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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> Cisinski model structure </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/1446/#Item_28" 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='#general'>General</a></li> <li><a href='#OnPresheafToposes'>On presheaf toposes</a></li> <ul> <li><a href='#PreliminaryNotions'>Preliminary notions</a></li> <li><a href='#the_model_structure'>The model structure</a></li> <li><a href='#proof_9'>Proof</a></li> <ul> <li><a href='#Lifting'>Lifting</a></li> <li><a href='#Factorization'>Factorization</a></li> <ul> <li><a href='#CofibrationFollowedByAcyclicFibration'>Cofibration followed by acyclic fibration</a></li> <li><a href='#AcyclicCofibrationFollowedByFibration'>Acyclic cofibration followed by fibration</a></li> </ul> <li><a href='#Completeness'>Completeness</a></li> </ul> <li><a href='#ALocalizers'>Localizers</a></li> <li><a href='#SimplicialCompletion'>Simplicial completion</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#properness'>Properness</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">Pursuing Stacks</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a> introduced the idea of a <a class="existingWikiWord" href="/nlab/show/test+category">test category</a>. These are by definition small categories on which the presheaves of sets are models for homotopy types of CW-complexes, thus generalising the situation for the category of simplices, for which the category of presheaves is that of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>. The resulting theory was completed and generalised by Cisinski: in order to prove, following Grothendieck’s prediction, that presheaves on a test category form a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, he constructed all possible cofibrantly generated model structures on a given topos for which the cofibrations are the monomorphisms.</p> <h2 id="general">General</h2> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>For <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">\mathcal{T}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, a <strong>Cisinski model structure</strong> on <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">\mathcal{T}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <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">\mathcal{T}</annotation></semantics></math> such that</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> are precisely the <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>;</p> </li> <li> <p>it is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a>.</p> </li> </ol> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Since every <a class="existingWikiWord" href="/nlab/show/topos">topos</a> is a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a>, a Cisinski model structure is in particular a <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a> structure.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Since every <a class="existingWikiWord" href="/nlab/show/topos">topos</a> is an <a class="existingWikiWord" href="/nlab/show/adhesive+category">adhesive category</a>, monomorphisms are automatically preserved by <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>.</p> </div> <div class="num_defn" id="Localizer"> <h6 id="definition_2">Definition</h6> <p>Say a class <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>𝒯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \subset Mor(\mathcal{T})</annotation></semantics></math> is an accessible <strong><a class="existingWikiWord" href="/nlab/show/localizer">localizer</a></strong> on <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">\mathcal{T}</annotation></semantics></math> if it is a class of <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> in a Cisinski model structure on <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">\mathcal{T}</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma \subset Mor(\mathcal{T})</annotation></semantics></math> is contained in a smallest localizer, def. <a class="maruku-ref" href="#Localizer"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\Sigma)</annotation></semantics></math>.</p> </div> <p>One says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\Sigma)</annotation></semantics></math> is the localiser <em>generated</em> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>.</p> <p>So in particular a Cisinski model structure always exists.</p> <h2 id="OnPresheafToposes">On presheaf toposes</h2> <p>We discuss how on a <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> equipped with a suitable notion of <a class="existingWikiWord" href="/nlab/show/cylinder+objects">cylinder objects</a> Cisinski model structures can be characterized fairly explicitly. After some <a href="#PreliminaryNotions">preliminaries</a><em>, the main statement is theorem <a class="maruku-ref" href="#ModelStructureFromHomotopicalStructure"></a> below.</em></p> <p>(This follows sections 1.2 and 1.3 of <a href="#Cisinski06">Cisinski 06</a>).</p> <h3 id="PreliminaryNotions">Preliminary notions</h3> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A^{op}, Set]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> over <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>. We introduce here, culminating in def. <a class="maruku-ref" href="#HomotopicalStructure"></a> below, the ingredients of a <em>homotopical structure</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>, which is a choice of functorial <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> together with a compatible notion of <em><a class="existingWikiWord" href="/nlab/show/anodyne+extensions">anodyne extensions</a></em>. Further below in def. <a class="maruku-ref" href="#ModelStructureMorphismsFromHomotopicalStructure"></a> this defines a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>.</p> <div class="num_defn" id="FunctorialCylinder"> <h6 id="definition_3">Definition</h6> <p>For <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>, a <strong>cylinder</strong> on <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> in the following means a <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊗</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">I \otimes X</annotation></semantics></math>, factoring the <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</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 lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>X</mi><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mi>X</mi> <mn>0</mn></msubsup><mo>,</mo><msubsup><mo>∂</mo> <mi>X</mi> <mn>1</mn></msubsup><mo stretchy="false">)</mo></mrow></mover><mi>I</mi><mo>⊗</mo><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>σ</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \coprod X \stackrel{(\partial_X^0, \partial_X^1)}{\to} I \otimes X \stackrel{\sigma_X}{\to} X </annotation></semantics></math></div> <p>such that the first morphism is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of such cyclinders is a pair of morphisms <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>I</mi><mo>⊗</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">I \otimes X \to I \otimes Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>, making the evident squares commute. This defines a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(A)</annotation></semantics></math> of cylinder objects on presheaves on <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>, equipped with a <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(A) \to PSh(A)</annotation></semantics></math> that sends a cylinder <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊗</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">I \otimes X</annotation></semantics></math> to its underlying 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>.</p> <p>A <strong>functorial cylinder object</strong> over <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> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of this functor.</p> </div> <p>This is (<a href="#Cisinski06">Cisinski 06, def. 1.3.1</a>).</p> <p>In the following, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J : PSh(A) \to Cyl(A)</annotation></semantics></math> be a choice of functorial cylinder object. Equivalently, this is a choice of endofunctor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I \otimes (-) : PSh(A) \to PSh(A) </annotation></semantics></math></div> <p>equipped with <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>∂</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mn>0</mn></msubsup><mo>,</mo><msubsup><mo>∂</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow> <mn>1</mn></msubsup><mo stretchy="false">)</mo></mrow></mover><mi>I</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>→</mo><mi>Id</mi></mrow><annotation encoding="application/x-tex"> (\partial I)\otimes (-) \stackrel{(\partial^0_{(-)}, \partial^1_{(-)})}{\to} I \otimes (-) \to Id </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>∂</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>X</mi><mo>:</mo><mo>=</mo><mi>X</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(\partial I) \otimes X := X \coprod X</annotation></semantics></math>, such that the composite is the functorial codiagonal, and where the first transformation is a monomorphism.</p> <div class="num_defn" id="ElementaryJHomotopy"> <h6 id="definition_4">Definition</h6> <p>For <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>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g : X \to Y</annotation></semantics></math> two morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>, we say an <strong>elementary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy</strong> <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 \Rightarrow g</annotation></semantics></math> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> with respect to the chosen cylinder object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>, hence a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mi>I</mi><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\eta : I \otimes X \to Y</annotation></semantics></math> fitting into 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>X</mi></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mi>X</mi> <mn>0</mn></msubsup></mrow></mpadded></msup></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mi>X</mi> <mn>1</mn></msubsup></mrow></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mi>g</mi></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X && \\ \downarrow^{\mathrlap{\partial^0_X}} & \searrow^f \\ I \otimes X &\stackrel{\eta}{\to}& Y \\ \uparrow^{\mathrlap{\partial^1_X}} & \nearrow_g \\ X } \,. </annotation></semantics></math></div> <p>We say <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy</strong> for the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> <em>generated</em> by this.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.3.3</a>).</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy is compatible with composition in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>It is sufficient to show that elementary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopies are compatible with composition.</p> <p>So for</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>X</mi></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mi>g</mi></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X && \\ \downarrow & \searrow^f \\ I \otimes X &\stackrel{\eta}{\to}& Y \\ \uparrow & \nearrow_g \\ X } </annotation></semantics></math></div> <p>an elementary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <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 \Rightarrow g</annotation></semantics></math>, and for</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></mtd> <mtd></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mrow><mi>f</mi><mo>′</mo></mrow></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mi>η</mi></mover></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>g</mi><mo>′</mo></mrow></msub></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y && \\ \downarrow & \searrow^{f'} \\ I \otimes Y &\stackrel{\eta}{\to}& Z \\ \uparrow & \nearrow_{g'} \\ Y } </annotation></semantics></math></div> <p>one <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>⇒</mo><mi>g</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f' \Rightarrow g'</annotation></semantics></math>, we obtain an elementary homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>∘</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo>′</mo><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">f' \circ f \Rightarrow g' \circ f</annotation></semantics></math> by forming</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>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mrow><mi>f</mi><mo>′</mo></mrow></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>I</mi><mo>⊗</mo><mi>f</mi></mrow></mover></mtd> <mtd><mi>I</mi><mo>⊗</mo><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mi>η</mi></mover></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>g</mi><mo>′</mo></mrow></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X & \stackrel{f}{\to} & Y && \\ \downarrow && \downarrow & \searrow^{f'} \\ I \otimes X &\stackrel{I \otimes f}{\to}& I \otimes Y &\stackrel{\eta}{\to}& Z \\ \uparrow && \uparrow & \nearrow_{g'} \\ X &\stackrel{f}{\to}& Y } </annotation></semantics></math></div> <p>and then an elementary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>′</mo><mo>∘</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo>′</mo><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">g' \circ f \Rightarrow g '\circ g</annotation></semantics></math> by forming</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>X</mi></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>g</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mi>g</mi></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X && \\ \downarrow & \searrow^f \\ I \otimes X &\stackrel{\eta}{\to}& Y &\stackrel{g'}{\to}& Z \\ \uparrow & \nearrow_g \\ X } \,. </annotation></semantics></math></div> <p>Together this generates a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>∘</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo>′</mo><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f' \circ f \Rightarrow g' \circ g</annotation></semantics></math>.