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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> proper 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/429/#Item_22" 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> </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='#the_rezk_criterion_for_properness'>The Rezk criterion for properness</a></li> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#left_proper_model_categories'>Left proper model categories</a></li> <li><a href='#nonleft_proper_model_categories'>Non-left proper model categories</a></li> <li><a href='#right_proper_model_categories'>Right proper model categories</a></li> <li><a href='#nonright_proper_model_categories'>Non-right proper model categories</a></li> <li><a href='#proper_model_categories'>Proper model categories</a></li> <li><a href='#reedy_model_structures'>Reedy model structures</a></li> <li><a href='#ProperEquivModels'>Proper Quillen equivalent model structures</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#HomotopyLimits'>Homotopy (co)limits in proper model categories</a></li> <li><a href='#SliceCategories'>Slice categories</a></li> <li><a href='#local_cartesian_closure'>Local cartesian closure</a></li> </ul> <li><a href='#related_pages'>Related pages</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> fibrations enjoy <a class="existingWikiWord" href="/nlab/show/pullback+stability">pullback stability</a> and cofibrations are stable under <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, but weak equivalences need not have either property. In a proper model category weak equivalences are also preserved under certain pullbacks and/or certain pushouts.</p> <p>Put differently, in a proper model category, <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a> and/or pushouts can be computed with less need for <a class="existingWikiWord" href="/nlab/show/fibrant+replacement">fibrant and/or cofibrant replacement</a>.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> is called</p> <ul> <li> <p><strong>right proper</strong> if <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are preserved by <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> along <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>,</p> </li> <li> <p><strong>left proper</strong> if weak equivalence are preserved by <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> along cofibrations,</p> </li> <li> <p><strong>proper</strong> if it is both left and right proper.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>More in detail this means the following. A model category is right proper if for every weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> in <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(C)</annotation></semantics></math> and every fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">h : C \to B</annotation></semantics></math> the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>h</mi> <mo>*</mo></msup><mi>f</mi><mo>:</mo><mi>A</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">h^* f : A \times_B C \to C</annotation></semantics></math> in</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>A</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><mo>⇒</mo><msup><mi>h</mi> <mo>*</mo></msup><mi>f</mi><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><mi>f</mi><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>h</mi><mo>∈</mo><mi>F</mi></mrow></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A \times_B C &\longrightarrow& A \\ \;\;\big\downarrow{}^{\mathrlap{\Rightarrow h^* f \in W}} && \big\downarrow{}^{\mathrlap{f \in W}} \\ C &\underset{h \in F}{\longrightarrow}& B } </annotation></semantics></math></div> <p>is a weak equivalence.</p> </div> <p> <div class='num_remark'> <h6>Remark</h6> <p>The above definition is the way it is usually phrased, but in fact it is equivalent to a seemingly weaker condition that is sometimes easier to check: for right properness it suffices to assume that weak equivalences are preserved by pullback along fibrations <em>between fibrant objects</em>. That is, in the more explicit version above, we are free to assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> (hence also <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>) is fibrant; this then implies the more general version without this hypothesis. See Proposition <a class="maruku-ref" href="#FibrationsBetweenFibrantObjectsSuffice"></a> below.</p> </div> </p> <h2 id="the_rezk_criterion_for_properness">The Rezk criterion for properness</h2> <p>The following criterion shows that the notion of left or right properness only depends on the underlying <a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a> of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, i.e., does not depend on fibrations or cofibrations. This is clear once we observe that the notion of a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> in the statement below can be replaced by the notion of a <a class="existingWikiWord" href="/nlab/show/Dwyer%E2%80%93Kan+equivalence">Dwyer–Kan equivalence</a> of underlying <a class="existingWikiWord" href="/nlab/show/relative+categories">relative categories</a>, or just ordinary equivalences of underlying <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a>.