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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> model category </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/1758/#Item_23" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </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> <p><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></strong>, <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></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/resolution">resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></p> </li> <li> <p><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></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a></p> </li> </ul> <p><strong>Refinements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a></p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> </ul> <p><strong>Producing new model structures</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a></p> </li> </ul> <p><strong>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</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a></p> </li> </ul> <p><strong>Model structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> <p><em>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</em></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a></p> </li> <li> <p><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></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a></p> </li> <li> <p><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></p> </li> <li> <p><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></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a></p> </li> </ul> <p><em>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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>)</p> </li> </ul> <p><em>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</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> <p><em>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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></p> </li> </ul> <p><em>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</em></p> <ul> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a></p> </li> </ul> <p><em>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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a></p> </li> </ul> <p><em>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</em></p> <p><em>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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a></p> </li> <li> <p><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</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><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></p> </li> </ul> <p><em>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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a></p> <ul> <li> <p><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></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a></p> </li> </ul> <p><em>for stable/spectrum objects</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a></p> </li> </ul> <p><em>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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a></p> </li> <li> <p><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>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a></p> </li> </ul> <p><em>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</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> <p><em>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</em></p> <ul> <li> <p><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></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><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></p> </li> <li> <p><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></p> </li> </ul> <p><em>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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a></p> </li> </ul> <p><em>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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a></p> <p><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></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks</p> </li> </ul> </div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><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></strong></p> <p>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>…</p> <p>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>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <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> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <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> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="category_theory"><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 theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <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</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </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> <li><a href='#variants'>Variants</a></li> <ul> <li><a href='#slight_variations_on_the_axioms'>Slight variations on the axioms</a></li> <li><a href='#enhancements_of_the_axioms'>Enhancements of the axioms</a></li> <li><a href='#weaker_axiom_systems'>Weaker axiom systems</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#ClosureOfMorphisms'>Closure of morphism classes under retracts</a></li> <li><a href='#RedundancyInTheAxioms'>Redundancy in the defining factorization systems</a></li> <li><a href='#BasicClosurePropertiesOfClassOfModelCategories'>Basic closure properties of class of model categories</a></li> <li><a href='#homotopy_and_homotopy_category'>Homotopy and Homotopy category</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#trivial_model_structure'>Trivial model structure</a></li> <li><a href='#classical_model_structures'>Classical model structures</a></li> <li><a href='#categorical_model_structures'>Categorical model structures</a></li> <li><a href='#parametrized_model_structures'>Parametrized model structures</a></li> <li><a href='#functor_and_localized_model_structures'>Functor and localized model structures</a></li> <li><a href='#limit_and_colimit_model_structures'>Limit and colimit model structures</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#monographs_and_textbooks'>Monographs and textbooks</a></li> <li><a href='#coverage_table_for_major_sources'>Coverage table for major sources</a></li> <li><a href='#book_chapters'>Book chapters</a></li> <li><a href='#survey_articles'>Survey articles</a></li> <li><a href='#other_sources'>Other sources</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>model category</strong> (sometimes called a <em>Quillen model category</em> or a <em>closed model category</em>, but <strong>not</strong> related to “<a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a>”) is a context for doing <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>. Quillen developed the definition of a model category to formalize the similarities between <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>: the key examples which motivated his definition were the category of <a class="existingWikiWord" href="/nlab/show/topological+space">topological spaces</a>, the category of <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial sets</a>, and the category of <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complexes</a>.