</p> </div> <p>Hence the following is well defined.</p> <div class="num_defn" id="JHomotopyCategory"> <h6 id="definition_5">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho_J(A)</annotation></semantics></math> for the category whose objects are those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>, and whose morphisms are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy equivalence classes of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> – the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy category</strong>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Q : PSh(A) \to Ho_J(A) </annotation></semantics></math></div> <p>for the projection functor.</p> </div> <div class="num_defn" id="JHomotopyEquivalence"> <h6 id="definition_6">Definition</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> is called a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy equivalence</strong> if it is sent by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> to an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p>An object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in PSh(A)</annotation></semantics></math> is called <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-contractible</strong> if <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> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy equivalence.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.3.4</a>).</p> <div class="num_defn" id="AcyclicFibration"> <h6 id="definition_7">Definition</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> is called an <strong>acyclic fibration</strong> if it has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against all monomorphisms.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.2.18</a>).</p> <div class="num_prop" id="SomePropertiesOfAcyclicFibrations"> <h6 id="proposition_3">Proposition</h6> <p>Every acyclic fibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy equivalence.</p> <p>More is true: every trivial fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math></p> <ol> <li> <p>has a <a class="existingWikiWord" href="/nlab/show/section">section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s : Y \to X</annotation></semantics></math>;</p> </li> <li> <p>which is also a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy left inverse;</p> </li> <li> <p>by an elementary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>id</mi><mo>⇒</mo><mi>s</mi><mo>∘</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">h : id \Rightarrow s \circ p</annotation></semantics></math> which satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>h</mi><mo>=</mo><mi>p</mi><mo>∘</mo><msub><mi>σ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">p \circ h = p \circ \sigma_X</annotation></semantics></math>.</p> </li> </ol> </div> <p>(<a href="#Cisinski06">Cisinski 06, lemma 1.3.5</a>).</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The existence of the section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s : Y \to X</annotation></semantics></math> follows by right lifting against the monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\emptyset \to Y</annotation></semantics></math> (out of the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>)</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>∅</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mi>s</mi></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \emptyset &\to& X \\ \downarrow &\nearrow_s& \downarrow^{\mathrlap{p}} \\ Y &\stackrel{=}{\to}& Y } \,. </annotation></semantics></math></div> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is obtained by lifting in the 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>X</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><mi>s</mi><mo>∘</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mi>X</mi> <mn>0</mn></msubsup><mo>,</mo><msubsup><mo>∂</mo> <mi>X</mi> <mn>1</mn></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo>∘</mo><msub><mi>σ</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \coprod X &\stackrel{(id_X, s \circ p)}{\to}& X \\ {}^{\mathllap{(\partial^0_X, \partial^1_X)}}\downarrow && \downarrow^{\mathrlap{p}} \\ I \otimes X &\stackrel{p \circ \sigma_X}{\to}& Y } \,. </annotation></semantics></math></div></div> <div class="num_defn" id="ElementaryHomotopicalDatum"> <h6 id="definition_8">Definition</h6> <p>An <strong>elementary homotopical datum</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> is a functorial cylinder <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>, def. <a class="maruku-ref" href="#FunctorialCylinder"></a>, such that</p> <ol> <li> <p>the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I \otimes (-)</annotation></semantics></math> commutes with small <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> and preserves monomorphisms;</p> </li> <li> <p>for all <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">f : K \to L</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a></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 stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>L</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>K</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>I</mi><mo>⊗</mo><mi>f</mi></mrow></mover></mtd> <mtd><mi>I</mi><mo>⊗</mo><mi>L</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ K &\stackrel{(id, f)}{\to}& L \\ {}^{\mathllap{(e,id)}}\downarrow && \downarrow^{\mathrlap{(e,id)}} \\ I \otimes K &\stackrel{I \otimes f}{\to}& I \otimes L } </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">e \in \{0,1\}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> square.</p> </li> </ol> </div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.3.6</a>).</p> <div class="num_remark" id="PullbackOfMonosAndUnions"> <h6 id="remark_3">Remark</h6> <p>For</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>R</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>T</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>U</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ R &\to & S \\ \downarrow && \downarrow \\ T &\to& U } </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> square in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> of two monomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>↪</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">S \hookrightarrow U</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>↪</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">T \hookrightarrow U</annotation></semantics></math>, the universal morphism out of the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>R</mi></munder><mi>S</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex"> T \coprod_R S \to U </annotation></semantics></math></div> <p>is also a monomorphism, usually written as the morphism out of the <em><a class="existingWikiWord" href="/nlab/show/union">union</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∪</mo><mi>S</mi><mo>↪</mo><mi>U</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T \cup S \hookrightarrow U \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>All this follows, for instance, from the corresponding statements in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, over each object of <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>.</p> </div> <div class="num_defn"> <h6 id="definition_9">Definition</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mi>ϵ</mi></msup><mo>:</mo><mo stretchy="false">{</mo><mi>ϵ</mi><mo stretchy="false">}</mo><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>I</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \partial^\epsilon : \{\epsilon\}\otimes(-) \hookrightarrow I \otimes(-) \,, </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\epsilon = 0,1</annotation></semantics></math>, be the <a class="existingWikiWord" href="/nlab/show/subfunctor">subfunctor</a> which is the <a class="existingWikiWord" href="/nlab/show/image">image</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mi>ϵ</mi></msup><mo>:</mo><msub><mi>Id</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo>→</mo><mi>I</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial^\epsilon : Id_{PSh(A)} \to I \otimes (-)</annotation></semantics></math>.</p> <p>This way for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in PSh(A)</annotation></semantics></math> the boundary inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∂</mo> <mi>X</mi> <mi>ϵ</mi></msubsup><mo>:</mo><mi>X</mi><mo>→</mo><mi>I</mi><mo>⊗</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\partial^\epsilon_X : X \to I \otimes X</annotation></semantics></math> are identified with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mi>ϵ</mi></msup><mo>⊗</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>:</mo><mo stretchy="false">{</mo><mi>ϵ</mi><mo stretchy="false">}</mo><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>I</mi><mo>⊗</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial^\epsilon \otimes id_X : \{\epsilon\} \otimes X \to I \otimes X \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>The second condition on an <em>elementary homotopical datum</em>, def. <a class="maruku-ref" href="#ElementaryHomotopicalDatum"></a> implies that the canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>⊗</mo><mi>K</mi><mo>∪</mo><mo stretchy="false">(</mo><mo>∂</mo><mi>J</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>L</mi><mo>→</mo><mi>J</mi><mo>⊗</mo><mi>L</mi></mrow><annotation encoding="application/x-tex"> J \otimes K \cup (\partial J) \otimes L \to J \otimes L </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, remark 1.3.7</a>).</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>The condition implies that</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><mo stretchy="false">{</mo><mi>ϵ</mi><mo stretchy="false">}</mo><mo>⊗</mo><mi>K</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>id</mi> <mo stretchy="false">{</mo></msub><mi>ϵ</mi><mo stretchy="false">}</mo><mo>⊗</mo><mi>j</mi></mrow></mover></mtd> <mtd><mo stretchy="false">{</mo><mi>ϵ</mi><mo stretchy="false">}</mo><mo>⊗</mo><mi>L</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mo>∂</mo> <mi>ϵ</mi></msup><mo>⊗</mo><msub><mi>id</mi> <mi>K</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mo>∂</mo> <mi>ϵ</mi></msup><mo>⊗</mo><msub><mi>id</mi> <mi>L</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>K</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>I</mi></msub><mo stretchy="false">)</mo><mo>⊗</mo><mi>j</mi></mrow></mover></mtd> <mtd><mi>I</mi><mo>⊗</mo><mi>L</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \{\epsilon\} \otimes K &\stackrel{id_\{\epsilon\} \otimes j}{\to}& \{\epsilon\} \otimes L \\ {}^{\mathllap{\partial^\epsilon \otimes id_K}}\downarrow && \downarrow^{\mathrlap{\partial^\epsilon \otimes id_L}} \\ I \otimes K &\stackrel{(id_I) \otimes j}{\to}& I \otimes L } </annotation></semantics></math></div> <p>is a pullback, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\epsilon = 0,1</annotation></semantics></math>, hence that</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><mo>∂</mo><mi>I</mi><mo>⊗</mo><mi>K</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>id</mi> <mrow><mo>∂</mo><mi>I</mi></mrow></msub><mo>⊗</mo><mi>j</mi></mrow></mover></mtd> <mtd><mo>∂</mo><mi>I</mi><mo>⊗</mo><mi>L</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>i</mi><mo>⊗</mo><msub><mi>id</mi> <mi>K</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>i</mi><mo>⊗</mo><msub><mi>id</mi> <mi>L</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>K</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>I</mi></msub><mo stretchy="false">)</mo><mo>⊗</mo><mi>j</mi></mrow></mover></mtd> <mtd><mi>I</mi><mo>⊗</mo><mi>L</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \partial I \otimes K &\stackrel{id_{\partial I} \otimes j}{\to}& \partial I \otimes L \\ {}^{\mathllap{i \otimes id_K}}\downarrow && \downarrow^{\mathrlap{i \otimes id_L}} \\ I \otimes K &\stackrel{(id_I) \otimes j}{\to}& I \otimes L } </annotation></semantics></math></div> <p>is a pullback. So the statement follows with remark <a class="maruku-ref" href="#PullbackOfMonosAndUnions"></a>.</p> </div> <div class="num_example" id="SegmentObject"> <h6 id="example">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I \in PSh(A)</annotation></semantics></math> be any object equipped with two points (global sections) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>0</mn></msup><mo>,</mo><msup><mo>∂</mo> <mn>1</mn></msup><mo>:</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\partial^0, \partial^1 : * \to I</annotation></semantics></math> which are disjoint in that</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>∅</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mo>∂</mo> <mn>0</mn></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mrow><msup><mo>∂</mo> <mn>1</mn></msup></mrow></mover></mtd> <mtd><mi>I</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \emptyset &\to& * \\ \downarrow && \downarrow^{\mathrlap{\partial^0}} \\ * &\stackrel{\partial^1}{\to}& I } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> square (here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi></mrow><annotation encoding="application/x-tex">\emptyset</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>). This induces a functorial cylinder by the assignment</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><mi>I</mi><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \mapsto I \times X \,, </annotation></semantics></math></div> <p>where on the right we have the cartesian product.