</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>(<a href="#Rezk02">Rezk</a>, Proposition 2.7 (arXiv), Proposition 2.5 (journal).) A model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is left proper if and only if for every weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X\to Y</annotation></semantics></math> the induced <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>M</mi><mo>⇆</mo><mi>Y</mi><mo stretchy="false">/</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X/M\leftrightarrows Y/M</annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>. A model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is right proper if and only if for every weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X\to Y</annotation></semantics></math> the induced <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">/</mo><mi>X</mi><mo>⇆</mo><mi>M</mi><mo stretchy="false">/</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">M/X\leftrightarrows M/Y</annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>.</p> </div> </p> <h2 id="Examples">Examples</h2> <h3 id="left_proper_model_categories">Left proper model categories</h3> <ul> <li> <p>by cor. <a class="maruku-ref" href="#AllObjectsFibrantImpliesRightProper"></a>, every model category in which all objects are cofibrant is left proper;</p> <p>this includes notably</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> </ul> <p>and many model structures derived from these, such as</p> <ul> <li>the injective global <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> over any (simplicial) category;</li> </ul> </li> <li> <p>the left <a class="existingWikiWord" href="/nlab/show/Bousfield+localization">Bousfield localization</a> of every left proper <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a> at a set of morphisms is again left proper.</p> <p>So in particular also the <em>local</em> injective model structures on simplicial presheaves over a <a class="existingWikiWord" href="/nlab/show/site">site</a> are left proper.</p> </li> </ul> <h3 id="nonleft_proper_model_categories">Non-left proper model categories</h3> <p>A class of model structures which tends to be <em>not</em> left proper are model structures on categories of not-necessarily commutative algebras.</p> <p>For instance</p> <ul> <li>the standard model structure on <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial</a> <a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a>s (weak equivalences and fibrations are those of the underlying simplicial sets) is <em>not</em> left proper (see <a href="#Rezk02">Rezk</a>, Example 2.11 (arXiv), Example 2.7 (journal)).</li> </ul> <p>But it is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to a model structure that <em>is</em> left proper. This is discussed <a href="#ProperEquivModels">below</a>.</p> <h3 id="right_proper_model_categories">Right proper model categories</h3> <ul> <li> <p>by cor. <a class="maruku-ref" href="#AllObjectsFibrantImpliesRightProper"></a> every model category in which each object is fibrant is right proper.</p> <p>This includes for instance the standard <a class="existingWikiWord" href="/nlab/show/Quillen+model+structure+on+topological+spaces">Quillen model structure on topological spaces</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a> is (automatically left proper, since all objects are cofibrant, but also) right proper, even though not all objects are fibrant. See <a href="classical+model+structure+on+simplicial+sets#Properness">there</a>.</p> </li> </ul> <h3 id="nonright_proper_model_categories">Non-right proper model categories</h3> <ul> <li>“Non-algebraic” models for higher categories (other than higher groupoids) are generally not right proper. For example, the <a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">model structure for complete Segal spaces</a> and the <a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a> are not right proper (see <a href="http://mathoverflow.net/questions/40938/is-the-model-category-of-complete-segal-spaces-right-proper">mathoverflow</a>).</li> </ul> <h3 id="proper_model_categories">Proper model categories</h3> <p>Model categories which are both left and right proper include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Top">Top</a>: <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> (<a href="classical+model+structure+on+topological+spaces#Properness">prop.</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>: <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a> (<a href="classical+model+structure+on+simplicial+sets#Properness">here</a>)</p> </li> <li> <p>The <em>global</em> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> and any local such model structure over a site with <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a> and weak equivalences the <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise weak equivalences.</p> </li> <li id="ModelStructureOnChainComplexesIsProper"> <p>The standard <a class="existingWikiWord" href="/nlab/show/model+structures+on+chain+complexes">model structures on chain complexes</a> (see <a href="model+structure+on+chain+complexes#Properness">there</a>).