</p> <p>So, what is a model category? For starters, it is a <a class="existingWikiWord" href="/nlab/show/category">category</a> equipped with three <a class="existingWikiWord" href="/nlab/show/classes">classes</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>, each closed under <a class="existingWikiWord" href="/nlab/show/composition">composition</a> and called <em><a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a></em>, <em><a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a></em> and <em><a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a></em>:</p> <ul> <li> <p>The weak equivalences play the role of ‘<a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a>’ or something a bit more general (such as <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>). Already in the case of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, it is useful to say that two spaces have the same <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> if there is a map from one to the other that induces <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> for any choice of basepoint in the first space. These maps are more general than homotopy equivalences, so they are called ‘weak equivalences’.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a> play the role of ‘nice surjections’. For example, in the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> with its usual <a class="existingWikiWord" href="/nlab/show/Quillen+model+structure+on+topological+spaces">Quillen model structure on topological spaces</a>, a locally trivial <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> is a fibration. More generally the fibrations here are the <a class="existingWikiWord" href="/nlab/show/Serre+fibrations">Serre fibrations</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> play the role of ‘nice inclusions’. For example, in the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> with its usual <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">model structure on topological spaces</a>, an <a class="existingWikiWord" href="/nlab/show/NDR+pair">NDR pair</a> is typically a cofibration.</p> </li> </ul> <p>A bit more technically: we can define an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> starting from any <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>. The idea is that this (∞,1)-category keeps track of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> in our original category, <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> between objects, <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> between morphisms, homotopies between homotopies, and so on, <em>ad infinitum</em>. However, the extra structure of a model category makes it easier to work with this (∞,1)-category. We can obtain this (∞,1)-category in various ways, such as <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of the underlying <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>, or (if the model category is simplical) the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a> of the <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicial subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>cf</mi></msub><mo>⊂</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">M_{cf}\subset M</annotation></semantics></math> of cofibrant-fibrant objects. We say this <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> is <em>presented</em> (or modeled) by the model category, and that the objects of the model category are <em>models</em> for the objects of this <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. Not every (∞,1)-category is obtained in this way (otherwise it would necessarily have all small <a class="existingWikiWord" href="/nlab/show/homotopy+limits">homotopy limits</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a>).</p> <p>In this sense model categories are ‘models for <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>’ or ‘categories of models for homotopy theory’. (The latter sense was the one intended by Quillen, but the former is also a useful way to think.)</p> <p>Recall that the idea of <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">categories with weak equivalences</a> is to work just with 1-morphisms instead of with <a class="existingWikiWord" href="/nlab/show/n-morphisms">n-morphisms</a> for all <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>, but to carry around extra information to remember which 1-morphisms are really <a class="existingWikiWord" href="/nlab/show/equivalence">equivalences</a> in the full <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>, i.e. <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> in the corresponding <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a>.</p> <p>In a model category the data of weak equivalences is accompanied by further auxiliary data that helps to compute the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a>, the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> and <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functors</a>. See <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a> for more on that.</p> <p>If the model category happens to be a <a class="existingWikiWord" href="/nlab/show/combinatorial+simplicial+model+category">combinatorial simplicial model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{A}</annotation></semantics></math> it <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presents</a> the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>A</mi></mstyle> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\mathbf{A}^\circ</annotation></semantics></math> in the form of a <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a> given by the full <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched subcategory</a> on objects that are both fibrant and cofibrant.</p> <h2 id="definition">Definition</h2> <p>The following is a somewhat terse account. For a more detailed exposition see at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Homotopy Theory</a></em> the section <em><a href="Introduction+to+Homotopy+Theory#ModelCategoryTheory">Abstract homotopy theory</a></em>.</p> <p><br /></p> <div class="num_defn" id="ModelStructure"> <h6 id="definition_2">Definition</h6> <p>A <strong>model structure</strong> on a <a class="existingWikiWord" href="/nlab/show/category">category</a> <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">\mathcal{C}</annotation></semantics></math> is a choice of three distinguished <a class="existingWikiWord" href="/nlab/show/classes">classes</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a></p> <ul> <li> <p><em>cofibrations</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cof</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cof \subset Mor(\mathcal{C})</annotation></semantics></math>,</p> </li> <li> <p><em>fibrations</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fib</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fib \subset Mor(\mathcal{C})</annotation></semantics></math>,</p> </li> <li> <p><em>weak equivalences</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \subset Mor(\mathcal{C})</annotation></semantics></math></p> </li> </ul> <p>satisfying the following conditions:</p> <ol> <li> <p><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> makes <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">\mathcal{C}</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>,</p> <p>(meaning that it contains all <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> and is closed under <strong><a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a></strong>: given a composable