</p> <p>This defines an <em>elementary homotopical datum</em> in the sense of def. <a class="maruku-ref" href="#ElementaryHomotopicalDatum"></a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, example 1.3.8</a>).</p> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>The disjointness of the two points ensures that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msup><mo>∂</mo> <mn>0</mn></msup><mo>,</mo><msup><mo>∂</mo> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></mover><mi>I</mi></mrow><annotation encoding="application/x-tex">* \coprod * \stackrel{(\partial^0, \partial^1)}{\to} I</annotation></semantics></math> is a monomorphism.</p> <p>The interval commutes with colimits as these and the product are computed objectwise, and products in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> commute with colimits. More abstractly: by the <a class="existingWikiWord" href="/nlab/show/Giraud+theorem">Giraud theorem</a> valid in the <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> we have “<a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a>”: they are preserved by pullback, and in particular by cartesian product. Therefore the first clause of def. <a class="maruku-ref" href="#ElementaryHomotopicalDatum"></a> is satisfied.</p> <p>Similarly, the second axiom of def. <a class="maruku-ref" href="#ElementaryHomotopicalDatum"></a> holds because <a class="existingWikiWord" href="/nlab/show/limits">limits</a> commute over each other.</p> </div> <div class="num_example" id="LawvereCylinder"> <h6 id="example_2">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Lawvere+cylinder">Lawvere cylinder</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>:</mo><mo>=</mo><mi>Ω</mi></mrow><annotation encoding="application/x-tex">I := \Omega</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a> in the <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>. This is the presheaf which to an object <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> of <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> assigns the set of <a class="existingWikiWord" href="/nlab/show/subobjects">subobjects</a> of the <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a> given by <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> (the <a class="existingWikiWord" href="/nlab/show/sieves">sieves</a> on <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>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo>:</mo><mi>U</mi><mo>↦</mo><mi>Sub</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega : U \mapsto Sub(U) </annotation></semantics></math></div> <p>and which to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f : U_1 \to U_2</annotation></semantics></math> assigns the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>Sub</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Sub</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^* : Sub(U_2) \to Sub(U_1)</annotation></semantics></math>.</p> <p>Let then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>0</mn></msup><mo>,</mo><msup><mo>∂</mo> <mn>1</mn></msup><mo>:</mo><mo>=</mo><mo>⊤</mo><mo>,</mo><mo>⊥</mo><mo>:</mo><mo>*</mo><mo>→</mo><mi>Ω</mi></mrow><annotation encoding="application/x-tex"> \partial^0, \partial^1 := \top, \bot: * \to \Omega </annotation></semantics></math></div> <p>be the morphisms that classify <a class="existingWikiWord" href="/nlab/show/top">top</a> and <a class="existingWikiWord" href="/nlab/show/bottom">bottom</a>, respectively, the terminal and the initial subobject of the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>.</p> <p>This is a segment object in the sense of example <a class="maruku-ref" href="#SegmentObject"></a> (the “<a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Lawvere</a>-segment”).</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, example 1.3.9</a>).</p> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>That the two points are separated, in that</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>∅</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>⊤</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mo>⊥</mo></mover></mtd> <mtd><mi>Ω</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \emptyset &\to& * \\ \downarrow && \downarrow^{\mathrlap{\top}} \\ * &\stackrel{\bot}{\to}& \Omega } </annotation></semantics></math></div> <p>is a pullback, is the defining property of the <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>.</p> </div> <div class="num_defn" id="AnodyneExtensions"> <h6 id="definition_10">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> an elementary homotopical datum on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>, a <strong><a class="existingWikiWord" href="/nlab/show/class">class</a> of <a class="existingWikiWord" href="/nlab/show/anodyne+extensions">anodyne extensions</a></strong> is a class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AnExt</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">AnExt \subset Mor(PSh(A))</annotation></semantics></math> such that</p> <ol> <li> <p>there exists a <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> of monomorphisms <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> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>AnExt</mi><mo>=</mo><mi>LLP</mi><mo stretchy="false">(</mo><mi>RLP</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> AnExt = LLP(RLP(S)) </annotation></semantics></math></div></li> <li> <p>for <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 \hookrightarrow L</annotation></semantics></math> a monomorphism, the <a class="existingWikiWord" href="/nlab/show/pushout-product+axiom">pushout product</a> morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>K</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>e</mi><mo stretchy="false">}</mo><mo>⊗</mo><mi>L</mi><mo stretchy="false">)</mo><mo>→</mo><mi>I</mi><mo>⊗</mo><mi>L</mi></mrow><annotation encoding="application/x-tex"> (I \otimes K) \cup (\{e\} \otimes L) \to I \otimes L </annotation></semantics></math></div> <p>(by <a class="existingWikiWord" href="/nlab/show/Joyal-Tierney+calculus">Joyal-Tierney calculus</a> to be thought of as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><mo>→</mo><mi>L</mi><mo stretchy="false">)</mo><mover><mo>⊗</mo><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>e</mi><mo stretchy="false">}</mo><mo>→</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K \to L) \bar \otimes (\{e\} \to I)</annotation></semantics></math>”)</p> <p>are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AnExt</mi></mrow><annotation encoding="application/x-tex">AnExt</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">e \in \{0,1\}</annotation></semantics></math>;</p> </li> <li> <p>if <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 in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AnExt</mi></mrow><annotation encoding="application/x-tex">AnExt</annotation></semantics></math>, then so is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>K</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo>∂</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>L</mi><mo stretchy="false">)</mo><mo>→</mo><mi>I</mi><mo>⊗</mo><mi>L</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (I \otimes K) \cup ((\partial I) \otimes L) \to I \otimes L \,. </annotation></semantics></math></div> <p>(hence “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><mo>→</mo><mi>L</mi><mo stretchy="false">)</mo><mover><mo>⊗</mo><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mo>∂</mo><mi>I</mi><mo>→</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K \to L) \bar \otimes (\partial I \to I)</annotation></semantics></math>”).</p> </li> </ol> </div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.3.10</a>).</p> <div class="num_prop" id="PropertiesOfAnodyneExtensions"> <h6 id="proposition_4">Proposition</h6> <p>A class of anodyne extensions</p> <ul> <li> <p>is generated under <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a>, <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> and <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> from the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>;</p> </li> <li> <p>is a subclass of the <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>;</p> </li> <li> <p>contains all morphisms of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>K</mi><mo>→</mo><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e : K \to (I \otimes K)</annotation></semantics></math>;</p> </li> <li> <p>is closed under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I\otimes(-)</annotation></semantics></math>;</p> </li> </ul> </div> <p>(<a href="#Cisinski06">Cisinski 06, remark. 1.3.11</a>).</p> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>By prop. <a class="maruku-ref" href="#PresheavesAreCompact"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in PSh(A)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|Mor(A/X)|</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/compact+object">compact</a>. Therefore the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> admits a <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>, which shows the first statement (see there).</p> <p>Since monomorphisms are closed under these operations, the second statement follows.</p> <p>The third statement follows by choosing the morphism in the second item of def. <a class="maruku-ref" href="#AnodyneExtensions"></a> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\emptyset \to K</annotation></semantics></math> and using that by def. <a class="maruku-ref" href="#ElementaryHomotopicalDatum"></a> the interval commutes with colimits, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊗</mo><mi>∅</mi><mo>≃</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">I \otimes \emptyset \simeq \emptyset</annotation></semantics></math>.</p> <p>Finally, to see that with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">j : K \to L</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊗</mo><mi>j</mi><mo>:</mo><mi>I</mi><mo>⊗</mo><mi>L</mi><mo>→</mo><mi>I</mi><mo>⊗</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">I \otimes j : I \otimes L \to I \otimes L</annotation></semantics></math> is anodyne, consider the naturality diagram of the endpoint inclusion</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><mi>j</mi></mover></mtd> <mtd><mi>L</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mi>k</mi> <mn>0</mn></msubsup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>K</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>j</mi><mo>′</mo></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>K</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>⊗</mo><mi>L</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><mi>I</mi><mo>⊗</mo><mi>j</mi></mrow></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>k</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>I</mi><mo>⊗</mo><mi>L</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ K &\stackrel{j}{\to}& L \\ \downarrow^{\mathrlap{\partial^0_k}} && \downarrow \\ I \otimes K &\stackrel{j'}{\to}& (I \otimes K) \cup \{0\}\otimes L \\ & {}_{I \otimes j} \searrow & \downarrow^{\mathrlap{k}} \\ && I \otimes L } \,, </annotation></semantics></math></div> <p>factored through the top pushout square, as indicated. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">j'</annotation></semantics></math> is anodyne, being a pushout of an anodyne morphism, and <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> is anodyne by the second clause in def. <a class="maruku-ref" href="#AnodyneExtensions"></a>. Therefore also their composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊗</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">I \otimes j</annotation></semantics></math> is anodyne.</p> </div> <div class="num_defn" id="HomotopicalStructure"> <h6 id="definition_11">Definition</h6> <p>A <strong>homotopical structure</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> is a choice of elementary homotopical datum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>, def. <a class="maruku-ref" href="#ElementaryHomotopicalDatum"></a> and a corresponding choice of a class of anodyne extensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AnExt</mi></mrow><annotation encoding="application/x-tex">AnExt</annotation></semantics></math>, def. <a class="maruku-ref" href="#AnodyneExtensions"></a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.3.14</a>).</p> <p> <div class='num_defn' id='DefinitionHomotopicalDatum'> <h6>Definition</h6> <p>A <em>homotopical datum</em>, or <em>donnée homotopique</em>, on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> is a choice of elementary homotopical datum (Definition <a class="maruku-ref" href="#ElementaryHomotopicalDatum"></a>) together with a set of <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>.</p> </div> </p> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.3.14</a>).