</p> </li> <li> <p>The projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+algebras">model structure on differential graded-commutative algebras</a> (unbounded). See <a href="http://mathoverflow.net/q/204414/381">this MO discussion</a>.</p> </li> </ul> <h3 id="reedy_model_structures">Reedy model structures</h3> <p> <div class='num_prop' id='ReedyStructureInheritsProperness'> <h6>Proposition</h6> <p>For <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">\mathcal{R}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Reedy+category">Reedy category</a> 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">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> which is left or right <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper</a>, then also the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>ℛ</mi><mo>,</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(\mathcal{R}, \mathcal{C})</annotation></semantics></math> is left or right <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper</a>, respectively.</p> </div> This appears as <a href="#Hirschorn02">Hirschorn (2002), Thm. 15.3.4 (2)</a>, there attributed to <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>.</p> <h3 id="ProperEquivModels">Proper Quillen equivalent model structures</h3> <p>While some model categories fail to be proper, often there is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> one that does enjoy properness.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Every model category whose acyclic cofibrations are <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>s is Quillen equivalent to its <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">model structure on algebraic fibrant objects</a>. In this all objects are fibrant, so that it is right proper.</p> </div> <p>(<a href="#Nikolaus10">Nikolaus 10, theorem 2.18</a>)</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>Let <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> be a simplicial (possibly multi-colored) <a class="existingWikiWord" href="/nlab/show/theory">theory</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>Alg</mi></mrow><annotation encoding="application/x-tex">T Alg</annotation></semantics></math> be the corresponding category of simplicial T-algebras. This carries a model category structure where the fibrations and weak equivalences are those of the underlying simplicial sets in the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>.</p> <p>Then there exists a morphism of simplicial theories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">T \to S</annotation></semantics></math> such that</p> <ol> <li> <p>the induced <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>S</mi><mi>Alg</mi><mover><mo>←</mo><mo>→</mo></mover><mi>T</mi><mi>Alg</mi></mrow><annotation encoding="application/x-tex">S Alg \stackrel{\to}{\leftarrow} T Alg</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>Alg</mi></mrow><annotation encoding="application/x-tex">S Alg</annotation></semantics></math> is a proper <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>.</p> </li> </ol> </div> <p>This is the content of (<a href="#Rezk02">Rezk 02</a>)</p> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <p>The following says that left/right properness holds <em>locally</em> in every model category, namely between cofibrant/fibrant objects.</p> <div class="num_prop" id="GoodPullbacksAndPushouts"> <h6 id="proposition">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>,</p> <ol> <li> <p>every <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> between <a class="existingWikiWord" href="/nlab/show/cofibrant+objects">cofibrant objects</a> along a <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a> is again a weak equivalence;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> between <a class="existingWikiWord" href="/nlab/show/fibrant+objects">fibrant objects</a> along a <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a> is again a weak equivalence.</p> </li> </ol> </div> <p>A proof is spelled out in <a href="#Hirschhorn">Hirschhorn, prop. 13.1.2</a>, there attributed to <a href="#Reedy">Reedy</a>.</p> <p>This gives a large class of examples of left/right proper model categories:</p> <p> <div class='num_cor' id='AllObjectsFibrantImpliesRightProper'> <h6>Corollary</h6> <p></p> <ul> <li> <p>A model category in which all objects are cofibrant is left proper.</p> </li> <li> <p>A model category in which all objects are fibrant is right proper.</p> </li> </ul> <p></p> </div> </p> <p>See in the list of <a href="#Examples">Examples</a> below for concrete examples.</p> <p>Notice that the prop. <a class="maruku-ref" href="#GoodPullbacksAndPushouts"></a> applies only (in the right proper case, for concreteness) to pullbacks of fibrations along weak equivalences in which <em>all three</em> objects are fibrant, since a fibration with fibrant codomain also has fibrant domain. The definition of right proper, on the other hand, states this property in the case when <em>none</em> of the objects are assumed to be fibrant.