pair of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f,g</annotation></semantics></math>, if two out of the three morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>g</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">f, g, g f</annotation></semantics></math> are in <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>, so is the third);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Cof</mi><mo>,</mo><mi>Fib</mi><mo>∩</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Cof, Fib \cap W)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Cof</mi><mo>∩</mo><mi>W</mi><mo>,</mo><mi>Fib</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Cof \cap W, Fib)</annotation></semantics></math> are two <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a> on <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">\mathcal{C}</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_defn" id="ModelCategory"> <h6 id="definition_3">Definition</h6> <p>A <strong>model category</strong> is a <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a> and <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete category</a> <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">\mathcal{C}</annotation></semantics></math> equipped with a model structure according to def. <a class="maruku-ref" href="#ModelStructure"></a>.</p> </div> <p>This equivalent version of the definition was observed in (<a href="#Joyal">Joyal, def. E.1.2</a>), highlighted in (<a href="#Riehl09">Riehl 09</a>). This definition already implies all the closure conditions on classes of morphisms which other definitions in the literature explicitly ask for, see <a href="#ClosureOfMorphisms">below</a>.</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p><strong>(terminology)</strong></p> <ul> <li> <p>The morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow><annotation encoding="application/x-tex">W \cap Fib</annotation></semantics></math> (the fibrations that are also weak equivalences) are called <strong>trivial fibrations</strong> or <strong><a class="existingWikiWord" href="/nlab/show/acyclic+fibrations">acyclic fibrations</a></strong></p> </li> <li> <p>The morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow><annotation encoding="application/x-tex">W \cap Cof</annotation></semantics></math> (the cofibrations that are also weak equivalences) are called <strong>trivial cofibrations</strong> or <strong><a class="existingWikiWord" href="/nlab/show/acyclic+cofibrations">acyclic cofibrations</a></strong>.</p> </li> <li> <p>An object is called <strong><a class="existingWikiWord" href="/nlab/show/cofibrant">cofibrant</a></strong> if the unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\emptyset \to X</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> is a cofibration</p> </li> <li> <p>An object is called <strong><a class="existingWikiWord" href="/nlab/show/fibrant">fibrant</a></strong> if the unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X\to *</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> is a fibration.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Often, the fibrant and cofibrant objects are the ones one is “really” interested in, but the category consisting only of these is not well-behaved (as a 1-category). The factorizations supply fibrant and cofibrant replacement functors which allow us to treat any object of the model category as a ‘model’ for its fibrant-cofibrant replacement.</p> </div> <h2 id="variants">Variants</h2> <h3 id="slight_variations_on_the_axioms">Slight variations on the axioms</h3> <p>Quillen’s original definition required only <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> and <a class="existingWikiWord" href="/nlab/show/finite+colimits">finite colimits</a>, which are enough for the basic constructions. Colimits of larger cardinality are sometimes required for the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>, however. This change was popularized by <a href="#DwyerHirschhornKan97">Dwyer, Hirschhorn & Kan 1997</a>, published as <a href="#DwyerHirschhornKanSmith2004">Dwyer, Hirschhorn, Kan & Smith 2004</a>.</p> <p><a class="existingWikiWord" href="/nlab/show/Robert+W.+Thomason">Robert W. Thomason</a> proposed to require that the factorizations given by (ii) are actually <em><a class="existingWikiWord" href="/nlab/show/functorial+factorization+systems">functorial factorization systems</a></em>,</p> <p>resulting in the notion of a <a class="existingWikiWord" href="/nlab/show/Thomason+model+category">Thomason model category</a>. <a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a> later included the data of such a functorial factorization (and not just its existence) into his definition of a model category. In practice, Quillen’s <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> means that many model categories can be made to have functorial factorizations. (But not all: an example of a model category with non-functorial factorizations can be found in <a href="https://arxiv.org/abs/math/0106152">Isaksen 2001</a>.)</p> <h3 id="enhancements_of_the_axioms">Enhancements of the axioms</h3> <p>There are several extra conditions that strengthen the notion of a model category:</p> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> is <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> that is also a model category in a compatible way.</p> </li> <li> <p>An <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> is an <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> over a monoidal category, that is also a model category in a compatible way.</p> </li> <li> <p>An <a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a> is one where the two defining <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a> are refined to <a class="existingWikiWord" href="/nlab/show/algebraic+weak+factorization+systems">algebraic weak factorization systems</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> is one with a good compatible notion of <a class="existingWikiWord" href="/nlab/show/cell+complexes">cell complexes</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a> is a cofibrantly generated one that in addition is a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a>.</p> </li> <li> <p>An <a class="existingWikiWord" href="/nlab/show/accessible+model+category">accessible model category</a> is one on a locally presentable category that admits <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible</a> factorizations, which can therefore be enhanced to <a class="existingWikiWord" href="/nlab/show/algebraic+weak+factorization+systems">algebraic weak factorization systems</a>.</p> </li> <li> <p>A left/right <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a> is one where the weak equivalences are stable under pushforward along cofibrations / pullback along fibrations</p> </li> </ul> <h3 id="weaker_axiom_systems">Weaker axiom systems</h3> <p>There are several notions of <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> with similar but less structure than a full model category.</p> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> has a notion of just weak equivalences and fibrations, none of cofibrations. As the name implies, all of its objects are fibrant; the canonical example is the subcategory of fibrant objects in a model category.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a> dually has a notion of weak equivalences and cofibrations, and all of its objects are cofibrant.