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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="category_theory">Category theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/category">category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="model_category_theory">Model category theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, <a class="existingWikiWord" href="/nlab/show/model+%28infinity%2C1%29-category">model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-category</a> Definitions <ul> <li> <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> (<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">of a model category</a> </li> </ul> Morphisms <ul> <li> <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> </li> </ul> </li> </ul> Universal constructions <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> <a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a> <a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a> </li> </ul> Refinements <ul> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+category">locally cartesian closed model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a> </li> </ul> Producing new model structures <ul> <li> <a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a> </li> </ul> Presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories <ul> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a> </li> </ul> Model structures <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids <a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+semi-simplicial+sets">on semi-simplicial sets</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-groupoids">on strict ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/natural+model+structure+on+groupoids">on groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">on chain complexes</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a> related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a> </li> </ul> for equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a> (<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>) </li> </ul> for rational <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> for rational equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-groupoids</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-types</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">on simplicial T-algebras</a>, on <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a> </li> </ul> specific <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">on dg-algebras</a> and on <a class="existingWikiWord" href="/nlab/show/simplicial+ring">on simplicial rings</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">on cosimplicial rings</a> related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a> </li> </ul> for stable/spectrum objects <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+quasi-categories">Joyal model for quasi-categories</a> (and its <a class="existingWikiWord" href="/nlab/show/model+structure+for+cubical+quasicategories">cubical version</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a> </li> </ul> for stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-operads <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">on operads</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">for Segal operads</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">for dendroidal complete Segal spaces</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">for dendroidal Cartesian fibrations</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-categories <ul> <li> <a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a> </li> </ul> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a> <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a>/<a class="existingWikiWord" href="/nlab/show/Cech+model+structure+on+simplicial+presheaves">Cech model structure</a>/<a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#internalization'>Internalization</a></li> <li><a href='#cofibrant_resolutions'>Cofibrant resolutions</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> In general, canonical model structures are <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures on the categories of some flavor of <a class="existingWikiWord" href="/nlab/show/n-category">n-categories</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">1\le n\le \infty</annotation></semantics></math> (note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">n=\infty</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is allowed), which are intended to capture the correct “categorical” theory of these categories. The archetypical examples are <ul> <li> the <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+categories">canonical model structure on categories</a> </li> <li> the <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">canonical model structure on groupoids</a>. </li> </ul> In <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a> of these, one has <ul> <li><a class="existingWikiWord" href="/nlab/show/canonical+model+structures+on+2-categories">canonical model structures on 2-categories</a>.</li> </ul> Canonical model structures are sometimes called “folk” model structures, but the appropriateness of this term is <a href="https://nforum.ncatlab.org/discussion/649">questionable</a>, especially in the cases <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \gt 1</annotation></semantics></math>. Other alternatives are ‘endogenous’, ‘standard’, ‘natural’, and ‘categorical’. While ultimately the collection of all <a class="existingWikiWord" href="/nlab/show/n-category"><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>-categories</a> should form an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-category, restricting that to just invertible higher morphisms will yield an <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n+1,1)-category</a>, and thus in particular an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>. It is this (∞,1)-category which is intended to be <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presented</a> by a canonical model structure. In particular, the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> in a canonical model structure