</p> <p>One might consider as an “in-between” assumption the situation when only the common codomain of the fibration and the weak equivalence (hence also the domain of the fibration) are fibrant; but it turns out that this apparently-weaker assumption is sufficient to imply full right properness. This can be found, for instance, as Lemma 9.4 of <a href="#Bousfield01">Bousfield 2001</a>.</p> <div class="num_prop" id="FibrationsBetweenFibrantObjectsSuffice"> <h6 id="proposition_2">Proposition</h6> <p>Suppose that in some model category, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\to Y</annotation></semantics></math> is a fibration and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Z\to Y</annotation></semantics></math> a weak equivalence, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> (hence also <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>) fibrant, then the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X\times_Y Z \to X</annotation></semantics></math> is a weak equivalence. Then the model category is right proper, i.e. the same statement is true without the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is fibrant.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Suppose given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi><mo>←</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">X\to Y\leftarrow Z</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\to Y</annotation></semantics></math> is a fibration and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Z\to Y</annotation></semantics></math> a weak equivalence. Choose a fibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>R</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y\to R Y</annotation></semantics></math>, and factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi><mo>→</mo><mi>R</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\to Y \to R Y</annotation></semantics></math> as a weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>R</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">X\to R X</annotation></semantics></math> followed by a fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>X</mi><mo>→</mo><mi>R</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">R X \to R Y</annotation></semantics></math>. The assumption now applies to the cospan <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>X</mi><mo>→</mo><mi>R</mi><mi>Y</mi><mo>←</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">R X \to R Y \leftarrow Y</annotation></semantics></math>, so that the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>X</mi><msub><mo>×</mo> <mrow><mi>R</mi><mi>Y</mi></mrow></msub><mi>Y</mi><mo>→</mo><mi>R</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">R X \times_{R Y} Y \to R X</annotation></semantics></math> is a weak equivalence. By 2-out-of-3, the induced map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>R</mi><mi>X</mi><msub><mo>×</mo> <mrow><mi>R</mi><mi>Y</mi></mrow></msub><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to R X \times_{R Y} Y</annotation></semantics></math> is also a weak equivalence.</p> <p>Now by <a class="existingWikiWord" href="/nlab/show/Ken+Brown%27s+lemma">Ken Brown's lemma</a>, the pullback functor along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Z\to Y</annotation></semantics></math> preserves weak equivalences between fibrant objects, and in particular preserves this weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>R</mi><mi>X</mi><msub><mo>×</mo> <mrow><mi>R</mi><mi>Y</mi></mrow></msub><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to R X \times_{R Y} Y</annotation></semantics></math>. Thus, the induced map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>Z</mi><mo>→</mo><mi>R</mi><mi>X</mi><msub><mo>×</mo> <mrow><mi>R</mi><mi>Y</mi></mrow></msub><mi>Z</mi></mrow><annotation encoding="application/x-tex">X\times_Y Z \to R X \times_{R Y} Z</annotation></semantics></math> is a weak equivalence. However, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>X</mi><msub><mo>×</mo> <mrow><mi>R</mi><mi>Y</mi></mrow></msub><mi>Z</mi><mo>→</mo><mi>R</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">R X \times_{R Y} Z \to R X</annotation></semantics></math> is a weak equivalence by the assumption, so by 2-out-of-3, the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X\times_Y Z \to X</annotation></semantics></math> is also a weak equivalence, as desired.</p> </div> <h3 id="HomotopyLimits">Homotopy (co)limits in proper model categories</h3> <div class="num_prop" id="PullbackAlongFibrationsAreHomotopyPullbacks"> <h6 id="proposition_3">Proposition</h6> <p>In a left proper model category, ordinary <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a> along cofibrations are <a class="existingWikiWord" href="/nlab/show/homotopy+pushouts">homotopy pushouts</a>.</p> <p>Dually, in a right proper model category, ordinary <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> along fibrations are <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is stated for instance in <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop A.2.4.4</a> or in prop. 1.19 in <a href="http://www.math.harvard.edu/~clarkbar/complete.pdf">Bar</a>. We follow the proof given in this latter reference.</p> <p>We demonstrate the first statement, the second is its direct formal dual.