</p> </li> <li> <p>There is also a slight variant of the full notion of model category by Thomason that is designed to make the <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+functors">global model structure on functors</a> more naturally accessible: this is the notion of <a class="existingWikiWord" href="/nlab/show/Thomason+model+category">Thomason model category</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/category+with+path+objects">category with path objects</a> is similar to a category of fibrant objects but has <a class="existingWikiWord" href="/nlab/show/path+space+objects">path space objects</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semimodel+category">Semimodel categories</a> relax some of the conditions on lifting properties.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+model+category">Weak model categories</a> relax these conditions even further.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/premodel+category">Premodel categories</a> are an even weaker notion that allows for a nice 2-categorical treatment.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="ClosureOfMorphisms">Closure of morphism classes under retracts</h3> <p>As a consequence of the definition, the classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cof</mi><mo>,</mo><mi>Fib</mi></mrow><annotation encoding="application/x-tex">Cof, Fib</annotation></semantics></math>, and <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> each</p> <ul> <li> <p>are closed under <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> in the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Arr</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">Arr C</annotation></semantics></math>,</p> </li> <li> <p>are closed under <a class="existingWikiWord" href="/nlab/show/composition">composition</a>,</p> </li> <li> <p>contain all the <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </li> </ul> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cof</mi></mrow><annotation encoding="application/x-tex">Cof</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fib</mi></mrow><annotation encoding="application/x-tex">Fib</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow><annotation encoding="application/x-tex">W \cap Cof</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow><annotation encoding="application/x-tex">W \cap Fib</annotation></semantics></math> this and further closure properties are discussed in detail at <em><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></em> in the section <em><a href="weak+factorization+system#ClosureProperties">Closure properties</a></em>.</p> <p>In the presence of <a class="existingWikiWord" href="/nlab/show/functorial+factorizations">functorial factorizations</a>, it follows immediately that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is closed under retracts: for any retract diagram may then be funtorially factored with the middle morphism factored through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow><annotation encoding="application/x-tex">W \cap Cof</annotation></semantics></math> followed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow><annotation encoding="application/x-tex">W \cap Fib</annotation></semantics></math>, and so the statement follows from the above closure of these classes under retracts.</p> <p>Without assuming functorial factorization the statement still holds:</p> <div class="num_prop" id="WeakEquivalencesAreClosedUnderRetracts"> <h6 id="proposition">Proposition</h6> <p>Given a model category in the sense of def. <a class="maruku-ref" href="#ModelCategory"></a>, then its class of weak equivalences is closed under forming <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> (in the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a>).</p> </div> <p>(<a href="#Joyal">Joyal, prop. E.1.3</a>), highlighted in (<a href="#Riehl09">Riehl 09</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>w</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ id \colon & A &\longrightarrow& X &\longrightarrow& A \\ & {}^{\mathllap{f}} \downarrow && \downarrow^{\mathrlap{w}} && \downarrow^{\mathrlap{f}} \\ id \colon & B &\longrightarrow& Y &\longrightarrow& B } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">w \in W</annotation></semantics></math> a weak equivalence. We need to show that then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f \in W</annotation></semantics></math>.</p> <p>First consider the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>Fib</mi></mrow><annotation encoding="application/x-tex">f \in Fib</annotation></semantics></math>.</p> <p>In this case, factor <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> as a cofibration followed by an acyclic fibration. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">w \in W</annotation></semantics></math> and by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> this is even a factorization through an acyclic cofibration followed by an acyclic fibration. Hence we obtain the commuting diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mi>s</mi></mover></mtd> <mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>t</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>A</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Fib</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{id}} \\ id \colon & A' &\overset{s}{\longrightarrow}& X' &\overset{\phantom{AA}t\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{f}}_{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> is uniquely defined and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> is any lift of the top middle vertical acyclic cofibration against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. This now exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as a retract of an acyclic fibration. These are closed under retract by <a href="weak+factorization+system#ClosuredPropertiesOfWeakFactorizationSystem">this prop.</a>.</p> <p>Now consider the general case. Factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as an acyclic cofibration followed by a fibration and form the pushout in the top left square of the following diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>Cof</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>A</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{\in W \cap Cof}}\downarrow &(po)& \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{\in W \cap Cof}} \\ id \colon & A' &\overset{}{\longrightarrow}& X' &\overset{\phantom{AA}\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W }} && \downarrow^{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,, </annotation></semantics></math></div> <p>where the other three squares are induced by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the pushout, as is the identification of the middle horizontal composite as the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">A'</annotation></semantics></math>. Since acyclic cofibrations are closed under forming pushouts by <a href="weak+factorization+system#ClosuredPropertiesOfWeakFactorizationSystem">this prop.</a>, the top middle vertical morphism is now an acyclic fibration, and hence by assumption and by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> so is the middle bottom vertical morphism.</p> <p>Thus the previous case now gives that the bottom left vertical morphism is a weak equivalence, and hence the total left vertical composite is.