should be the <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">category-theoretic equivalences</a>. This is to be contrasted with <a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structures</a> in which the weak equivalences are the morphisms that induce a <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">weak homotopy equivalence</a> of <a class="existingWikiWord" href="/nlab/show/nerve">nerves</a>. This amounts to regarding each category, or rather its <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>, as a placeholder for its groupoidification (Kan fibrant replacement) and then considering the standard notion of equivalence. In a canonical model structure for some flavor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories, usually <ul> <li>a fibration is a functor that lifts <a class="existingWikiWord" href="/nlab/show/equivalence">equivalences</a> in all dimensions,</li> <li>an acyclic fibration is a functor which is <a class="existingWikiWord" href="/nlab/show/k-surjective+functor">k-surjective</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0\le k\le n</annotation></semantics></math>,</li> <li>a weak equivalence is a functor which is <a class="existingWikiWord" href="/nlab/show/k-surjective+functor">essentially k-surjective</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0\le k\le n</annotation></semantics></math>, and</li> <li>a cofibration is a functor which is injective on objects and “relatively free” on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-morphisms for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1\le k \lt n</annotation></semantics></math>. These can also be described as the morphisms <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+structure">generated</a> by the inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><msub><mi>G</mi> <mi>k</mi></msub><mo>↪</mo><msub><mi>G</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\partial G_k \hookrightarrow G_k</annotation></semantics></math> of the boundary of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/globe">globe</a> into the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/globe">globe</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo><</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">0\le k \lt \infty</annotation></semantics></math>.</li> </ul> <h2 id="internalization">Internalization</h2> A common problem is to transport the (a) model structure on plain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories <a class="existingWikiWord" href="/nlab/show/internal+category">internal to</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sets</mi></mrow><annotation encoding="application/x-tex">Sets</annotation></semantics></math> to another <a class="existingWikiWord" href="/nlab/show/internalization">internal context</a>, notably for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sets</mi></mrow><annotation encoding="application/x-tex">Sets</annotation></semantics></math> is replaced with some kind of category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spaces</mi></mrow><annotation encoding="application/x-tex">Spaces</annotation></semantics></math>. This is relevant for the discussion of the homotopy theory of topological and smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories. Usually, such internalization of model structures has the consequence that some properties invoked in the description of the original model structure, notably some of the lifting properties, will only continue to hold “locally”. One way to deal with this is to pass to a notion slightly weaker than that of a <a class="existingWikiWord" href="/nlab/show/homotopy+theory">model category</a> called a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> as used in <a class="existingWikiWord" href="/nlab/show/homotopical+cohomology+theory">homotopical cohomology theory</a>. But there are also full model structures for such situations. Notice that under a suitable <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> operation all <a class="existingWikiWord" href="/nlab/show/n-category">n-categories</a> usually embed into <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s. The <a class="existingWikiWord" href="/nlab/show/models+for+infinity-stack+%28infinity%2C1%29-toposes">models for infinity-stack (infinity,1)-toposes</a> given by the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> then serves to present the corresponding <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>-category of parameterized or internal <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>-categories. See for instance also <a class="existingWikiWord" href="/nlab/show/smooth+infinity-stack">smooth infinity-stack</a>. <h2 id="cofibrant_resolutions">Cofibrant resolutions</h2> In <ul> <li><a class="existingWikiWord" href="/nlab/show/Francois+M%C3%A9tayer">Francois Métayer</a>, Cofibrant complexes are free (<a href="http://arxiv.org/abs/math.CT/0701746">arXiv</a>)</li> </ul> it is shown that cofibrant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories with respect to the canonical model structure are precisely the “free” ones, where “free” here means “generated from a <a class="existingWikiWord" href="/nlab/show/polygraph">polygraph</a>” as described in <ul> <li><a class="existingWikiWord" href="/nlab/show/Francois+M%C3%A9tayer">Francois Métayer</a>, Resolutions by <a class="existingWikiWord" href="/nlab/show/polygraphs">polygraphs</a> (<a href="http://www.tac.mta.ca/tac/volumes/11/7/11-07.pdf">tac</a>)</li> </ul> We had some blog discussion about this at <a href="http://golem.ph.utexas.edu/category/2008/10/freely_generated_categories.html">Freely generated omega-categories</a>. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/basic+localizer">basic localizer</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+Cat">canonical model structure on Cat</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+Grpd">canonical model structure on Grpd</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+2Cat">canonical model structure on 2Cat</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+Operad">canonical model structure on Operad</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+omega-categories">model structure on strict omega-categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/2-trivial+model+structure">2-trivial model structure</a> </li> </ul> <h2 id="References">References</h2> The <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+categories">canonical model structure on categories</a> was known to experts for some time before being written down formally (this is the origin of the adjective “folk”, as in “<a class="existingWikiWord" href="/nlab/show/folklore">folklore</a>”). The <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">canonical model structure on groupoids</a> is claimed (but not proven) <ul> <li id="Anderson78"><a class="existingWikiWord" href="/nlab/show/Donald+W.+Anderson">Donald W. Anderson</a>, p. 783 (12 of 37) in: Fibrations and geometric realization, Bull. Amer. Math. Soc. 84 5 (1978) 765-788 [<a href="http://projecteuclid.org/euclid.bams/1183541139">euclid:1183541139</a>]</li> </ul> A published proof of this and the <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+categories">canonical model structure on categories</a>, in the further generality of <a class="existingWikiWord" href="/nlab/show/internal+category">categories internal</a> to a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> (hence of <a class="existingWikiWord" href="/nlab/show/2-sheaves">2-sheaves</a>) is due to <ul> <li id="JoyalTierney91"><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, Thm. 2 (p. 221) Strong stacks and classifying spaces, in: Category Theory Lecture Notes in Mathematics 1488, Springer (1991) 213-236 [<a href="https://doi.org/10.1007/BFb0084222">doi:10.1007/BFb0084222</a>]</li> </ul> and more elementary proofs for the plain case (internal to <a class="existingWikiWord" href="/nlab/show/Set">Set</a>) are given in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, A model category for categories (1996) [<a href="http://www.math.uiuc.edu/~rezk/cat-ho.dvi">dvi</a>, <a class="existingWikiWord" href="/nlab/files/Rezk_ModelCategoryForCategories.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/joyalscatlab/published/Model+structures+on+Cat">Model structures on Cat</a> (2010) </li> </ul> and (for the <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">case of groupoids</a>) in <ul> <li id="Strickland00"><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, §6.1 of: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(n)</annotation></semantics></math>-local duality for finite groups and groupoids , Topology 39 4 (2000) [<a href="https://arxiv.org/abs/math/0011109">arXiv:math/0011109</a>, <a href="https://doi.org/10.1016/S0040-9383(99)00031-2">doi:10.1016/S0040-9383(99)00031-2</a>]</li> </ul> Further generalization to <a class="existingWikiWord" href="/nlab/show/internal+category">categories internal</a> to <a class="existingWikiWord" href="/nlab/show/finitely+complete+categories">finitely complete categories</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Tomas+Everaert">Tomas Everaert</a>, <a class="existingWikiWord" href="/nlab/show/Rudger+W.+Kieboom">Rudger W. Kieboom</a>, <a class="existingWikiWord" href="/nlab/show/Tim+Van+der+Linden">Tim Van der Linden</a>, Model structures for homotopy of internal categories, Theory and Applications of Categories, Vol. 15, CT2004, No. 3, pp 66-94. (<a href="http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html">tac:15-03</a>).</li> </ul> though it seems that the assumptions in this article apply to ambient categories that are <a class="existingWikiWord" href="/nlab/show/semiabelian+categories">semiabelian categories</a>, but do not apply to ambient categories like <a class="existingWikiWord" href="/nlab/show/Top">Top</a>. A brief summary, together with a generalization when one assumes only weaker versions of the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>, can be found at <a class="existingWikiWord" href="/nlab/show/folk+model+structure+on+Cat">folk model structure on Cat</a>. The <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+2-categories">canonical model structures for 2-categories and bicategories</a> are due to <ul> <li> <a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, A Quillen Model Structure for 2-Categories, K-Theory 26: 171–205, 2002. (<a href="http://web.science.mq.edu.au/~slack/papers/qmc2cat.html">website</a>) A Quillen Model Structure for Bicategories, K-Theory 33: 185-197, 2004. (<a href="http://web.science.mq.edu.au/~slack/papers/qmcbicat.html">website</a>) </li> </ul> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n=3</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Gray-categories">Gray-categories</a>) in <ul> <li><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, A Quillen model structure for Gray-categories (<a href="http://arxiv.org/abs/1001.2366">arxiv:1001.2366</a>)</li> </ul> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">n = \omega</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/strict+omega-categories">strict omega-categories</a>) <ul> <li><a class="existingWikiWord" href="/nlab/show/Yves+Lafont">Yves Lafont</a>, <a class="existingWikiWord" href="/nlab/show/Francois+M%C3%A9tayer">Francois Métayer</a>, Krzysztof Worytkiewicz, A folk model structure on omega-cat (<a href="http://arxiv.org/abs/0712.0617">arXiv</a>)</li> </ul> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">n = \omega</annotation></semantics></math> and all morphisms invertible, there is the <a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+omega-groupoids">model structure on strict omega-groupoids</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a> and M. Golasinski, A model structure for the homotopy theory of crossed complexes, Cah. Top. Géom. Diff. Cat. 30 (1989) 61-82 (<a href="http://www.bangor.ac.uk/~mas010/pdffiles/RB-golskyrev.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Dimitri+Ara">Dimitri Ara</a>, <a class="existingWikiWord" href="/nlab/show/Francois+Metayer">Francois Metayer</a>, The Brown-Golasinski model structure on ∞-groupoids revisited (pdf) Homology, Homotopy Appl. 13 (2011), no. 1, 121–142. </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on November 2, 2023 at 12:18:54. 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