</p> <p> <div class='num_remark'> <h6>Remark</h6> <p>A homotopical datum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>J</mi><mo>,</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(J, S)</annotation></semantics></math> <em>generates</em> a homotopical structure for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AnExt</mi></mrow><annotation encoding="application/x-tex">AnExt</annotation></semantics></math> is the smallest class of anodyne extensions relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> which contains <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> <h3 id="the_model_structure">The model structure</h3> <div class="num_defn" id="ModelStructureMorphismsFromHomotopicalStructure"> <h6 id="definition_12">Definition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>J</mi><mo>,</mo><mi>AnExt</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(J, AnExt)</annotation></semantics></math> a homotopical structure, def. <a class="maruku-ref" href="#HomotopicalStructure"></a>, on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>. Define the following classes of objects and morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>:</p> <ul> <li> <p>the <strong>cofibrations</strong> are the <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>;</p> </li> <li> <p>the <strong>fibrant objects</strong> are those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in PSh(A)</annotation></semantics></math> for which the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> 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> has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against the anodyne extensions (the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AnExt</mi></mrow><annotation encoding="application/x-tex">AnExt</annotation></semantics></math>, def. <a class="maruku-ref" href="#AnodyneExtensions"></a>);</p> </li> <li> <p>the <strong>weak equivalences</strong> are those morphisms <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>, such that for all fibrant objects <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 induced morphism in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy category, def. <a class="maruku-ref" href="#JHomotopyCategory"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho_J(f,X) : Ho_J(B,X) \to Ho_J(A, X) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> (a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> of sets).</p> </li> </ul> </div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.3.21</a>).</p> <div class="num_theorem" id="ModelStructureFromHomotopicalStructure"> <h6 id="theorem">Theorem</h6> <p>With the classes of morphisms as in def. <a class="maruku-ref" href="#ModelStructureMorphismsFromHomotopicalStructure"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, theorem 1.3.22</a>).</p> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>The retract properties are clear, as is the 2-out-of-3 property for weak equivalences, see lemma <a class="maruku-ref" href="#JHomotopyEquivsAreWeakEquivs"></a> below.</p> <p>The lifting properties hold by prop. <a class="maruku-ref" href="#AcyclicFibrationsAreAcyclicFibrations"></a> below, the proof of which is in the section <em><a href="#Lifting">Lifting</a></em> below.</p> <p>The factorization properties hold by cor. <a class="maruku-ref" href="#FirstFactorizationEstablishes"></a> and cor. <a class="maruku-ref" href="#FactorizationAcyclicCofibFib"></a> below, which are in the section <em><a href="#Factorization">Factorization</a></em> below.</p> <p>The existence of a set of generating cofibrations is prop. <a class="maruku-ref" href="#CellularStructuresExist"></a> below, that of generating acyclic cofibrations is prop. <a class="maruku-ref" href="#GeneratingAcyclicCofibrationsExist"></a> below.</p> </div> <div class="num_prop" id="HomotopyCatIsFullSubcatgOfJHo"> <h6 id="proposition_5">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(PSh(A), W)</annotation></semantics></math> is (up to <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a>) the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub></mrow><annotation encoding="application/x-tex">Ho_J</annotation></semantics></math>, def. <a class="maruku-ref" href="#JHomotopyCategory"></a>, on the fibrant objects.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, 1.3.23</a>).</p> <p>Before coming to the proof of these lemmas, the following two statements say that the terminology introduced so far is indeed consistent with the meaning of this theorem.</p> <div class="num_prop" id="AcyclicFibrationsAreAcyclicFibrations"> <h6 id="proposition_6">Proposition</h6> <p>The morphisms called <em>acyclic fibrations</em> in def. <a class="maruku-ref" href="#AcyclicFibration"></a> are indeed precisely the acyclic fibrations with respect to the model structure of theorem <a class="maruku-ref" href="#ModelStructureFromHomotopicalStructure"></a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, theorem 1.3.27</a>).</p> <div class="num_prop" id="EveryAnodyneExtensionIsWeakEquivalence"> <h6 id="proposition_7">Proposition</h6> <p>Every anodyne extension, def. <a class="maruku-ref" href="#AnodyneExtensions"></a>, is a weak equivalence in the model structure of theorem <a class="maruku-ref" href="#ModelStructureFromHomotopicalStructure"></a>.</p> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>What is not true in general is the converse of prop. <a class="maruku-ref" href="#EveryAnodyneExtensionIsWeakEquivalence"></a>, that every acyclic cofibration is an anodyne extension. (A counterexample derives from chapter X, remark 2.4 in <a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Jardine</a> <em><a class="existingWikiWord" href="/nlab/show/Simplicial+homotopy+theory">Simplicial homotopy theory</a></em>.)</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, remark 1.3.46</a>).</p> <div class="num_prop" id="ConditionsForCompleteness"> <h6 id="proposition_8">Proposition</h6> <p>For the model structure from theorem <a class="maruku-ref" href="#ModelStructureFromHomotopicalStructure"></a>, the following are equivalent:</p> <ol> <li> <p>every acyclic cofibration is an anodyne extension;</p> </li> <li> <p>every morphism with right lifting against anodyne extensions is a fibration;</p> </li> <li> <p>every weak equivalence with right lifting against anodyne extensions is an acyclic fibration;</p> </li> <li> <p>every morphism with right lifting against anodyne extensions factors as an anodyne extension followed by a fibration.</p> </li> </ol> </div> <p>(<a href="#Cisinski06">Cisinski 06, prop. 1.3.47</a>).</p> <div class="num_defn"> <h6 id="definition_13">Definition</h6> <p>A homotopical structure, def. <a class="maruku-ref" href="#HomotopicalStructure"></a>, on a presheaf category is called <strong>complete</strong> if the model structure from theorem <a class="maruku-ref" href="#ModelStructureFromHomotopicalStructure"></a> satisfies the equivalent conditions of prop. <a class="maruku-ref" href="#ConditionsForCompleteness"></a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.3.48</a>).</p> <p>We discuss the proof of prop. <a class="maruku-ref" href="#ConditionsForCompleteness"></a> below in <em><a href="#Completeness">Completeness</a></em>.</p> <h3 id="proof_9">Proof</h3> <p>We collect lemmas to prove theorem <a class="maruku-ref" href="#ModelStructureFromHomotopicalStructure"></a> and related statements. A little bit of work is required for demonstrating the lifting axioms, which we do below in <em><a href="#Lifting">Lifting</a></em>. A little bit more work is required for demonstrating the factorization axioms, which we do below in <em><a href="#Factorization">Factorization</a></em>. Finally, the proof of the equivalence of the conditions of completeness is in <em><a href="#Completeness">Completeness</a></em>.</p> <div class="num_lemma" id="JHomotopyEquivsAreWeakEquivs"> <h6 id="lemma">Lemma</h6> <p>Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy equivalence, def. <a class="maruku-ref" href="#JHomotopyEquivalence"></a>, is a weak equivalence.</p> <p>The weak equivalences satisfy the <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a>-property and are stable under <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, remark 1.3.24</a>).</p> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>The first statement holds by definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub></mrow><annotation encoding="application/x-tex">Ho_J</annotation></semantics></math>.</p> <p>The second statement also follows directly from the definition. If for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>B</mi><mover><mo>→</mo><mi>g</mi></mover><mi>C</mi></mrow><annotation encoding="application/x-tex">A \stackrel{f}{\to} B \stackrel{g}{\to} C</annotation></semantics></math> and fibrant <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> in the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow></mover><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho_J(C,X) \stackrel{g^*}{\to} Ho_J(B,X) \stackrel{f^*}{\to} Ho_J(A,X) </annotation></semantics></math></div> <p>two of three are isomorphisms, then so is the third.</p> </div> <h4 id="Lifting">Lifting</h4> <p>We discuss the lifting properties in the model structure of def. <a class="maruku-ref" href="#ModelStructureMorphismsFromHomotopicalStructure"></a>.</p> <p>Since fibrations are defined to be the morphisms satisfying the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against acyclic cofibration, we only need to show that the fibrations which are also weak equivalences have the right lifting property against the monomorphisms. For this it is sufficient to show prop. <a class="maruku-ref" href="#AcyclicFibrationsAreAcyclicFibrations"></a>. This we do now, after a lemma.</p> <div class="num_defn" id="DualDeformationRetract"> <h6 id="definition_14">Definition</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> with retraction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">g : Y \to X</annotation></semantics></math>, which is also a <a class="existingWikiWord" href="/nlab/show/section">section</a> of <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> up to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><msub><mi>id</mi> <mi>Y</mi></msub><mo>⇒</mo><mi>f</mi><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">h : id_Y \Rightarrow f \circ g</annotation></semantics></math>. It is <em>strong</em> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>σ</mi> <mi>Y</mi></msub><mo>∘</mo><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h \circ (I \otimes f) = \sigma_Y \circ (I \otimes f)</annotation></semantics></math>.</p> <p>A <em>dual deformation restract</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/section">section</a> by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">g : Y \to X</annotation></semantics></math> and is also a retract up to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>⇒</mo><mi>g</mi><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">k : id_X \Rightarrow g \circ f</annotation></semantics></math>. Is is <em>strong</em> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>k</mi><mo>=</mo><mi>f</mi><mo>∘</mo><msub><mi>σ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">f \circ k = f \circ \sigma_X</annotation></semantics></math>.</p> </div> <div class="num_lemma" id="SectionsOfAcyclicFibrationsAreDefRetracts"> <h6 id="lemma_2">Lemma</h6> <p>Every acyclic fibration is a dual <a class="existingWikiWord" href="/nlab/show/strong+deformation+retract">strong deformation retract</a>, def. <a class="maruku-ref" href="#DualDeformationRetract"></a>.</p> <p>Every <a class="existingWikiWord" href="/nlab/show/section">section</a> of an acyclic fibration is a <a class="existingWikiWord" href="/nlab/show/strong+deformation+retract">strong deformation retract</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, prop. 1.3.26</a>).</p> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>The first statement is a direct consequence of prop <a class="maruku-ref" href="#SomePropertiesOfAcyclicFibrations"></a>.</p> <p>For the second statement, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> be an acyclic fibration, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s : Y \to X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/section">section</a>. This induces a commuting square</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><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo>∂</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>∘</mo><msub><mi>σ</mi> <mi>Y</mi></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><mi>s</mi><mo>∘</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mi>h</mi></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo>∘</mo><msub><mi>σ</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ (I \otimes Y) \cup ((\partial I) \otimes X) &\stackrel{(s\circ \sigma_Y,(id_X, s \circ p))}{\to}& X \\ \downarrow &\nearrow_{h}& \downarrow^{\mathrlap{p}} \\ I \otimes X &\stackrel{ p \circ \sigma_X}{\to}& Y } \,, </annotation></semantics></math></div> <p>where the lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> exists by assumption on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> (<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 necessarily a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>, being a <a class="existingWikiWord" href="/nlab/show/section">section</a>).</p> <p>The resulting component triangle</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><mo stretchy="false">(</mo><mo>∂</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><mi>s</mi><mo>∘</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mi>h</mi></msub></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (\partial I) \otimes X &\stackrel{(id_X, s \circ p)}{\to}& X \\ \downarrow &\nearrow_{h}& \\ I \otimes X } </annotation></semantics></math></div> <p>exhibits <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> as a <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a>, and the other resulting component triangle</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>I</mi><mo>⊗</mo><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>s</mi><mo>∘</mo><msub><mi>σ</mi> <mi>Y</mi></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>I</mi><mo>⊗</mo><mi>s</mi></mrow></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mi>h</mi></msub></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ I \otimes Y &\stackrel{s\circ \sigma_Y}{\to}& X \\ \downarrow^{\mathrlap{I \otimes s}} &\nearrow_{h} \\ I \otimes X } </annotation></semantics></math></div> <p>says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mo>∘</mo><msub><mi>σ</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">h\circ (I \otimes s) = s \circ \sigma_Y</annotation></semantics></math>, hence by <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> of cylinders <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><msub><mi>σ</mi> <mi>X</mi></msub><mo>∘</mo><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cdots = \sigma_X \circ (I \otimes s)</annotation></semantics></math>, hence that the deformation retract is indeed strong.