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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> model structure on functors </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/1054/#Item_13" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="model_category_theory">Model category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></strong>, <a class="existingWikiWord" href="/nlab/show/model+%28infinity%2C1%29-category">model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-category</a></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/resolution">resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">of a model category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a></p> </li> </ul> <p><strong>Refinements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+category">locally cartesian closed model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> </ul> <p><strong>Producing new model structures</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a></p> </li> </ul> <p><strong>Presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a></p> </li> </ul> <p><strong>Model structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+semi-simplicial+sets">on semi-simplicial sets</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-groupoids">on strict ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/natural+model+structure+on+groupoids">on groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">on chain complexes</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a></p> </li> </ul> <p><em>for equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>)</p> </li> </ul> <p><em>for rational <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> <p><em>for rational equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-groupoids</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-types</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <p><em>general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">on simplicial T-algebras</a>, on <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> </ul> <p><em>specific <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">on dg-algebras</a> and on <a class="existingWikiWord" href="/nlab/show/simplicial+ring">on simplicial rings</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">on cosimplicial rings</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a></p> </li> </ul> <p><em>for stable/spectrum objects</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+quasi-categories">Joyal model for quasi-categories</a> (and its <a class="existingWikiWord" href="/nlab/show/model+structure+for+cubical+quasicategories">cubical version</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a></p> </li> </ul> <p><em>for stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-operads</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">on operads</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">for Segal operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">for dendroidal complete Segal spaces</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">for dendroidal Cartesian fibrations</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a>/<a class="existingWikiWord" href="/nlab/show/Cech+model+structure+on+simplicial+presheaves">Cech model structure</a>/<a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#existence'>Existence</a></li> <ul> <li><a href='#projective_case'>Projective case</a></li> <li><a href='#accessible_case'>Accessible case</a></li> <li><a href='#CombinatorialCase'>Combinatorial case</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#relation_to_other_model_structures'>Relation to other model structures</a></li> <li><a href='#functoriality_in_domain_and_codomain'>Functoriality in domain and codomain</a></li> <li><a href='#relation_to_homotopy_kan_extensionslimitscolimits'>Relation to homotopy Kan extensions/limits/colimits</a></li> <li><a href='#RelationToInfinityFunctorCategories'>Relation to <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>-functor categories</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures on <a class="existingWikiWord" href="/nlab/show/functor+category">functor categories</a> are models for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-categories of (∞,1)-functors</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> 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> any <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> there are two “obvious” ways to put a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math>, called the <em>projective</em> and the <em>injective</em> model structures. For completely general <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>, neither one need exist, but there are rather general conditions that ensure their existence. In particular, the projective model structure exists as long as <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 class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a>, while both model structures exist if <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 class="existingWikiWord" href="/nlab/show/accessible+model+category">accessible</a> (and in particular if it is <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial</a>). In the case of <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> diagrams, additional cofibrancy-type conditions are required on <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> <p>A related kind of model structure is the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a>/<a class="existingWikiWord" href="/nlab/show/generalized+Reedy+model+structure">generalized Reedy model structure</a> on functor categories, which applies for <em>any</em> model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, but requires <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 a very special sort of category, namely a <a class="existingWikiWord" href="/nlab/show/Reedy+category">Reedy category</a>/<a class="existingWikiWord" href="/nlab/show/generalized+Reedy+category">generalized Reedy category</a>.</p> <p>In the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> is the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a> the projective and injective model structure on the functor categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>SSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,SSet]</annotation></semantics></math> are described in more detail at <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure on simplicial presheaves</a> and <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">model structure on sSet-enriched presheaves</a>.</p> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>, 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 an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/model+category">model category</a> that is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>, and let <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> be a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-enriched category. Usually we have either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathbf{S}=Set</annotation></semantics></math> or else <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math> denote the enriched <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>, whose objects are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-enriched functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>⟶</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">D\longrightarrow C</annotation></semantics></math>.