</p> </div> <h3 id="RedundancyInTheAxioms">Redundancy in the defining factorization systems</h3> <p>It is clear that:</p> <div class="num_remark" id="AnyTwooClassesDetermineTheThird"> <h6 id="remark_2">Remark</h6> <p>Given a model category structure, any two of the three classes of special morphisms (cofibrations, fibrations, weak equivalences) determine the third:</p> <ul> <li> <p>given <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mi>RLP</mi><mo stretchy="false">(</mo><mi>W</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F = RLP(W \cap C)</annotation></semantics></math>;</p> </li> <li> <p>given <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>LLP</mi><mo stretchy="false">(</mo><mi>W</mi><mo>∩</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = LLP(W \cap F)</annotation></semantics></math>;</p> </li> <li> <p>given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>, we find <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> as the class of morphisms which factor into a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LLP</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">LLP(F)</annotation></semantics></math> followed by a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RLP</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RLP(C)</annotation></semantics></math>.</p> </li> </ul> <p>(Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RLP</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RLP(S)</annotation></semantics></math> denotes the class of morphisms with the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LLP</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">LLP(S)</annotation></semantics></math> denotes the class of morphisms with the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.)</p> </div> <p>But, in fact, already the cofibrations and the fibrant objects determine the model structure.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>A model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,W,F)</annotation></semantics></math> on a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is determined by its class of cofibrations and its class of fibrant objects.</p> </div> <p>This statement appears for instance as (<a href="#Joyal">Joyal, prop. E.1.10</a>)</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>W</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C,F,W \subset Mor(\mathcal{E})</annotation></semantics></math> be a model category.</p> <p>By remark <a class="maruku-ref" href="#AnyTwooClassesDetermineTheThird"></a> it is sufficient to show that the cofibrations and the fibrant objects determine the class of weak equivalences. Moreover, these are already determined by the weak equivalences between cofibrant objects, because for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">u : A \to B</annotation></semantics></math> any morphism, <a class="existingWikiWord" href="/nlab/show/functorial+factorization">functorial</a> cofibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>↪</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mo>≃</mo></mover><mi>A</mi></mrow><annotation encoding="application/x-tex">\emptyset \hookrightarrow \hat A \stackrel{\simeq}{\to} A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>↪</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mo>≃</mo></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">\emptyset \hookrightarrow \hat B \stackrel{\simeq}{\to} B</annotation></semantics></math> with 2-out-of-3 implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is a weak equivalence precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo>:</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat u : \hat A \to \hat B </annotation></semantics></math> is.</p> <p>By the nature of the <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><mi>Ho</mi></mrow><annotation encoding="application/x-tex">Ho</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> and by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo>:</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat u : \hat A \to \hat B</annotation></semantics></math> between cofibrant objects is a weak equivalence precisely if for every fibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mover><mi>u</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho(\hat u, X) : Ho(\hat B, X) \to Ho(\hat A, X) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, namely a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> of sets. The <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> that defines <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(\hat A,X)</annotation></semantics></math> may be taken to be given by <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> induced by <a class="existingWikiWord" href="/nlab/show/cylinder+objects">cylinder objects</a>, which in turn are obtained by factoring <a class="existingWikiWord" href="/nlab/show/codiagonals">codiagonals</a> into cofibrations followed by acyclic fibrations. So all this is determined already by the class of cofibrations, and hence weak equivalences are determined by the cofibrations and the fibrant objects.</p> </div> <h3 id="BasicClosurePropertiesOfClassOfModelCategories">Basic closure properties of class of model categories</h3> <p> <div class='num_prop' id='ProductModelStructure'> <h6>Proposition</h6> <p><strong>(product model structure)</strong> <br /> For <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">\mathcal{C}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>, the <a class="existingWikiWord" href="/nlab/show/product+category">product category</a> of their <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/categories">categories</a> carries the <em>product model structure</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>×</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C} \times \mathcal{D}</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>/<a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>/<a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> are the <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> of the corresponding such morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>.</p> <p>More generally, for an <a class="existingWikiWord" href="/nlab/show/indexed+set">indexed set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>𝒞</mi> <mi>s</mi></msub><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\big(\mathcal{C}_s\big)_{s \in S}</annotation></semantics></math> of model categories, there is the analogous product model structure on their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-indexed <a class="existingWikiWord" href="/nlab/show/product+category">product category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\underset{s \in S}{\prod} \mathcal{C}</annotation></semantics></math>, whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> and morphisms are <a class="existingWikiWord" href="/nlab/show/dependent+coproduct">dependent</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/tuples">tuples</a> of objects and morphisms in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_i</annotation></semantics></math>.</p> </div> (e.g. <a href="#Hovey99">Hovey (1999), Exp. 1.1.6</a>)</p> <p> <div class='num_prop' id='OppositeModelStructure'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/opposite+model+structure">opposite model structure</a>)</strong> <br /> If a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> carries a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a>, then the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> carries the <a class="existingWikiWord" href="/nlab/show/opposite+model+structure">opposite model structure</a>:</p> <p>its weak equivalences are those morphisms whose dual was a weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, its fibrations are those morphisms that were cofibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and similarly for its cofibrations.