</p> </div> <div class="proof"> <h6 id="proof_12">Proof</h6> <p><strong>of prop. <a class="maruku-ref" href="#AcyclicFibrationsAreAcyclicFibrations"></a></strong></p> <p>First to see that the acyclic fibrations of def. <a class="maruku-ref" href="#AcyclicFibration"></a> are indeed fibrations and weak equivalences:</p> <p>By lemma <a class="maruku-ref" href="#SectionsOfAcyclicFibrationsAreDefRetracts"></a> every acyclic fibration is in particular a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy equivalence, hence by lemma <a class="maruku-ref" href="#JHomotopyEquivsAreWeakEquivs"></a> a weak equivalence. Moreover, by def. <a class="maruku-ref" href="#AcyclicFibration"></a> the acyclic fibrations right-lift against monomorphisms, hence in particular against the acyclic cofibrations, hence are fibrations.</p> <p>Conversely, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> be a fibration which is also a weak equivalence. We need to show that it has the right lifting property against all monomorphisms.</p> <p>By prop. <a class="maruku-ref" href="#CellularStructuresExist"></a>, proven below, we may apply the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> to factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mi>q</mi><mo>∘</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">p = q \circ j</annotation></semantics></math> as a monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> followed by an acyclic fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>. By the previous argument, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> is a weak equivalence, and so by lemma <a class="maruku-ref" href="#JHomotopyEquivsAreWeakEquivs"></a> so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>. Therefore, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a fibration, we have a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</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></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mi>σ</mi></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>p</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mo>→</mo><mi>q</mi></mover></mtd> <mtd></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ &\stackrel{id}{\to}& \\ \downarrow^{\mathrlap{j}} & \nearrow_\sigma& \downarrow^{p} \\ &\stackrel{q}{\to}& } \,. </annotation></semantics></math></div> <p>This equivalently exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></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></mtd> <mtd><mover><mo>→</mo><mi>j</mi></mover></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>σ</mi></mover></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>q</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ &\stackrel{j}{\to} & & \stackrel{\sigma}{\to} & \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{q}} && \downarrow^{\mathrlap{p}} \\ & \stackrel{id}{\to} & & \stackrel{id}{\to} & } \,. </annotation></semantics></math></div> <p>So by lemma <a class="maruku-ref" href="#PropertiesOfAnodyneExtensions"></a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is an acyclic fibration.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, remark 1.3.28</a>).</p> <h4 id="Factorization">Factorization</h4> <p>We discuss the two factorization axioms for the <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure from def. <a class="maruku-ref" href="#ModelStructureMorphismsFromHomotopicalStructure"></a> to be established. First for factorizations into <a href="#CofibrationFollowedByAcyclicFibration">cofibrations followed by acyclic fibrations</a>, then for factorizations into <a href="AcyclicCofibrationFollowedByFibration">acyclic cofibrations followed by fibrations</a>.</p> <h5 id="CofibrationFollowedByAcyclicFibration">Cofibration followed by acyclic fibration</h5> <p>Every <a class="existingWikiWord" href="/nlab/show/partial+map+classifier">partial map classifier</a> is an injective object, and so we have a functoral factorization of <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> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>X</mi> <mo>⊥</mo></msub><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to X_\bot \times Y \to Y</annotation></semantics></math>. However, we can say more by appealing to general machinery.</p> <p>For showing that every morphism factors as a monomorphism followed by an acyclic fibration, it is by prop. <a class="maruku-ref" href="#AcyclicFibrationsAreAcyclicFibrations"></a> sufficient to show that the monomorphisms are generated by a <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> that admits the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>. This we do now.</p> <p>This section follows (<a href="#Cisinski06">Cisinski 06, section 1.2</a>).</p> <p>We start with some entirely general statements about <a class="existingWikiWord" href="/nlab/show/compact+objects">compact objects</a>.</p> <div class="num_prop" id="LimitsOverASmallDiagramAreCompact"> <h6 id="proposition_9">Proposition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">\alpha = {\vert Mor(A)\vert}</annotation></semantics></math> be the smallest <a class="existingWikiWord" href="/nlab/show/regular+cardinal">regular cardinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\geq</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> of the set of morphisms of <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>. Then the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mo>←</mo></munder><mo>:</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> \lim_\leftarrow : Func(A,Set) \to Set </annotation></semantics></math></div> <p>commutes with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a> / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/directed+colimits">directed colimits</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, prop. 1.2.9</a>).</p> <div class="num_prop" id="CompactnessOfColimitsOverCompacts"> <h6 id="proposition_10">Proposition</h6> <p>Let <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> be a category with all small <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/cardinal">cardinal</a>, let <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> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and finally let <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>C</mi></mrow><annotation encoding="application/x-tex">F : A \to C</annotation></semantics></math> be a functor with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/compact+objects">compact objects</a> 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>. Then the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mo>→</mo></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">\lim_\to F</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-compact object, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/maximum">maximum</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Mor</mi><mi>A</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert Mor A\vert}</annotation></semantics></math> of the set of morphisms of <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>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, prop. 1.2.10</a>).</p> <div class="proof"> <h6 id="proof_13">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">G : I \to C</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/filtered+category">filtered diagram</a>.</p> <p>Then by prop. <a class="maruku-ref" href="#LimitsOverASmallDiagramAreCompact"></a> we have <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><munder><mi>lim</mi> <munder><mo>→</mo><mi>I</mi></munder></munder><mi>C</mi><mo stretchy="false">(</mo><munder><mi>lim</mi> <munder><mo>→</mo><mi>A</mi></munder></munder><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><munder><mi>lim</mi> <munder><mo>→</mo><mi>I</mi></munder></munder><munder><mi>lim</mi> <munder><mo>←</mo><mi>A</mi></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><munder><mi>lim</mi> <munder><mo>←</mo><mi>A</mi></munder></munder><munder><mi>lim</mi> <munder><mo>→</mo><mi>I</mi></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \lim_{\underset{I}{\to}} C(\lim_{\underset{A}{\to}} F, G) & \simeq \lim_{\underset{I}{\to}} \lim_{\underset{A}{\leftarrow}} C(F, G) \\ & \simeq \lim_{\underset{A}{\leftarrow}} \lim_{\underset{I}{\to}} C(F, G) \end{aligned} \,, </annotation></semantics></math></div> <p>because the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-filtered diagram is at least <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert Mor(A)\vert}</annotation></semantics></math>-filtered and hence, by prop. <a class="maruku-ref" href="#LimitsOverASmallDiagramAreCompact"></a>, its colimit commutes with the limit over <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>.</p> <p>Now since each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(a)</annotation></semantics></math> is assumed to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-compact and hence is also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-compact, we conclude with the natural isomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>≃</mo><munder><mi>lim</mi> <munder><mo>←</mo><mi>A</mi></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><munder><mi>lim</mi> <munder><mo>→</mo><mi>I</mi></munder></munder><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>C</mi><mo stretchy="false">(</mo><munder><mi>lim</mi> <munder><mo>→</mo><mi>A</mi></munder></munder><mi>F</mi><mo>,</mo><munder><mi>lim</mi> <munder><mo>→</mo><mi>I</mi></munder></munder><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \cdots & \simeq \lim_{\underset{A}{\leftarrow}} C(F, \lim_{\underset{I}{\to}} G) \\ & \simeq C(\lim_{\underset{A}{\to}} F, \lim_{\underset{I}{\to}} G) \end{aligned} \,. </annotation></semantics></math></div></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in PSh(A)</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A/X</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+elements">category of elements</a> of <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>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert Mor(A/X)\vert}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> of the set of morphisms of the category of elements (throughout assuming <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> to be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>).</p> <div class="num_prop" id="PresheavesAreCompact"> <h6 id="proposition_11">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in PSh(A)</annotation></semantics></math> any object, the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Hom(X, -) : PSh(A) \to Set</annotation></semantics></math> preserves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert Mor(A/X)\vert}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>.</p> <p>In other words: <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> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert Mor(A/X)\vert}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, cor. 1.2.11</a>).</p> <div class="proof"> <h6 id="proof_14">Proof</h6> <p>By the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a> <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> is the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> over its elements</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><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>X</mi><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \simeq \lim_\to (A/X \to PSh(A)) \,. </annotation></semantics></math></div> <p>Since the image of the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mi>X</mi><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A/X \to PSh(A)</annotation></semantics></math> is in representables, which are maximally compact, the stament follows with prop. <a class="maruku-ref" href="#CompactnessOfColimitsOverCompacts"></a>.</p> </div> <div class="num_defn" id="CellularModel"> <h6 id="definition_15">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/cellular+model">cellular model</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> is a choice of a <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I \subset Mor(PSh(A))</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>, such that the <a class="existingWikiWord" href="/nlab/show/class">class</a> of all monomorphisms is generated from it</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Monos</mi><mo>=</mo><mi>LLP</mi><mo stretchy="false">(</mo><mi>RLP</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Monos = LLP(RLP(I)) \,. </annotation></semantics></math></div></div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.2.26</a>).</p> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>By prop. <a class="maruku-ref" href="#PresheavesAreCompact"></a> we may apply the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> and so it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LLP</mi><mo stretchy="false">(</mo><mi>RLP</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">LLP(RLP(I))</annotation></semantics></math> is the smallest class containing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> that is closed under <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/retract">retract</a> and <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a>.</p> </div> <p>The following lemma will be used to show that cellular structures always exist.