</p> <p>So consider a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> 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>K</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo maxsize="1.2em" minsize="1.2em">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mi>L</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ K &\longrightarrow& Y \\ \big\downarrow^{\mathrlap{\in cof}} && \big\downarrow \\ L &\longrightarrow& X } </annotation></semantics></math></div> <p>in a left proper model category, where the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">K \to L</annotation></semantics></math> is a cofibration, as indicated. We need to exhibit a weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mover><mo>→</mo><mrow></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">X' \stackrel{}{\to} X</annotation></semantics></math> from an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">X'</annotation></semantics></math> that is manifestly a <a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>←</mo><mi>K</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">L \leftarrow K \to Y</annotation></semantics></math>.</p> <p>The standard procedure to produce this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">X'</annotation></semantics></math> is to pass to a weakly equivalent diagram with the property that all objects are cofibrant and one of the morphisms is a cofibration. The ordinary pushout of that diagram is well known to be the <a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a>, as described there.</p> <p>So pick a cofibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>↪</mo><mi>K</mi><mo>′</mo><mover><mo>→</mo><mo>≃</mo></mover></mrow><annotation encoding="application/x-tex">\emptyset \hookrightarrow K' \stackrel{\simeq}{\to}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>′</mo><mo>→</mo><mi>K</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">K' \to K \to Y</annotation></semantics></math> as a cofibration followed by a weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>′</mo><mo>↪</mo><mi>Y</mi><mo>′</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">K' \hookrightarrow Y' \stackrel{\simeq}{\to} Y</annotation></semantics></math> and similarly factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>′</mo><mo>→</mo><mi>K</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">K' \to K \to L</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>′</mo><mo>↪</mo><mi>L</mi><mo>′</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>L</mi></mrow><annotation encoding="application/x-tex">K' \hookrightarrow L' \stackrel{\simeq}{\to} L</annotation></semantics></math></p> <p>This yields a weak equivalence of diagrams</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>Y</mi></mtd> <mtd><mover><mo>←</mo><mo>≃</mo></mover></mtd> <mtd><mi>Y</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>K</mi></mtd> <mtd><mover><mo>←</mo><mo>≃</mo></mover></mtd> <mtd><mi>K</mi><mo>′</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>L</mi></mtd> <mtd><mover><mo>←</mo><mo>≃</mo></mover></mtd> <mtd><mi>L</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ Y &\stackrel{\simeq}{\leftarrow}& Y' \\ \uparrow && \uparrow^{\mathrlap{\in cof}} \\ K &\stackrel{\simeq}{\leftarrow}& K' \\ \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} \\ L &\stackrel{\simeq}{\leftarrow}& L' } \,, </annotation></semantics></math></div> <p>where now the diagram on the right is cofibrant as a diagram, so that its ordinary pushout</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mo>≔</mo><mi>L</mi><mo>′</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>K</mi><mo>′</mo></mrow></munder><mi>Y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex"> X' \coloneqq L' \coprod_{K'} Y' </annotation></semantics></math></div> <p>is a homotopy colimit of the original diagram. To obtain the weak equivalence from there to <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>, first form the further pushouts</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>K</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo>↖</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mo>≃</mo></mpadded></msub></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>K</mi><mo>′</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi><mo>′</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>L</mi><mo>′</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>′</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>L</mi><mo>″</mo><mo>≔</mo><mi>K</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>K</mi><mo>′</mo></mrow></munder><mi>L</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>L</mi><mo>″</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>K</mi></munder><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>L</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ K &&&\to&&& Y \\ & \nwarrow^{\mathrlap{\in W}} &&&& \nearrow_{\mathrlap{\simeq}} & \\ && K' &\to& Y' && \\ \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow \\ && L' &\to& X' && \\ & {}^{\mathllap{\in W}} \swarrow &&&& \searrow^{\mathrlap{\simeq}} & \\ L'' \coloneqq K \coprod_{K'} L &&&\to&&& L'' \coprod_{K} Y \\ \downarrow^{\mathrlap{\in W}} &&&&&& \downarrow \\ L &&&\to&&& X } \,, </annotation></semantics></math></div> <p>where the total outer diagram is the original pushout diagram. Here the cofibrations are as indicated by the above factorization and by their stability under pushouts, and the weak equivalences are as indicated by the above factorization and by the left properness of the model category. The weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>″</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>L</mi></mrow><annotation encoding="application/x-tex">L'' \stackrel{\simeq}{\to} L</annotation></semantics></math> is by the <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">2-out-of-3 property</a>.