</p> </div> (e.g. <a href="#Hovey99">Hovey (1999), Rem. 1.1.7</a>)</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> along <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a>)</strong> <br /> Given an <a class="existingWikiWord" href="/nlab/show/right+adjoint">right</a> <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo lspace="verythinmathspace">:</mo><mi>𝒟</mi><mover><mo>⟶</mo><mo>∼</mo></mover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">R \colon \mathcal{D} \overset{\sim}{\longrightarrow} \mathcal{C}</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/categories">categories</a> and a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> on <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>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> carries the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> whose <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>/<a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>/<a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> are the <a class="existingWikiWord" href="/nlab/show/preimages">preimages</a> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> of the corresponding morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. With respect to this transferred structure, the original <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>⊣</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">L \dashv R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>.</p> </div> (cf. <a href="transferred+model+structure#RightTransferAlongAdjointEquivalence">this Example</a> at <em><a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a></em>)</p> <h3 id="homotopy_and_homotopy_category">Homotopy and Homotopy category</h3> <p>See at</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+in+a+model+category">homotopy in a model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a></p> </li> </ul> <h2 id="examples">Examples</h2> <h3 id="trivial_model_structure">Trivial model structure</h3> <p> <div class='num_remark' id='TrivialModelStructure'> <h6>Example</h6> <p><strong>(trivial model structure)</strong> <br /> Every <a class="existingWikiWord" href="/nlab/show/bicomplete+category">bicomplete category</a> carries its <em><a class="existingWikiWord" href="/nlab/show/trivial+model+structure">trivial model structure</a></em> whose weak equivalences are the <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> and all morphisms are both cofibrations and fibrations.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>Trivial as this example <a class="maruku-ref" href="#TrivialModelStructure"></a> may be, to the extent that model categories are presentations of certain <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-categories"><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>-categories</a> it shows that/how ordinary (<a class="existingWikiWord" href="/nlab/show/bicomplete+category">bicomplete</a>) <a class="existingWikiWord" href="/nlab/show/1-categories">1-categories</a> are subsumed as a special case.</p> </div> </p> <h3 id="classical_model_structures">Classical model structures</h3> <p>The archetypical model structures are the</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></li> </ul> <p>and the</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>.</li> </ul> <p>These model categories are <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> and encapsulate much of “classical” <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>. From a higher-categorical viewpoint, they can be regarded as models for <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> (in terms of <a class="existingWikiWord" href="/nlab/show/CW+complexes">CW complexes</a> or <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>, respectively).</p> <p>The passage to <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> is given by <a class="existingWikiWord" href="/nlab/show/model+structures+on+spectra">model structures on spectra</a> built out of either of these two classical model structures. See at <em><a class="existingWikiWord" href="/nlab/show/Model+categories+of+diagram+spectra">Model categories of diagram spectra</a></em> for a unified treatment.</p> <p>Accordingly <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> with its <a class="existingWikiWord" href="/nlab/show/derived+categories">derived categories</a> and <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> (which may be thought of as a sub-topic of <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> via the <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a>) is reflected by</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structures+on+chain+complexes">model structures on chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structures+on+dg-algebras">model structures on dg-algebras</a></p> </li> </ul> <p>In fact, the original definition of model categories in <a href="#Quillen67">Quillen67</a> was motivated by the <a class="existingWikiWord" href="/nlab/show/analogy">analogy</a> between constructions in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>.</p> <h3 id="categorical_model_structures">Categorical model structures</h3> <p>Of interest to category theorists is that many notions of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categories</a> come equipped with model structures, witnessing the fact that when retaining only invertible <a class="existingWikiWord" href="/nlab/show/transfors">transfors</a> between <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 they should form an <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>-<a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">category</a>. Many of these are called</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/canonical+model+structure">canonical model structure</a>s, including “categorical” model structures for <ul> <li>categories</li> <li>(strict or weak) <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-categories">strict ∞-categories</a>, and</li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoids">strict ∞-groupoids</a>.</li> </ul> </li> </ul> <p>Model categories have successfully been used to compare many different notions of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>. The following definitions 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>-category all form Quillen equivalent model categories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">simplicially enriched categories</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a> (via the Joyal <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>)</li> <li><a class="existingWikiWord" href="/nlab/show/Segal+categories">Segal categories</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+spaces">complete Segal spaces</a></li> </ul> <p>There are related model structures for enriched higher categories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+enriched+categories">model structure on enriched categories</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">model structure on dg-categories</a></li> </ul> <p>Other “higher categorical structures” can also be expected to form model categories, such as the</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">model structure on dendroidal sets</a></li> </ul> <p>which generalizes the Joyal model structure from <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operads">(∞,1)-operads</a>.