</p> <div class="num_lemma" id="LemmaForGenerationOfMonos"> <h6 id="lemma_3">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \subset Mor(PSh(A))</annotation></semantics></math> be a class of morphisms, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>⊂</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D \subset PSh(A)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> of objects, such that</p> <ol> <li> <p><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 closed under <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a>, <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> and <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a>;</p> </li> <li> <p>if <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>Mor</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g \in Mor(PSh(A))</annotation></semantics></math> are two composable morphisms with <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">g \circ f</annotation></semantics></math> 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>, then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is 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>;</p> </li> <li> <p>every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in PSh(A)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/union">union</a> of those of its <a class="existingWikiWord" href="/nlab/show/sub-objects">sub-objects</a> isomorphic to an object in <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>;</p> </li> <li> <p>for every <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> 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 every sub-object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> of <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> in <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>, there is a sub-object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">T \hookrightarrow Y</annotation></semantics></math> from <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>, which contains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> and such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∩</mo><mi>X</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">T \cap X \to T</annotation></semantics></math> is 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>.</p> </li> </ol> <p>Then the <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I \subset Mor(PSh(A))</annotation></semantics></math> of 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">C</annotation></semantics></math> with codomain in <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>, generates <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> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>LLP</mi><mo stretchy="false">(</mo><mi>RLP</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C = LLP(RLP(I)) \,. </annotation></semantics></math></div></div> <p>(<a href="#Cisinski06">Cisinski 06, lemma. 1.2.24</a>).</p> <div class="proof"> <h6 id="proof_15">Proof</h6> <p>A bit of work…</p> </div> <div class="num_prop" id="CellularStructuresExist"> <h6 id="proposition_12">Proposition</h6> <p>There exists a cellular structure, def. <a class="maruku-ref" href="#CellularModel"></a>, on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>.</p> <p>The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> can be chosen to consist of morphisms into <a class="existingWikiWord" href="/nlab/show/quotient+objects">quotient objects</a> of <a class="existingWikiWord" href="/nlab/show/representable+functors">representables</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, prop. 1.2.27</a>), also sketched at <a class="existingWikiWord" href="/nlab/show/cellular+model">cellular model</a>.</p> <div class="proof"> <h6 id="proof_16">Proof</h6> <p>Take in lemma <a class="maruku-ref" href="#LemmaForGenerationOfMonos"></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">C</annotation></semantics></math> to be the class of monomorphisms 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">D</annotation></semantics></math> to be the class of quotients of representables.</p> </div> <div class="num_cor" id="FirstFactorizationEstablishes"> <h6 id="corollary">Corollary</h6> <p>There exists a functorial factorization of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> into a monomorphism followed by an acyclic fibration.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, cor. 1.2.28</a>).</p> <div class="proof"> <h6 id="proof_17">Proof</h6> <p>By prop. <a class="maruku-ref" href="#PresheavesAreCompact"></a> every object is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/compact+object">small</a>, for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>. Therefore by prop. <a class="maruku-ref" href="#CellularStructuresExist"></a> we can apply the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>.</p> </div> <h5 id="AcyclicCofibrationFollowedByFibration">Acyclic cofibration followed by fibration</h5> <p>We show now for def. <a class="maruku-ref" href="#ModelStructureMorphismsFromHomotopicalStructure"></a> that every morphism factors as an acyclic cofibration followed by a fibration. Since the fibrations are defined by right lifting against acylcic cofibrations, for this it is sufficient to establish a set of <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">generating acyclic cofibrations</a>. This is the statement of prop. <a class="maruku-ref" href="#GeneratingAcyclicCofibrationsExist"></a> below. Establishing this takes a few technical lemmas.</p> <div class="num_lemma" id="FactorAnodyneFollowedByRLPAnodyne"> <h6 id="lemma_4">Lemma</h6> <p>Every morphism admits a factorization into an anodyne extension, followed by a morphism having the right lifting property against anodyne extensions.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, remark 1.3.29</a>).</p> <div class="proof"> <h6 id="proof_18">Proof</h6> <p>By the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>, in view of prop. <a class="maruku-ref" href="#PropertiesOfAnodyneExtensions"></a>.</p> </div> <div class="num_lemma" id="ElementaryJHomotopyIntoFibIsEquivRel"> <h6 id="lemma_5">Lemma</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \in PSh(A)</annotation></semantics></math> is fibrant, then for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K \in PSh(A)</annotation></semantics></math> elementary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy, def. <a class="maruku-ref" href="#ElementaryJHomotopy"></a>, is already an equivalence relation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{PSh(A)}(K,T)</annotation></semantics></math> and coincides with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, lemma 1.3.30</a>).</p> <div class="num_prop" id="EveryAnodyneExtensionIsWeakEquivalence"> <h6 id="proposition_13">Proposition</h6> <p>Every anodyne extension is a weak equivalence.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, lemma 1.3.31</a>).</p> <div class="proof"> <h6 id="proof_19">Proof</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">j : K \to L</annotation></semantics></math> an anodyne extension and <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> a fibrant object, we need to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Ho</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho_J(j,T) : Ho_J(L,T) \to Ho_J(K,T) </annotation></semantics></math></div> <p>is a bijection.</p> <p>It is surjective by the defining lifting property, which provides <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</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>K</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>T</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>σ</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>L</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ K &\stackrel{}{\to}& T \\ \downarrow^{\mathrlap{j}} & \nearrow_{\mathrlap{\sigma}} \\ L } \,. </annotation></semantics></math></div> <p>To see injectivity, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>l</mi> <mn>1</mn></msub><mo>:</mo><mi>L</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">l_0, l_1 : L \to T</annotation></semantics></math> be two morphisms such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mn>0</mn></msub><mo>∘</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">l_0 \circ j</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mn>1</mn></msub><mo>∘</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">l_1 \circ j</annotation></semantics></math> coincide in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub></mrow><annotation encoding="application/x-tex">Ho_J</annotation></semantics></math>. By lemma <a class="maruku-ref" href="#ElementaryJHomotopyIntoFibIsEquivRel"></a> this is the case precisely if there is an elementary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>I</mi><mo>⊗</mo><mi>K</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">h : I \otimes K \to T</annotation></semantics></math> relating them. This induces the horizontal morphism in the 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><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>K</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo>∂</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>L</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>h</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>l</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>l</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>T</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>η</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>L</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ (I \otimes K) \cup ((\partial I) \otimes L) &\stackrel{(h,(l_0,l_1))}{\to}& T \\ \downarrow & \nearrow_{\mathrlap{\eta}} \\ I \otimes L } \,, </annotation></semantics></math></div> <p>where the left morphism is anodyne, by the second clause of def. <a class="maruku-ref" href="#AnodyneExtensions"></a>, so that the lift denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> exists. This lift exhibits a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mn>0</mn></msub><mo>⇒</mo><msub><mi>l</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">l_0 \Rightarrow l_1</annotation></semantics></math>, hence shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">l_0</annotation></semantics></math> was already equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">l_1</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub></mrow><annotation encoding="application/x-tex">Ho_J</annotation></semantics></math>, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">j^*</annotation></semantics></math> is injective.</p> </div> <div class="num_lemma" id="WeakEquivalencesBetweenFibrantObjects"> <h6 id="lemma_6">Lemma</h6> <p>A morphism between fibrant objects is a weak equivalence precisely if it is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy equivalence, def. <a class="maruku-ref" href="#JHomotopyEquivalence"></a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, lemma 1.3.32</a>).</p> <div class="proof"> <h6 id="proof_20">Proof</h6> <p>Is is clear that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy is a weak equivalence. Conversely, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> be a weak equivalence between fibrant objects. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ho</mi> <mi>J</mi> <mi>fib</mi></msubsup><mo>↪</mo><msub><mi>Ho</mi> <mi>J</mi></msub></mrow><annotation encoding="application/x-tex">Ho_J^{fib} \hookrightarrow Ho_J</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub></mrow><annotation encoding="application/x-tex">Ho_J</annotation></semantics></math>, def. <a class="maruku-ref" href="#JHomotopyEquivalence"></a> on the fibrant objects. The localization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(f)</annotation></semantics></math> is by definition in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ho</mi> <mi>J</mi> <mi>fib</mi></msubsup></mrow><annotation encoding="application/x-tex">Ho_J^{fib}</annotation></semantics></math> and for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∈</mo><msubsup><mi>Ho</mi> <mi>J</mi> <mi>fib</mi></msubsup></mrow><annotation encoding="application/x-tex">T \in Ho_J^{fib}</annotation></semantics></math> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ho</mi> <mi>J</mi> <mi>fib</mi></msubsup><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho_J^{fib}(Q(f), T)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. By the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(f)</annotation></semantics></math> itself is an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ho</mi> <mi>J</mi> <mi>fib</mi></msubsup></mrow><annotation encoding="application/x-tex">Ho_J^{fib}</annotation></semantics></math>, hence also in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>J</mi></msub></mrow><annotation encoding="application/x-tex">Ho_J</annotation></semantics></math>, hence is a weak equivalence.</p> </div> <div class="num_lemma" id="NaiveFibAndDualStrongDefRetractIsAcyclFib"> <h6 id="lemma_7">Lemma</h6> <p>If a morphism has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against the anodyne extensions, then it is an acyclic fibration precisely if it is a dual <a class="existingWikiWord" href="/nlab/show/strong+deformation+retract">strong deformation retract</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, lemma 1.3.33</a>).</p> <div class="proof"> <h6 id="proof_21">Proof</h6> <p>That the former implies the latter was the statement of lemma <a class="maruku-ref" href="#SectionsOfAcyclicFibrationsAreDefRetracts"></a>. Conversely, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> be a dual strong deformation retract, meaning that there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s : Y \to X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>s</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">p \circ s = id</annotation></semantics></math>, as well as a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>I</mi><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">k : I \otimes X \to X</annotation></semantics></math> exhibiting a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo>⇒</mo><mi>s</mi><mo>∘</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">id \Rightarrow s \circ p</annotation></semantics></math>. This being <em>strong</em> means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>k</mi><mo>=</mo><mi>p</mi><mo>∘</mo><msub><mi>σ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">p \circ k = p \circ \sigma_X</annotation></semantics></math>.