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? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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="topology">Topology</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/topology">topology</a> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>) see also <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a> and <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a> Basic concepts <ul> <li> <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/convergence">convergence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <a href="Top#UniversalConstructions">Universal constructions</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>, </li> <li> fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a> <a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> </li> </ul> Examples <ul> <li> <a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a> </li> </ul> Basic statements <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a> </li> </ul> Analysis Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a> </li> <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/homotopy+group">homotopy group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> </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='#definition'>Definition</a></li> <ul> <li><a href='#QuillenModelStructure'>Classical Quillen Model Structure</a></li> <li><a href='#StromModelStructure'>Hurewicz (or Strøm) Model Structure</a></li> <li><a href='#mixed_model_structure'>Mixed Model Structure</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#OnNiceTopologicalSpaces'>Restriction to convenient categories of topological spaces</a></li> <li><a href='#relation_between__and_'>Relation between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math></a></li> <li><a href='#relation_between__and__2'>Relation between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Strom</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Strom}</annotation></semantics></math></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> Philosophically, <a class="existingWikiWord" href="/nlab/show/model+category">model structures</a> allow one to <a class="existingWikiWord" href="/nlab/show/localization">localize</a> a <a class="existingWikiWord" href="/nlab/show/category">category</a> at a particular collection of <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">weak equivalences</a>, which one would like to formally invert. For <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s, there are two natural candidates for the collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> of weak equivalences: <ol> <li> the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>s </li> <li> and the <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>s. </li> </ol> Both of these have accompanying model structures. Interestingly, these two model structures can also be combined to form what’s known as the mixed model structure. All of these model structures exist not only on the category of all topological spaces, but also on most <a class="existingWikiWord" href="/nlab/show/convenient+categories+of+topological+spaces">convenient categories of topological spaces</a>. Using a nice category instead is sometimes important, such as if we want the model structure to be <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal</a>. <h2 id="definition">Definition</h2> <h3 id="QuillenModelStructure">Classical Quillen Model Structure</h3> The first, and most prevalent, <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, called the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> (<a href="#Quillen67">Quillen67, II.3</a>) has <ul> <li> weak equivalences are the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>; </li> <li> fibrations are the <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a>, maps which have the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> with respect to all inclusions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>:</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i_0 : D^n \hookrightarrow D^n \times I</annotation></semantics></math> that include the <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>-disk as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">D^n \times \{0\}</annotation></semantics></math>. </li> <li> cofibrations are the “<a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> of <a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a>”. </li> </ul> This is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a>. The cofibrations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> are generated by the set of <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^{n-1} \hookrightarrow D^n</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> in the sense that they are the smallest saturated class containing these morphisms. As a consequence, of Quillen’s <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>, all cofibrations have the form described above, where a <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> is a <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composite</a> of <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a> of <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> of these generating maps. This model structure is sometimes called the <a class="existingWikiWord" href="/nlab/show/Quillen+model+structure+on+topological+spaces">Quillen model structure on topological spaces</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>-model structure on <a class="existingWikiWord" href="/nlab/show/Top">Top</a>. <h3 id="StromModelStructure">Hurewicz (or Strøm) Model Structure</h3> A second model structure – the <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a> – has <ul> <li> weak equivalences are the <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a>; </li> <li> fibrations are the <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a>s, which are defined to be maps that have the right lifting property with respect to all inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>:</mo><mi>A</mi><mo>↪</mo><mi>A</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i_0 : A \hookrightarrow A \times I</annotation></semantics></math> for any topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. </li> <li> cofibrations are determined by these classes and are called the closed <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a>s. </li> </ul> This model structure is sometimes called the Hurewicz model structure, since it uses Hurewicz fibrations and cofibrations, or