</p> <p>There is also another class of model structures on categorical structures, often called <a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a>s (not to be confused with the notion of “Thomason model category”). In the “categorical” or “canonical” model structures, the weak equivalences are the categorical <a class="existingWikiWord" href="/nlab/show/equivalences">equivalences</a>, but in the Thomason model structures, the weak equivalences are those that induce weak homotopy equivalences of <a class="existingWikiWord" href="/nlab/show/nerves">nerves</a>. Thomason model structures are known to exist on 1-categories and 2-categories, at least, and are generally <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/Quillen+model+structure+on+topological+spaces">Quillen model structure on topological spaces</a> and thus (via the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> and <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>) to the and <a class="existingWikiWord" href="/nlab/show/Quillen+model+structure+on+simplicial+sets">Quillen model structure on simplicial sets</a>.</p> <h3 id="parametrized_model_structures">Parametrized model structures</h3> <p>The <em>parameterized</em> version of the model structure on simplicial sets is a</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> or</li> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">model structure on simplicial sheaves</a> or</li> </ul> <p>which serves as a <a class="existingWikiWord" href="/nlab/show/models+for+infinity-stack+%28infinity%2C1%29-toposes">model for ∞-stack (∞,1)-toposes</a> (for <a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a>es, more precisely).</p> <h3 id="functor_and_localized_model_structures">Functor and localized model structures</h3> <p>Many model structures, including those for complete Segal spaces, simplicial presheaves, and diagram spectra, are constructed by starting with a model structure on a functor category, such as a</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/projective+model+structure">projective model structure</a>,</li> <li><a class="existingWikiWord" href="/nlab/show/injective+model+structure">injective model structure</a>, or</li> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a>,</li> </ul> <p>and applying a general technique called <a class="existingWikiWord" href="/nlab/show/Bousfield+localization">Bousfield localization</a> which forces a certain class of morphisms to become weak equivalences. It can also be thought of as forcing a certain class of objects to become fibrant.</p> <h3 id="limit_and_colimit_model_structures">Limit and colimit model structures</h3> <p>Model structures can be induced on certain (usually <a class="existingWikiWord" href="/nlab/show/lax+limit">lax</a>) limits and colimits of diagrams of model categories.</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a> (lax colimits)</li> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+sections">model structure on sections</a> (lax limit)</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category+of+model+categories">2-category of model categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+homotopy+theory">synthetic homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+model+categories">localization of model categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/double+category+of+model+categories">double category of model categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ho%28CombModCat%29">Ho(CombModCat)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/premodel+category">premodel category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28%E2%88%9E%2C1%29-category">model (∞,1)-category</a></p> </li> </ul> <div> <p><strong>Algebraic model structures:</strong> <a class="existingWikiWord" href="/nlab/show/Quillen+model+structures">Quillen model structures</a>, mainly on <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a>, and their constituent <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a> and <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a>, that can be equipped with further algebraic structure and “freely generated” by small data.</p> <table><thead><tr><th>structure</th><th>small-set-generated</th><th>small-category-generated</th><th>algebraicized</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+wfs">combinatorial wfs</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+wfs">accessible wfs</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+wfs">algebraic wfs</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+model+category">accessible model category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/transfinite+construction+of+free+algebras">construction method</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></td><td style="text-align: left;">same as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+small+object+argument">algebraic small object argument</a></td></tr> </tbody></table> </div> <h2 id="References">References</h2> <p>The concept of a model category originates with</p> <ul> <li id="Quillen67"><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, Chapter I, <em>Axiomatic homotopy theory</em> in: <em><a class="existingWikiWord" href="/nlab/show/Homotopical+Algebra">Homotopical Algebra</a></em>, Lecture Notes in Mathematics <strong>43</strong> Springer (1967) [<a href="https://doi.org/10.1007/BFb0097438">doi:10.1007/BFb0097438</a>]</li> </ul> <p>and the modern form of the axioms (replacing requirement of finite (co-)limits by small (co-)limits ) is due to:</p> <ul> <li id="DwyerHirschhornKan97"><a class="existingWikiWord" href="/nlab/show/William+G.+Dwyer">William G. Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Philip+S.+Hirschhorn">Philip S. Hirschhorn</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+M.+Kan">Daniel M. Kan</a>, <em><a class="existingWikiWord" href="/nlab/show/Model+Categories+and+More+General+Abstract+Homotopy+Theory">Model Categories and More General Abstract Homotopy Theory</a></em> (1997) [<a href="https://people.math.rochester.edu/faculty/doug/otherpapers/dhk.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DwyerHirschhornKan-ModelCategories.pdf" title="pdf">pdf</a>]</li> </ul> <h4 id="monographs_and_textbooks">Monographs and textbooks</h4> <ul> <li id="Hovey99"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em><a class="existingWikiWord" href="/nlab/show/Model+Categories">Model Categories</a></em>, Mathematical Surveys and Monographs, <strong>63</strong> 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>]</p> </li> <li id="Hovey99Err"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Errata to Model Categories</em> (1999) [<a href="https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats-errata.