</p> <p>We need to show that this implies for</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><mi>a</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>L</mi></mtd> <mtd><mover><mo>→</mo><mi>b</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ K &\stackrel{a}{\to}& X \\ \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{p}} \\ L &\stackrel{b}{\to}& Y } </annotation></semantics></math></div> <p>a commuting diagram with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> a monomorphism, there is a lift. To this end, observe that the given structures induce a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>K</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>⊗</mo><mi>L</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> (I \otimes K) \cup (\{1\} \otimes L) \to X </annotation></semantics></math></div> <p>such that</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><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>K</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>⊗</mo><mi>L</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo>∘</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>L</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>σ</mi> <mi>L</mi></msub></mrow></mover></mtd> <mtd><mi>L</mi></mtd> <mtd><mover><mo>→</mo><mi>b</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (I \otimes K) \cup (\{1\} \otimes L) &&\stackrel{(k \circ (I \otimes a), s \circ b)}{\to}&& X \\ \downarrow^{\mathrlap{j}} && && \downarrow^{\mathrlap{p}} \\ I \otimes L &\stackrel{\sigma_L}{\to}& L &\stackrel{b}{\to}& Y } \,. </annotation></semantics></math></div> <p>By the second clause of def. <a class="maruku-ref" href="#AnodyneExtensions"></a> the morphism on the left is an anodyne extension, and so this diagram admits a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>I</mi><mo>⊗</mo><mi>L</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">h : I \otimes L \to X</annotation></semantics></math>. One see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>:</mo><mo>=</mo><mi>h</mi><mo>∘</mo><msubsup><mo>∂</mo> <mi>L</mi> <mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">l := h \circ \partial^0_L</annotation></semantics></math> is a lift of the original square above.</p> </div> <div class="num_lemma" id="NaiveFibIntoFibIsWeakEquivIffItIsAcyclicFib"> <h6 id="lemma_8">Lemma</h6> <p>A morphism into a fibrant object with <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against anodyne extensions is a weak equivalence precisely if it is an acyclic fibration.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, lemma 1.3.34</a>).</p> <div class="proof"> <h6 id="proof_22">Proof</h6> <p>We already know from prop. <a class="maruku-ref" href="#AcyclicFibrationsAreAcyclicFibrations"></a> that acyclic fibrations are weak equivalences.</p> <p>So let <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> be fibrant and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> be a weak equivalence that has rlp against anodyne extensions. We need to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is an acyclic fibration. By lemma <a class="maruku-ref" href="#NaiveFibAndDualStrongDefRetractIsAcyclFib"></a> it is sufficient to show that it is a dual strong deformation retract.</p> <p>By lemma <a class="maruku-ref" href="#WeakEquivalencesBetweenFibrantObjects"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is also a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy equivalence. By lemma <a class="maruku-ref" href="#ElementaryJHomotopyIntoFibIsEquivRel"></a> this is exhibited by an elementary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>I</mi><mo>⊗</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">k : I \otimes Y \to Y</annotation></semantics></math>, which in particular gives a 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>Y</mi></mtd> <mtd><mover><mo>→</mo><mi>t</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mi>Y</mi> <mn>1</mn></msubsup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mi>k</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ Y &\stackrel{t}{\to}& X \\ \downarrow^{\mathrlap{\partial^1_Y}} && \downarrow^{\mathrlap{p}} \\ I \otimes Y &\stackrel{k}{\to}& Y } \,, </annotation></semantics></math></div> <p>from which we obtain a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>′</mo><mo>:</mo><mi>I</mi><mo>⊗</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">k' : I \otimes Y \to X</annotation></semantics></math>. Set then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mo>=</mo><mi>k</mi><mo>′</mo><mo>∘</mo><msubsup><mo>∂</mo> <mi>Y</mi> <mn>0</mn></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> s := k' \circ \partial^0_Y \,. </annotation></semantics></math></div> <p>One finds then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>s</mi><mo>=</mo><msub><mi>id</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">p \circ s = id_Y</annotation></semantics></math>. As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy equivalence, it follows that <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 its homotopic inverse, in particular, there is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>⇒</mo><mi>s</mi><mo>∘</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">h : id_X \Rightarrow s \circ p</annotation></semantics></math>.</p> <p>Next we lift the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⇒</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">p \Rightarrow p</annotation></semantics></math> to transform <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> into a dual stronf deformation:</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><mo>∂</mo><mi>I</mi><mo>⊗</mo><mi>I</mi><mo>⊗</mo><mi>X</mi><mo>∪</mo><mi>I</mi><mo>⊗</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>h</mi><mo>,</mo><mi>s</mi><mo>∘</mo><mi>p</mi><mo>∘</mo><mi>h</mi><mo stretchy="false">]</mo><mo>,</mo><mi>s</mi><mo>∘</mo><mi>p</mi><mo>∘</mo><msub><mi>σ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mi>H</mi><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi><mo>⊗</mo><mi>I</mi><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo>∘</mo><mi>h</mi><mo>∘</mo><msub><mi>σ</mi> <mrow><mi>I</mi><mo>⊗</mo><mi>X</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \partial I \otimes I \otimes X \cup I \otimes \{1\} \otimes X & \stackrel{([h, s \circ p \circ h], s \circ p \circ \sigma_X)}{\to} & X \\ \downarrow & H \nearrow & \downarrow^{\mathrlap{p}} \\ I \otimes I \otimes X & \stackrel{p \circ h \circ \sigma_{I \otimes X}}{\to} & Y } </annotation></semantics></math></div> <p>Now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>I</mi><mo>⊗</mo><msubsup><mo>∂</mo> <mi>X</mi> <mn>0</mn></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \circ (I \otimes \partial^0_{X})</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-homotopy showing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a dual strong deformation retract.</p> </div> <div class="num_cor"> <h6 id="corollary_2">Corollary</h6> <p>A cofibration into a fibrant object is a weak equivalence precisely if it is an anodyne extension.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, cor 1.3.35</a>).</p> <div class="proof"> <h6 id="proof_23">Proof</h6> <p>By lemma <a class="maruku-ref" href="#EveryAnodyneExtensionIsWeakEquivalence"></a> we already know that every anodyne extension is a weak equivalence. So we need to show that a cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">i : A \to T</annotation></semantics></math> into a fibrant object <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> which is a weak equivalence is also an anodyne extension. By prop. <a class="maruku-ref" href="#FactorAnodyneFollowedByRLPAnodyne"></a> we may factor this as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mi>q</mi><mo>∘</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i = q \circ j</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> an anodyne extension and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> having RLP against anodyne extensions. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> is a weak equivalence, by 2-out-of-3 so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>. By lemma <a class="maruku-ref" href="#NaiveFibIntoFibIsWeakEquivIffItIsAcyclicFib"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> is an acyclic fibration.</p> <p>Therefore we have a lift <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> 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></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>q</mi></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mi>s</mi></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mo>→</mo><mi>j</mi></mover></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ &\stackrel{id}{\to}& \\ \downarrow^{\mathrlap{q}} &\nearrow_s& \downarrow^{\mathrlap{i}} \\ &\stackrel{j}{\to}& } </annotation></semantics></math></div> <p>and this exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>. Hence with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is an anodyne extension.</p> </div> <div class="num_prop"> <h6 id="proposition_14">Proposition</h6> <p>A cofibration is a weak equivalence precisely if it has the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against morphisms into a fibrant object that have the right lifting property against anodyne extensions.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, prop. 1.3.36</a>).</p> <div class="proof"> <h6 id="proof_24">Proof</h6> <p>(…)</p> </div> <div class="num_cor"> <h6 id="corollary_3">Corollary</h6> <p>The acyclic cofibrations are stable under <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> and <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a></p> </div> <p>(<a href="#Cisinski06">Cisinski 06, cor. 1.3.37</a>).</p> <div class="num_lemma"> <h6 id="lemma_9">Lemma</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/strong+deformation+retract">strong deformation retract</a> is an anodyne extension.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, cor. 1.3.38</a>).</p> <div class="num_lemma"> <h6 id="lemma_10">Lemma</h6> <p>Every anodyne extension between fibrant objects is a <a class="existingWikiWord" href="/nlab/show/strong+deformation+retract">strong deformation retract</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, lemma 1.3.39</a>).</p> <div class="num_prop"> <h6 id="proposition_15">Proposition</h6> <p>Pour tout cardinal assez grand <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>, si on pose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><msup><mn>2</mn> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">\beta = 2^\alpha</annotation></semantics></math>, pour toute cofibration triviale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">i : C \to D</annotation></semantics></math>, et pour tout <a class="existingWikiWord" href="/nlab/show/subobject">sous-objet</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>-accessible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> de <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>, il existe un sous-objet <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>-accessible <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> de <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>, qui contient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>, tel que l’inclusion canonique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∩</mo><mi>K</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">C \cap K \to K</annotation></semantics></math> soit une cofibration triviale.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, prop. 1.3.40</a>).</p> <div class="proof"> <h6 id="proof_25">Proof</h6> <p>Three pages of work…</p> </div> <div class="num_prop" id="GeneratingAcyclicCofibrationsExist"> <h6 id="proposition_16">Proposition</h6> <p>There exists a set of generating acyclic cofibrations.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, prop. 1.3.42</a>).</p> <div class="proof"> <h6 id="proof_26">Proof</h6> <p>Use lemma <a class="maruku-ref" href="#LemmaForGenerationOfMonos"></a> with <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> the class of acyclic cofibrations and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><msub><mi>Acc</mi> <mi>α</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D = Acc_\alpha(A)</annotation></semantics></math> the set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-accessible presheaves for a sufficiently large cardinal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>.</p> </div> <div class="num_cor" id="FactorizationAcyclicCofibFib"> <h6 id="corollary_4">Corollary</h6> <p>There is a functorial factorization of every morphism into an acyclic cofibration followed by a fibration.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, cor. 1.3.43</a>).</p> <div class="proof"> <h6 id="proof_27">Proof</h6> <p>By prop. <a class="maruku-ref" href="#GeneratingAcyclicCofibrationsExist"></a> and prop. <a class="maruku-ref" href="#PresheavesAreCompact"></a> we may apply the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>.</p> </div> <h4 id="Completeness">Completeness</h4> <p>We list lemmas to show prop. <a class="maruku-ref" href="#ConditionsForCompleteness"></a>.</p> <p>(…)</p> <h3 id="ALocalizers">Localizers</h3> <p>Continuing to let <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> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a>.</p> <div class="num_defn" id="ALocalizer"> <h6 id="definition_16">Definition</h6> <p>An <strong><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>-localizer</strong> is a <a class="existingWikiWord" href="/nlab/show/class">class</a> of morphisms <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>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \subset Mor(PSh(A))</annotation></semantics></math> satisfying the following axioms</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/2-out-of-3">2-out-of-3</a>;</p> </li> <li> <p>every acyclic fibration, def. <a class="maruku-ref" href="#AcyclicFibration"></a>, is 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>;</p> </li> <li> <p>The class of <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> that is 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> is stable under <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> and <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a>.