations</a> from: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Arne+Str%C3%B8m">Arne Strøm</a>, Note on cofibrations, Math. Scand. 19 (1966) 11-14 (<a href="https://www.jstor.org/stable/24490229">jstor:24490229</a>, <a href="https://eudml.org/doc/165952">dml:165952</a>, MR0211403) </li> <li> <a class="existingWikiWord" href="/nlab/show/Arne+Str%C3%B8m">Arne Strøm</a>, Note on cofibrations II, Math. Scand. 22 (1968) 130–142 (1969) (<a href="https://www.jstor.org/stable/24489730">jstor:24489730</a>, <a href="https://eudml.org/doc/166037">dml:166037</a>, MR0243525) </li> </ul> A new proof using <a class="existingWikiWord" href="/nlab/show/algebraic+weak+factorization+systems">algebraic weak factorization systems</a>, and its generalization to any <a class="existingWikiWord" href="/nlab/show/bicomplete+category">bicomplete category</a> which is <a class="existingWikiWord" href="/nlab/show/power">powered</a>, <a class="existingWikiWord" href="/nlab/show/copower">copowered</a> and <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in <a class="existingWikiWord" href="/nlab/show/TopSp">TopSp</a> is due to: <ul> <li id="BarthelRiehl13"><a class="existingWikiWord" href="/nlab/show/Tobias+Barthel">Tobias Barthel</a>, <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, On the construction of functorial factorizations for model categories, Algebr. Geom. Topol. 13 (2013) 1089-1124 (<a href="http://arxiv.org/abs/1204.5427">arXiv:1204.5427</a>, <a href="http://dx.doi.org/10.2140/agt.2013.13.1089">doi:10.2140/agt.2013.13.1089</a>, <a href="https://projecteuclid.org/euclid.agt/1513715550">euclid:agt/1513715550</a>)</li> </ul> Beware that a proof of the Strøm model structure was also claimed in <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+May">May</a>, <a class="existingWikiWord" href="/nlab/show/Johann+Sigurdsson">Sigurdsson</a>, <a class="existingWikiWord" href="/nlab/show/Parametrized+Homotopy+Theory">Parametrized Homotopy Theory</a>,</li> </ul> but relying on <ul> <li id="Cole06"> <a class="existingWikiWord" href="/nlab/show/Michael+Cole">Michael Cole</a>, Prop. 5.3 in Many homotopy categories are homotopy categories, Topology and its Applications 153 (2006) 1084–1099 (<a href="https://doi.org/10.1016/j.topol.2005.02.006">doi:10.1016/j.topol.2005.02.006</a>) </li> <li id="MayPonto12"> <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Kate+Ponto">Kate Ponto</a>, Lemma 17.1.7 in: <a class="existingWikiWord" href="/nlab/show/More+concise+algebraic+topology">More concise algebraic topology</a> – Localization, Completion, and Model Categories, University of Chicago Press (2012) (<a href="https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html">ISBN:9780226511795</a>, <a href="https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf">pdf</a>) </li> </ul> which later was noticed to be false, by <a class="existingWikiWord" href="/nlab/show/Richard+Williamson">Richard Williamson</a>, see <a href="#BarthelRiehl13">Barthel & Riehl, p. 2 and Rem 5.12 and Sec. 6.1</a> for details. </body></html> </div> <div class="revisedby"> Last revised on September 20, 2021 at 09:30:44. 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