</p> <p>This establishes in particular a weak equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>L</mi><mo>″</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>K</mi></munder><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X' \stackrel{\simeq}{\to} L'' \coprod_K Y \,. </annotation></semantics></math></div> <p>It remains to get a weak equivalence further to <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>. For that, take the two outer squares from the above</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>K</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>L</mi><mo>″</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>L</mi><mo>″</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>K</mi><mo>′</mo></mrow></munder><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>L</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ K &\to& Y \\ \downarrow^{\mathrlap{\in cof}} && \downarrow \\ L'' &\to& L'' \coprod_{K'} Y \\ \downarrow^{\mathrlap{\in W}} && \downarrow \\ L &\to& X } \,. </annotation></semantics></math></div> <p>Notice that the top square is a pushout by construction, and the total one by assumption. Therefore by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>, also the lower square is a pushout.</p> <p>Then factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">K \to Y</annotation></semantics></math> as a cofibration followed by a weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>↪</mo><mi>Z</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">K \hookrightarrow Z \stackrel{\simeq}{\to} Y</annotation></semantics></math> and push that factorization through the double diagram, to obtain</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>K</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo>∈</mo><mi>cof</mi></mrow></mover></mtd> <mtd><mi>Z</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mo lspace="0em" rspace="thinmathspace">cof</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>L</mi><mo>″</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>∈</mo><mi>cof</mi></mrow></mover></mtd> <mtd><mi>L</mi><mo>″</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>K</mi></munder><mi>Z</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><mi>L</mi><mo>″</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>K</mi><mo>′</mo></mrow></munder><mi>Y</mi></mtd></mtr> <mtr><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>W</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>L</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>L</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>K</mi></munder><mi>Z</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ K &\stackrel{\in cof}{\to}& Z &\stackrel{\in W}{\to}& Y \\ \downarrow^{\mathrlap{\in \cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow \\ L'' &\stackrel{\in cof}{\to}& L'' \coprod_{K} Z &\stackrel{\in W}{\to}& L'' \coprod_{K'} Y \\ \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} && \downarrow \\ L & \to& L \coprod_K Z &\stackrel{\in W}{\to}& X } \,. </annotation></semantics></math></div> <p>Again by the behaviour of pushouts under pasting, every single square and composite rectangle in this diagram is a pushout. Using this, the cofibration and weak equivalence properties from before push through the diagram as indicated. This finally yields the desired weak equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>″</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>K</mi><mo>′</mo></mrow></munder><mi>Y</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> L'' \coprod_{K'} Y \stackrel{\simeq}{\to} X </annotation></semantics></math></div> <p>by <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">2-out-of-3</a>.</p> </div> <p>If we had allowed ourselved to assume in addition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> itself is already cofibrant, then the above statement has a much simpler proof, which we list just for fun, too.