also the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>-model structure, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> can stand for either “Hurewicz” or “homotopy equivalence.” However, it is also sometimes called the <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a>, since it was first proven to exist by <a class="existingWikiWord" href="/nlab/show/Arne+Str%C3%B8m">Arne Strøm</a>. <h3 id="mixed_model_structure">Mixed Model Structure</h3> From the definitions, <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a>s are necessarily <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>s. It is well-known that <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>s are <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>s. If we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>F</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>W</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C_1, F_1, W_1)</annotation></semantics></math> for the classes of the first model structure and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>F</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>W</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C_2, F_2, W_2)</annotation></semantics></math> for the classes of the second, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mn>2</mn></msub><mo>⊂</mo><msub><mi>W</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">W_2 \subset W_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub><mo>⊂</mo><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_2 \subset F_1</annotation></semantics></math>. In general given two model structures with these inclusions, we get a third <a class="existingWikiWord" href="/nlab/show/mixed+model+structure">mixed model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>m</mi></msub><mo>,</mo><msub><mi>F</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>W</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C_m, F_2, W_1)</annotation></semantics></math> where the cofibrations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">C_m</annotation></semantics></math> are determined by the other two classes. On topological spaces, this model structure has <ul> <li> weak equivalences the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>s </li> <li> fibrations the <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a>s; </li> <li> cofibrant spaces (the <a class="existingWikiWord" href="/nlab/show/m-cofibrant+space">m-cofibrant space</a>s) are precisely those spaces that are homotopy equivalent to <a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a>es. </li> </ul> <h2 id="properties">Properties</h2> <h3 id="OnNiceTopologicalSpaces">Restriction to convenient categories of topological spaces</h3> For the discussion of the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> given by the model structure on topological spaces, it is necessary or at least useful to pass to <a class="existingWikiWord" href="/nlab/show/convenient+categories+of+topological+spaces">convenient categories of topological spaces</a>. <div class="num_defn"> <h6 id="definition_2">Definition</h6> Write <ul> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Top</mi><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">k Top \hookrightarrow Top</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/k-spaces">k-spaces</a>; </li> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CGTop</mi><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">CGTop \hookrightarrow Top</annotation></semantics></math> for the full subcategory of <a class="existingWikiWord" href="/nlab/show/compactly+generated+spaces">compactly generated spaces</a>. </li> </ul> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">model structure on compactly generated topological spaces</a>) There is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>kTop</mi> <mi>Qu</mi></msub></mrow><annotation encoding="application/x-tex">kTop_{Qu}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">k Top</annotation></semantics></math> in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Top</mi><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">k Top \hookrightarrow Top</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>. This inclusion is the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> in a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Top</mi><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mo>≃</mo> <mpadded width="0"><mi>Qu</mi></mpadded></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟵</mo><mi>k</mi></munder><mover><mo>↪</mo><mrow></mrow></mover></munderover><mi>Top</mi></mrow><annotation encoding="application/x-tex"> k Top \underoverset {\underset{k}{\longleftarrow}} {\overset{}{\hookrightarrow}} {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;} Top </annotation></semantics></math></div></div> This appears for instance as (<a href="#Hovey">Hovey, theorem 2.4.23</a>) <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> There is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CGTop</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">CGTop_{Quillen}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CGTop</mi></mrow><annotation encoding="application/x-tex">CGTop</annotation></semantics></math> in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CGTop</mi><mo>↪</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">CGTop \hookrightarrow Top</annotation></semantics></math>. And this inclusion is the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> in a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>CGTop</mi> <mi>Qu</mi></msub><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mo>≃</mo> <mpadded width="0"><mi>Qu</mi></mpadded></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>↪</mo><mrow></mrow></munder><mover><mo>⟵</mo><mi>h</mi></mover></munderover><mi>k</mi><msub><mi>Top</mi> <mi>Qu</mi></msub><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mrow><annotation encoding="application/x-tex"> CGTop_{Qu} \underoverset {\underset{}{\hookrightarrow}} {\overset{ h }{\longleftarrow}} {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\; } k Top_{Qu} \mathrlap{\,.