</p> <div class="num_defn" id="ProjectiveAndInjectiveStructure"> <h6 id="definition_2">Definition</h6> <p>We define the following classes of maps in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math>:</p> <ul> <li>the <strong>projective weak equivalences</strong> and <strong>projective fibrations</strong> are the <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> that are objectwise such morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</li> <li>the <strong>injective weak equivalences</strong> and <strong>injective cofibrations</strong> are the <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> that are objectwise such morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</li> </ul> <p>If either of these choices defines a model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math>, we call it the <strong>projective model structure</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{proj}</annotation></semantics></math> or <strong>injective model structure</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{inj}</annotation></semantics></math> respectively. Of course, the projective cofibrations and injective fibrations can then be characterized by lifting properties.</p> </div> <h2 id="existence">Existence</h2> <h3 id="projective_case">Projective case</h3> <p>The projective model structure can be regarded as a right-<a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a>. This yields the following basic result on its existence.</p> <p> <div class='num_theorem' id='ExistenceOfProjectiveStructureOnEnrichedFunctors'> <h6>Theorem</h6> <p><strong>(existence and cofibrant generation of projective structure on enriched functors)</strong> <br /> Given</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cosmos+for+enrichment">cosmos for enrichment</a>,</p> </li> <li> <p><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>, <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> a pair of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-enriched categories</p> </li> </ul> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> carries the <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a>,</p> </li> <li> <p>which admits <a class="existingWikiWord" href="/nlab/show/copowers">copowers</a> by the <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">D(x,y)\in \mathbf{S}</annotation></semantics></math>, which preserve <a class="existingWikiWord" href="/nlab/show/acyclic+cofibrations">acyclic cofibrations</a>.</p> <p>(This is the case for instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{S}=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, or if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/enriched+model+category"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mstyle mathvariant="bold"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">\mathbf{S}</annotation> </semantics> </math>-model category</a>, and the <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(x,y)</annotation></semantics></math> are cofibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>.)</p> </li> </ol> <p>Then the projective model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{proj}</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a> exists, and is again cofibrantly generated.</p> </div> (For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{S} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Sets">Sets</a> this is <a href="#Hirschhorn02">Hirschhorn 2002, Thm. 11.6.1</a>.). <div class='proof'> <h6>Proof</h6> <p>Assuming the existence of such copowers, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ob</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\in ob(D)</annotation></semantics></math> the “<a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> at <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>” functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mi>x</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo>⟶</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">ev_x \colon [D,C]\longrightarrow C</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">F_x</annotation></semantics></math> sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A\in C</annotation></semantics></math> to the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>↦</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊙</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">y\mapsto D(x,y)\odot A</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊙</mo></mrow><annotation encoding="application/x-tex">\odot</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/copower">copower</a>. Now if <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> and <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> are generating sets of cofibrations and trivial cofibrations for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">I^D</annotation></semantics></math> be the set of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_x(i)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math>, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i\in I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ob</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\in ob(D)</annotation></semantics></math>, and similarly 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>. Then the projective fibrations and trivial fibrations are characterized by having the right lifting property with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">J^D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">I^D</annotation></semantics></math> respectively, while both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">I^D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">J^D</annotation></semantics></math> permit the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> since <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> and <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> do and colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math> are pointwise. Since the trivial fibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math> clearly coincide with the fibrations that are weak equivalences, it remains only to show that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">J^D</annotation></semantics></math>-cell complexes are weak equivalences. But a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">J^D</annotation></semantics></math>-cell complex is objectwise a cell complex built from cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊙</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">D(x,y)\odot j</annotation></semantics></math> for maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j\in J</annotation></semantics></math>, and the assumption ensures that these are trivial cofibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, hence so is any cell complex built from them.</p> </div> </p> <p>There do exist projective model structures that do not fall under this theorem, however, such as the following.</p> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>If <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 <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a> <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> with its <a class="existingWikiWord" href="/nlab/show/2-trivial+model+structure">2-trivial model structure</a> 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> is a small 2-category, then the projective model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math> exists.