</p> <div class="proof"> <h6 id="proof_of_prop__assuming_that_the_domain_of_the_cofibration_is_cofibrant">Proof of prop. <a class="maruku-ref" href="#PullbackAlongFibrationsAreHomotopyPullbacks"></a> assuming that the domain of the cofibration is cofibrant</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow B</annotation></semantics></math> be a cofibration with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> cofibrant and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \to C</annotation></semantics></math> be any other morphism. Factor this morphism as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>C</mi><mo>′</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>C</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow C' \stackrel{\simeq}{\to} C</annotation></semantics></math> by a cofibration followed by an acyclic fibration. This give a weak equivalence of pushout diagrams</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>C</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>C</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C' &\stackrel{\simeq}{\to}& C \\ \uparrow && \uparrow \\ A &\stackrel{=}{\to}& A \\ \downarrow && \downarrow \\ B &\stackrel{=}{\to}& B } \,. </annotation></semantics></math></div> <p>In the diagram on the left all objects are cofibrant and one morphism is a cofibration, hence this is a cofibrant diagram and its ordinary colimit is the homotopy colimit. Using that <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> diagrams compose to pushout diagrams, that cofibrations are preserved under pushout and that in a left proper model category weak equivalences are preserved under pushout along cofibrations, we find a weak equiovalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hocolim</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>B</mi><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>A</mi></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">hocolim \stackrel{\simeq}{\to} B \coprod_A C</annotation></semantics></math></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>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo>∈</mo><mi>cof</mi></mrow></mover></mtd> <mtd><mi>C</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi><mo>∩</mo><mi>fib</mi></mrow></mover></mtd> <mtd><mi>C</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>cof</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>hocolim</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover></mtd> <mtd><mi>B</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>A</mi></munder><mi>C</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{\in cof}{\to}& C' &\stackrel{\in W \cap fib}{\to}& C \\ \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} \\ B &\to& hocolim &\stackrel{\in W}{\to}& B \coprod_A C } \,. </annotation></semantics></math></div> <p>The proof for the second statement is the precise formal dual.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>A model category is <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper</a> if and only if every fibration is a <a class="existingWikiWord" href="/nlab/show/sharp+map">sharp map</a>.</p> </div> <p>(<a href="#Rezk98">Rezk 98</a>)</p> <h3 id="SliceCategories">Slice categories</h3> <p>For any model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, and any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f \colon A\to B</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> of <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a> along <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>f</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>M</mi><mo stretchy="false">/</mo><mi>A</mi><mo>⇄</mo><mi>M</mi><mo stretchy="false">/</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> \Sigma_f \;\colon\; M/A \rightleftarrows M/B \;\colon\; f^* </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> between <a class="existingWikiWord" href="/nlab/show/slice+model+categories">slice model categories</a> (<a href="slice+model+structure#LeftBaseChangeQuillenAdjunction">this Prop.</a>).</p> <p>If this adjunction is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>, then <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> must be a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>. In general, the converse can be proven only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> are fibrant.</p> <p> <div class='num_prop' id='ViaSliceCategories'> <h6>Proposition</h6> <p>The following are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is right proper.</p> </li> <li> <p>If <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> is any weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>f</mi></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\Sigma_f \dashv f^*</annotation></semantics></math> is a Quillen equivalence.</p> </li> </ol> <p></p> </div> </p> <p>This is due to <a href="#Rezk02">Rezk 02, Prop. 2.5</a>.</p> <p> <div class='num_remark'> <h6>Remark</h6> <p>The statement of Prop. <a class="maruku-ref" href="#ViaSliceCategories"></a> may be read as saying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is right proper iff all <a class="existingWikiWord" href="/nlab/show/slice+model+categories">slice model categories</a> have the “correct” Quillen equivalence type.</p> <p>Since whether or not a Quillen adjunction is a Quillen equivalence depends only on the classes of weak equivalences, not the fibrations and cofibrations, it follows that being right proper is really a property of a <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>. In particular, if one model structure is right proper, then so is any other model structure on the same category with the same weak equivalences.