</p> </li> </ol> <p>The elements of <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> we call <strong><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>-equivalences</strong>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \subset Mor(PSh(A))</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/class">class</a> of morphisms, the smallest <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>-localizer containing <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 called the <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>-localized <strong>generated</strong> by <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>. If an <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>-localizer is generated from a <a class="existingWikiWord" href="/nlab/show/small+set">small set</a>, we call it <strong>accessible</strong>. The <strong>minimal <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>-localizer</strong> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\emptyset)</annotation></semantics></math>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, def. 1.4.1</a>))</p> <div class="num_prop"> <h6 id="proposition_17">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>J</mi><mo>,</mo><mi>AnExt</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(J,AnExt)</annotation></semantics></math> a homotopical structure, def. <a class="maruku-ref" href="#HomotopicalStructure"></a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>, the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> of weak equivalences of the induced model category structure of theorem <a class="maruku-ref" href="#ModelStructureFromHomotopicalStructure"></a> is an <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>-localizer.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> is a small set generating the anodyne extensions, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AnExt</mi><mo>=</mo><mi>LLP</mi><mo stretchy="false">(</mo><mi>RLP</mi><mo stretchy="false">(</mo><mi>Λ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">AnExt = LLP(RLP(\Lambda))</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>=</mo><mi>W</mi><mo stretchy="false">(</mo><mi>Λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W = W(\Lambda)</annotation></semantics></math>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, prop. 1.4.2</a>))</p> <p> <div class='num_theorem' id='TheoremCharacterisationOfAccessibleLocalisersForPresheafCategories'> <h6>Theorem</h6> <p>Let <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>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \subset Mor(PSh(A))</annotation></semantics></math>. 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>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is an accessible <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>-localizer, def. <a class="maruku-ref" href="#ALocalizer"></a>;</p> </li> <li> <p>There is a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \subset Mor(PSh(A))</annotation></semantics></math> of monomorphisms, such that <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 the class of weak equivalences of the model structure induced by theorem <a class="maruku-ref" href="#ModelStructureFromHomotopicalStructure"></a> from the homotopical structure, def. <a class="maruku-ref" href="#HomotopicalStructure"></a>, given by the <a class="existingWikiWord" href="/nlab/show/Lawvere+cylinder">Lawvere cylinder</a>, def. <a class="maruku-ref" href="#LawvereCylinder"></a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AnExt</mi><mo>:</mo><mo>=</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">AnExt := S</annotation></semantics></math>.</p> </li> <li> <p>There is some homotopical structure, def. <a class="maruku-ref" href="#HomotopicalStructure"></a>, on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>, such that <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 the class of weak equivalences of the model structure corresponding to it by theorem <a class="maruku-ref" href="#ModelStructureFromHomotopicalStructure"></a>.</p> </li> <li> <p>There exists a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> such that <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 its class of weak equivalences, and such that the cofibrations are the monomorphisms.</p> </li> </ol> <p>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math> admits a model structure whose cofibrations are the monomorphisms and whose weak equivalences are the minimal localizer, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\emptyset)</annotation></semantics></math>. This is called the <strong>minimal model structure</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>. It is generated from the homotopical datum given by the Lawvere cylinder, example <a class="maruku-ref" href="#LawvereCylinder"></a> and the empty set.</p> <p></p> </div> </p> <p>(<a href="#Cisinski06">Cisinski 06, theorem 1.4.3</a>)</p> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>This may be compared to <a class="existingWikiWord" href="/nlab/show/Jeff+Smith%27s+theorem">Jeff Smith's theorem</a>, which constructs a model structure on a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a>.</p> </div> <p>(<a href="#Cisinski06">Cisinski 06, scholie 1.4.6</a>))</p> <h3 id="SimplicialCompletion">Simplicial completion</h3> <p>Give a localizer <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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A)</annotation></semantics></math>, there is a localizer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>horiz</mi></msub></mrow><annotation encoding="application/x-tex">W_{horiz}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>A</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(A \times \Delta)</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>W</mi> <mi>horiz</mi></msub><mo>≔</mo><mo stretchy="false">{</mo><mi>X</mi><mo>×</mo><msup><mi>q</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> W_{horiz} \coloneqq \{ X \times q^* (\Delta_1) \to X \} \,. </annotation></semantics></math></div> <p>(…)</p> <p>See (<a href="#Ara">Ara, p. 9</a>).</p> <h2 id="Examples">Examples</h2> <ul> <li> <p>The archetypical and motivating example is the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>, which is a Cisinski model structure on the <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> over the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> (<a href="#Cisinski06">Cisinski 06, section 2</a>).</p> </li> <li> <p>Accordingly, the <em>injective</em> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> over a site <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 a Cisinski model structure, namely on the <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(C \times \Delta)</annotation></semantics></math>. Moreover, every <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">left Bousfield localization</a> of such a model structure is still a Cisinski model structure, since left Bousfield localization preserves the class of cofibrations.</p> <p>Notice that, as discussed there, every <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentable (infinity,1)-category</a> has a presentation by such a localization, hence by a Cisinski model structure.</p> </li> <li> <p>Also the <a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a> is a Cisinski model structure on <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>, induced by the localizer given by the <a class="existingWikiWord" href="/nlab/show/spine">spine</a> inclusions.</p> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">model structure for complete Segal spaces</a> is the <a href="#SimplicialCompletion">simplicial completion</a> of this model structure. (see <a href="#Ara">Ara</a>).</p> </li> <li> <p>As a <a class="existingWikiWord" href="/nlab/show/cellular+set">cellular set</a>-variant of this, the <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">model structure on cellular sets</a></em> is a Cisinski model structure on the category of presheaves over the <a class="existingWikiWord" href="/nlab/show/Theta+category">Theta category</a> restricted to <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>-cells.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">model structure on dendroidal sets</a> is not exactly a Cisinki model structure, but is <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred</a> from one that is.</p> </li> <li> <p>For any small category <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>, the homotopical datum (Definition <a class="maruku-ref" href="#DefinitionHomotopicalDatum"></a>) given by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>J</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(J, \emptyset)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Lawvere+cylinder">Lawvere cylinder</a>(Definition <a class="maruku-ref" href="#LawvereCylinder"></a>), generates a Cisinki model structure on the category of presheaves on <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>. By (<a href="#Cisinski06">Cisinski 06, rem. 1.3.15</a>), a set of generating anodyne extensions is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>Ω</mi><msub><mo>∪</mo> <mi>A</mi></msub><mi>B</mi><mo>→</mo><mi>B</mi><mo>×</mo><mi>Ω</mi></mrow><annotation encoding="application/x-tex">A\times \Omega \cup_A B \to B \times \Omega</annotation></semantics></math> where <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 \to B</annotation></semantics></math> are generators for the class of monomorphisms. This is sometimes known as the <span class="newWikiWord">minimal Cisinski model structure<a href="/nlab/new/minimal+Cisinski+model+structure">?</a></span>. See also Theorem <a class="maruku-ref" href="#TheoremCharacterisationOfAccessibleLocalisersForPresheafCategories"></a>.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="properness">Properness</h3> <div class="num_theorem" id="RightProper"> <h6 id="theorem_2">Theorem</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\Sigma)</annotation></semantics></math> is the localizer generated by a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> in a Grothendieck topos, then (the Cisinski model structure whose weak equivalences are) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\Sigma)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper</a> if and only if pullback along fibrations between fibrant objects in this model structure takes morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to weak equivalences.</p> </div> <p>This is <a href="#Cisinski02">Cisinski 02, Théorème 4.8</a>. This is closely related to (but not a consequence of) a theorem of Bousfield that applies to any model category whatsoever; see <a class="existingWikiWord" href="/nlab/show/proper+model+category#FibrationsBetweenFibrantObjectsSuffice">this proposition</a>.</p> <p>We can use this to prove that a <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a> is <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+%28%E2%88%9E%2C1%29-category">locally cartesian closed (∞,1)-category</a> if and only if it has a presentation by a right proper Cisinski model structure. See <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+%28%E2%88%9E%2C1%29-category">locally cartesian closed (∞,1)-category</a> for details.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/test+topos">test topos</a></li> </ul> <h2 id="References">References</h2> <p>The original articles are</p> <ul> <li id="Cisinski02"> <p><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a> , <em>Théories homotopiques dans les topos</em>, JPAA, Volume 174 (2002), p.43-82</p> </li> <li id="Cisinski06"> <p><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em>Les préfaisceaux comme types d’homotopie</em>, Astérisque <strong>308</strong> Soc. Math. France (2006), 392 pages [<a href="http://www.numdam.org/item/?id=AST_2006__308__R1_0">numdam:AST_2006__308__R1_0</a> <a href="http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf">pdf</a>]</p> </li> </ul> <p>A more recent, English presentation of much of this material appears in section 2.4 of</p> <ul> <li id="Cisinski20"><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em>Higher category theory and homotopical algebra</em> (<a href="http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf">pdf</a>)</li> </ul> <p>A textbook account:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Garth+Warner">Garth Warner</a>, <em>Homotopical Topos Theory</em> (2012) [<a href="https://sites.math.washington.edu//~warner/HTT_Warner.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Waner-HomotopicalTopos.pdf" title="pdf">pdf</a>]</li> </ul> <p>Further developments:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>, <em>Categorical homotopy theory</em> (2003) (<a href="http://www.math.uiuc.edu/K-theory/0669/">K-theory</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marc+Olschok">Marc Olschok</a>, <em>On constructions of left determined model structures</em>, PhD thesis (2009) (<a href="http://is.muni.cz/th/183259/prif_d/diss.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Henry">Simon Henry</a>, <em>Minimal model structures</em>, <a href="https://arxiv.org/abs/2011.13408">arXiv:2011.13408</a>.</p> </li> </ul> <p>Some work on generalizing from presheaf toposes to all toposes is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em>Faisceaux localement asphériques</em> (preliminary version), 2003, <a href="http://www.mathematik.uni-regensburg.de/cisinski/mtest2.pdf">pdf</a></li> </ul> <p>See also</p> <ul id="Ara"> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <em><a class="existingWikiWord" href="/joyalscatlab/published/Cisinski%27s+theory">Cisinski's theory</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dimitri+Ara">Dimitri Ara</a>, <em>Higher quasi-categories vs higher Rezk spaces</em> (<a href="http://arxiv.org/abs/1206.4354">arXiv:1206.4354</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 27, 2024 at 13:13:50. 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