} </annotation></semantics></math></div></div> (e.g. <a href="#Hovey99">Hovey 1999, Thm. 2.4.25</a>) To appreciate the following, notice that <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>Top</mi> <mi>Qu</mi></msub> </mrow> <annotation encoding="application/x-tex">Top_{Qu}</annotation> </semantics> </math></a> is not a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a>, because <a class="existingWikiWord" href="/nlab/show/Top">Top</a> itself is not (<a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian</a>) <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a>. <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> Both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><msub><mi>Top</mi> <mi>Qu</mi></msub></mrow><annotation encoding="application/x-tex">k Top_{Qu}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CGTop</mi> <mi>Qu</mi></msub></mrow><annotation encoding="application/x-tex">CGTop_{Qu}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/monoidal+model+categories">monoidal model categories</a>. </div> This appears as (<a href="#Hovey">Hovey, prop. 4.2.11</a>). <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> In fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CGTop</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">CGTop_{Quillen}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>. </div> (This is briefly mentioned in <a href="#BergerMoerdijk03">Berger-Moerdijk 2003, Sec. 3.3.2</a>.) <h3 id="relation_between__and_">Relation between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math></h3> The <a href="#QuillenModelStructure">Quillen model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Qullen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Qullen}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to the standard (Quillen) <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a> via the total <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid">singular complex</a> and <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> functors. <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>⊣</mo><mi>Sing</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><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"> (\vert-\vert \dashv Sing) \colon Top_{Quillen} \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen} \,. </annotation></semantics></math></div> Since the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a> is a <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentation</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s realized as <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es, this identifies topological spaces with <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-categorical</a> sense. Notably it says that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid is, up to equivalence, the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> of some topological space. This statement is called the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a> (which here is a theorem). See there for more details. <h3 id="relation_between__and__2">Relation between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Strom</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Strom}</annotation></semantics></math></h3> The identity functor constitutes a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>⊣</mo><mi>Id</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Top</mi> <mi>Strom</mi></msub><mover><mo>→</mo><mo>←</mo></mover><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> (Id \dashv Id) : Top_{Strom} \stackrel{\leftarrow}{\to} Top_{Quillen} </annotation></semantics></math></div> between the <a href="#QuillenModelStructure">Quillen model structure</a> and the <a href="#StromModelStructure">Strom model structure</a> on <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>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Strom</mi></msub><mo>→</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Strom} \to Top_{Quillen}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">right Quillen functor</a>. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopical+structure+on+C%2A-algebras">homotopical structure on C*-algebras</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> </ul> <h2 id="references">References</h2> The original “Quillen” or “q-” model structure is due to <ul> <li id="Quillen67"><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, Ch. II, Sec. 3 of <a class="existingWikiWord" href="/nlab/show/Homotopical+Algebra">Homotopical Algebra</a>, Lecture Notes in Mathematics 43, Springer 1967(<a href="https://doi.org/10.1007/BFb0097438">doi:10.1007/BFb0097438</a>)</li> </ul> An expository, concise and comprehensive writeup is in <ul> <li><a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, The Quillen model category of topological spaces, 2015 (<a href="http://arxiv.org/abs/1508.01942">arXiv:1508.01942</a>)</li> </ul> Standard textbooks references include <ul> <li id="Hovey"> <a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <a class="existingWikiWord" href="/nlab/show/Model+Categories">Model Categories</a>, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (<a href="https://bookstore.ams.org/surv-63-s">ISBN:978-0-8218-4361-1</a>, <a href="https://doi.org/http://dx.doi.org/10.1090/surv/063">doi:10.1090/surv/063</a>, <a href="https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf">pdf</a>, <a href="http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover">Google books</a>) </li> <li id="Hirschhorn02"> <a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (<a href="https://bookstore.ams.org/surv-99-s/">ISBN:978-0-8218-4917-0</a>, <a href="http://www.gbv.de/dms/goettingen/360115845.pdf">pdf toc</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Kate+Ponto">Kate Ponto</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, section 17 of More concise algebraic topology (<a href="http://www.maths.ed.ac.uk/~aar/papers/mayponto.pdf">pdf</a>) </li> </ul> For the “Hurewicz,” “h-” or “<a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a>”: <ul> <li><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> For the “m-” or “<a class="existingWikiWord" href="/nlab/show/mixed+model+structure">mixed model structure</a>”: <ul> <li>Michael Cole, Mixing model structures, Topology Appl. 153 no. 7 (2006) <a href="http://dx.doi.org/10.1016/j.topol.2005.02.004">doi</a>.</li> </ul> The generalization to the <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+operads">model structure on topological operads</a> is due to <ul> <li id="BergerMoerdijk03"><a class="existingWikiWord" href="/nlab/show/Clemens+Berger">Clemens Berger</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, Axiomatic homotopy theory for operads, Comment. Math. Helv. Vol. 78 (2003), no. 4 (<a href="http://arxiv.org/abs/math/0206094">arXiv:math/0206094</a>)</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on November 30, 2021 at 13:22:28. 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