</p> </div> </p> <h3 id="local_cartesian_closure">Local cartesian closure</h3> <p>Since most well-behaved model categories are equivalent to a model category in which all objects are fibrant — namely, the model category of <a class="existingWikiWord" href="/nlab/show/algebraically+fibrant+objects">algebraically fibrant objects</a> — they are in particular equivalent to one which is right proper. Thus, right properness by itself is not a property of 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>-category, only of a particular presentation of it via a model category.</p> <p>However, if a <a class="existingWikiWord" href="/nlab/show/Cisinski+model+category">Cisinski model category</a> is right proper, then the <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 which it presents must be <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+%28%E2%88%9E%2C1%29-category">locally cartesian closed</a>. Conversely, any locally cartesian closed (∞,1)-category has a presentation by a right proper Cisinski model category; see <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+%28%E2%88%9E%2C1%29-category">locally cartesian closed (∞,1)-category</a> for the proof.</p> <h2 id="related_pages">Related pages</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/semi-left-exact+left+Bousfield+localization">semi-left-exact left Bousfield localization</a></li> </ul> <h2 id="references">References</h2> <p>The concept originates in</p> <ul> <li id="BousfieldFriedlander78"><a class="existingWikiWord" href="/nlab/show/Aldridge+Bousfield">Aldridge Bousfield</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Friedlander">Eric Friedlander</a>, def. 1.1.6 in <em>Homotopy theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-spaces, spectra, and bisimplicial sets</em>, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (<a href="https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf">pdf</a>)</li> </ul> <p>Textbook account:</p> <ul> <li id="Hirschhorn02"><a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, Chapter 13 of: <em><a class="existingWikiWord" href="/nlab/show/Model+Categories+and+Their+Localizations">Model Categories and Their Localizations</a></em>, AMS Math. Survey and Monographs Vol 99 (2002) [<a href="https://bookstore.ams.org/surv-99-s/">ISBN:978-0-8218-4917-0</a>, <a href="http://www.gbv.de/dms/goettingen/360115845.pdf">pdf toc</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf">pdf</a>]</li> </ul> <p>The usefulness of right properness for constructions of <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy categories</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>, <em>Cocycle categories</em> (<a href="http://arxiv.org/PS_cache/math/pdf/0605/0605198v1.pdf">pdf</a>)</li> </ul> <p>The general theory can be found in</p> <ul> <li id="Hirschhorn"><a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, chapter 13 of <em>Model Categories and Their Localizations</em>, 2003 (<a href="http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99">AMS</a>, <a href="http://www.gbv.de/dms/goettingen/360115845.pdf">pdf toc</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf">pdf</a>)</li> </ul> <p>also in</p> <ul> <li id="Reedy"><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>Proposition <a class="maruku-ref" href="#FibrationsBetweenFibrantObjectsSuffice"></a> can be found in</p> <ul> <li id="Bousfield01"><a class="existingWikiWord" href="/nlab/show/Aldridge+Bousfield">Aldridge Bousfield</a>, <em>On the telescopic homotopy theory of spaces</em>, Trans. Amer. Math. Soc. 353 (2001), 2391-2426, <a href="https://www.ams.org/journals/tran/2001-353-06/S0002-9947-00-02649-0/">web with fulltext</a></li> </ul> <p>See also:</p> <ul> <li id="Rezk98"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Fibrations and homotopy colimits of simplicial sheaves</em> (<a href="http://arxiv.org/abs/math/9811038">arXiv:9811038</a>)</p> </li> <li id="Rezk02"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Every homotopy theory of simplicial algebras admits a proper model</em>, Topology and its Applications, Volume 119, Issue 1, 2002, Pages 65-94, (<a href="https://doi.org/10.1016/S0166-8641(01)00057-8">doi:10.1016/S0166-8641(01)00057-8</a>, <a href="http://arxiv.org/abs/math/0003065">arXiv:math/0003065</a>)</p> </li> <li id="Nikolaus10"> <p><a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <em>Algebraic models for higher categories</em> (<a href="http://arxiv.org/abs/1003.1342">arXiv:1003.1342</a>)</p> </li> </ul> <p>Examples of nonproper model structures can be found in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Henry">Simon Henry</a>, <em>Counter-example to the existence of left Bousfield localization of combinatorial model category</em> <a href="https://mathoverflow.net/questions/325383/counter-example-to-the-existence-of-left-bousfield-localization-of-combinatorial">MathOverflow</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Henry">Simon Henry</a>, <em>Examples of non-proper model structure</em>,</p> <p><a href="https://mathoverflow.net/questions/339084/examples-of-non-proper-model-structure">MathOverflow</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 4, 2024 at 13:46:30. 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