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Hovey-ModelCategories-Errata.pdf" title="pdf">pdf</a>]</p> </li> <li id="Hirschhorn02"> <p><a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, <em><a class="existingWikiWord" href="/nlab/show/Model+Categories+and+Their+Localizations">Model Categories and Their Localizations</a></em>, AMS Math. Survey and Monographs <strong>99</strong> (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="https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf">pdf</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf">pdf</a>]</p> </li> <li id="DwyerHirschhornKanSmith04"> <p><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Smith">Jeff Smith</a>, <strong><a class="existingWikiWord" href="/nlab/show/Homotopy+Limit+Functors+on+Model+Categories+and+Homotopical+Categories">Homotopy Limit Functors on Model Categories and Homotopical Categories</a></strong>, Mathematical Surveys and Monographs <strong>113</strong>, AMS 2004 (<a href="https://bookstore.ams.org/surv-113-s">ISBN: 978-1-4704-1340-8</a>, <a href="http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J.+Peter+May">J. Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Kate+Ponto">Kate Ponto</a>, <em><a class="existingWikiWord" href="/nlab/show/More+Concise+Algebraic+Topology">More Concise Algebraic Topology</a></em>, The University of Chicago Press, 2012. Part 4.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em><a class="existingWikiWord" href="/nlab/show/Categorical+Homotopy+Theory">Categorical Homotopy Theory</a></em>, Cambridge University Press, 2014.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Scott+Balchin">Scott Balchin</a>, <em>A Handbook of Model Categories</em>, Algebra and Applications, <strong>27</strong> Springer (2021) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/978-3-030-75035-0">doi:10.1007/978-3-030-75035-0</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <h4 id="coverage_table_for_major_sources">Coverage table for major sources</h4> <table><thead><tr><th>Topic</th><th>Quillen</th><th>Hovey</th><th>Hirschhorn</th><th>DHKS</th><th>May-Ponto</th><th>Riehl</th><th>Lurie</th><th>Balchin</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a></td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">no*</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+model+categories">monoidal model categories</a></td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/enriched+model+categories">enriched model categories</a></td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a></td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">no*</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bousfield+localizations">Bousfield localizations</a></td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/transferred+model+structures">transferred model structures</a></td><td style="text-align: left;">yes</td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">no</td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Reedy+model+structures">Reedy model structures</a></td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">no</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td><td style="text-align: left;">yes</td></tr> </tbody></table> <h4 id="book_chapters">Book chapters</h4> <ul> <li id="GoerssJardine99"><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>, Chapters 1 and 2 of <em><a class="existingWikiWord" href="/nlab/show/Simplicial+homotopy+theory">Simplicial homotopy theory</a></em>, Birkhäuser, 1999, 2009.</li> </ul> <p>For yet another introduction to model categories, with an eye towards their use as <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentations</a> 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>-<a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">categories</a> see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, Appendix A.2 and A.3 of <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> <h4 id="survey_articles">Survey articles</h4> <ul> <li id="DwyerSpalinski95"> <p><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Spalinski">Jan Spalinski</a>, <em><a class="existingWikiWord" href="/nlab/show/Homotopy+theories+and+model+categories">Homotopy theories and model categories</a></em> (<a class="existingWikiWord" href="/nlab/files/DwyerSpalinski_HomotopyTheories.pdf" title="pdf">pdf</a>)</p> <p>in: <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">I. M. James</a>, <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em>, North Holland 1995 (<a href="https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2">ISBN:9780080532981</a>, <a href="https://doi.org/10.1016/B978-0-444-81779-2.X5000-7">doi:10.1016/B978-0-444-81779-2.X5000-7</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Kirsten+Schemmerhorn">Kirsten Schemmerhorn</a>, <em>Model categories and simplicial methods</em>, Notes from lectures given at the University of Chicago, August 2004, in: <em>Interactions between Homotopy Theory and Algebra</em>, Contemporary Mathematics 436, AMS 2007(<a href="http://arxiv.org/abs/math.AT/0609537">arXiv:math.AT/0609537</a>, <a href="http://dx.doi.org/10.1090/conm/436">doi:10.1090/conm/436</a>)</p> </li> </ul> <h4 id="other_sources">Other sources</h4> <p>An account is in</p> <ul> <li><a class="existingWikiWord" href="/joyalscatlab/published/HomePage">Joyal's CatLab</a>, <em><a class="existingWikiWord" href="/joyalscatlab/published/Model+categories">Model categories</a></em></li> </ul> <p>and appendix E of</p> <ul> <li id="Joyal"><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em>The theory of quasi-categories and its applications</em> (<a href="http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JoyalTheoryOfQuasicategories.pdf" title="pdf">pdf</a>)</li> </ul> <p>The version of the definition in (<a href="#Joyal">Joyal</a>) is also highlighted in</p> <ul> <li id="Riehl09"><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em>A concise definition of model category</em>, 2009 (<a href="http://www.math.jhu.edu/~eriehl/modelcat.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/RiehlConciseDefinitionOfModelCategory.pdf" title="pdf">pdf</a>)</li> </ul> <p>An introductory survey of some key concepts is in the set of slides</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em><a class="existingWikiWord" href="/nlab/files/ModelCatPrimer.pdf" title="Model Category Primer">Model Category Primer</a></em></li> </ul> <p>There is an unpublished manuscript of <a class="existingWikiWord" href="/nlab/show/Chris+Reedy">Chris Reedy</a> from around 1974 that’s been circulating as an increasingly faded photocopy. It’s been typed into LaTeX, and the author has given <a href="http://www-math.mit.edu/~psh/#Other%20mathematics">permission</a> for it to be posted on the net:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Reedy">Chris Reedy</a>, <em>Homotopy theory of model categories</em> (<a href="http://www-math.mit.edu/~psh/reedy.pdf">pdf</a>)</li> </ul> <p>See</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, personal website: <em><a href="http://www-math.mit.edu/~psh/#Mathematics">Mathematics</a></em></li> </ul> <p>for errata and more.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on March 14, 2024 at 04:57:38. See the <a href="/nlab/history/model+category" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/model+category" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/1758/#Item_23">Discuss</a><span class="backintime"><a href="/nlab/revision/model+category/124" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/model+category" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/model+category" accesskey="S" class="navlink" id="history" rel="nofollow">History (124 revisions)</a> <a href="/nlab/show/model+category/cite" style="color: black">Cite</a> <a href="/nlab/print/model+category" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/model+category" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>