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This follows from the result of <a href="#Lack06">Lack</a> on <a class="existingWikiWord" href="/nlab/show/transferred+model+structures">transferred model structures</a> for algebras over <a class="existingWikiWord" href="/nlab/show/2-monads">2-monads</a>, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math> is the category of algebras for an accessible 2-monad on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><mi>ob</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">C^{ob(D)}</annotation></semantics></math>.</p> </div> <p>Note that <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> need not be cofibrantly generated (and the 2-trivial model structure often fails to be cofibrantly generated), so the generality of this result is not entirely included in the previous one.</p> <h3 id="accessible_case">Accessible case</h3> <p>In the case when <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 an <a class="existingWikiWord" href="/nlab/show/accessible+model+category">accessible model category</a>, i.e. it is a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> and its constituent <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a> have <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible</a> realizations as <a class="existingWikiWord" href="/nlab/show/functorial+factorizations">functorial factorizations</a>, we have the following general result from <a href="#Moser">Moser</a> (the unenriched case appears in <a href="#HKRS15">HKRS15</a> and <a href="#GKR18">GKR18</a>).</p> <p> <div class='num_theorem' id='Accessible'> <h6>Theorem</h6> <p>Let</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a> <a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a></p> </li> <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> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-cocomplete locally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-presentable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> that is an <a class="existingWikiWord" href="/nlab/show/accessible+model+category">accessible model category</a>,</p> </li> <li> <p><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> a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-category.</p> </li> </ul> <p>Then:</p> <ol> <li> <p>If <a class="existingWikiWord" href="/nlab/show/copowers">copowers</a> by the <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(x,y)</annotation></semantics></math> preserve <a class="existingWikiWord" href="/nlab/show/acyclic+cofibrations">acyclic cofibrations</a>, then the projective model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math> exists and is accessible.</p> </li> <li> <p>If <a class="existingWikiWord" href="/nlab/show/copowers">copowers</a> by the hom-objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(x,y)</annotation></semantics></math> preserve <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a>, then the injective model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math> exists and is accessible</p> </li> </ol> <p></p> </div> </p> <h3 id="CombinatorialCase">Combinatorial case</h3> <p>Every <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a> (i.e. locally presentable and cofibrantly generated) is accessible, so Theorem <a class="maruku-ref" href="#Accessible"></a> shows that both model structures exist, and Theorem <a class="maruku-ref" href="#CofGenProj"></a> shows that the projective model structure is cofibrantly generated, hence (by <a href="locally+presentable+category#PresheavesWithValuesInLocPresAreLocPres">this Prop.</a>) also <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial</a>.</p> <p>In fact the injective model structure is also combinatorial, although the proof is much more involved, because there is no explicit description of the generating cofibrations and acyclic cofibrations; they have to be produced by a cardinality argument. This was first proven by in <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. A.2.8.2 and A.3.3.2</a> under strong assumptions on the enriching category (in particular that all objects are cofibrant), and later generalized by <a href="#MakkaiRosický14">Makkai & Rosický 2014</a> to essentially the following:</p> <p> <div class='num_theorem' id='Combinatorial'> <h6>Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a> <a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</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> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-cocomplete locally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-presentable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> that is a <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a>, 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> a small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-category. Then:</p> <ol> <li> <p>If <a class="existingWikiWord" href="/nlab/show/copowers">copowers</a> by the hom-objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(x,y)</annotation></semantics></math> preserve trivial cofibrations, then the projective model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math> exists and is combinatorial.</p> </li> <li> <p>If <a class="existingWikiWord" href="/nlab/show/copowers">copowers</a> by the hom-objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(x,y)</annotation></semantics></math> preserve cofibrations, then the injective model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,C]</annotation></semantics></math> exists and is combinatorial.</p> </li> </ol> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>It suffices to construct the factorizations, which follows from <a href="#MakkaiRosický14">Makkai & Rosický 2014, Remark 3.8</a> about left-lifting combinatorial weak factorization systems.</p> </div> </p> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The projective and injective structures <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{proj}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{inj}</annotation></semantics></math>, def. <a class="maruku-ref" href="#ProjectiveAndInjectiveStructure"></a>, are (insofar as they exist):</p> <ul> <li> <p>right or left <a class="existingWikiWord" href="/nlab/show/proper+model+categories">proper model categories</a> if <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 right or left proper, respectively.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+model+categories">enriched model categories</a> if <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 an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-model category.</p> </li> </ul> </div> <p>The statement about properness appears as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, remark A.2.8.4</a>.</p> <div class="num_prop" id="PresentationOfInfinityFunctors"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/combinatorial+simplicial+model+category">combinatorial simplicial model category</a>, the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented by</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{proj}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{inj}</annotation></semantics></math> under the above assumptions is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><msup><mi>C</mi> <mo>∘</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(D,C^\circ)</annotation></semantics></math> from the ordinary category <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 the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category presented by <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> </div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></em> for more.</p> <h3 id="relation_to_other_model_structures">Relation to other model structures</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>If <a class="existingWikiWord" href="/nlab/show/copowers">copowers</a> by the hom-objects of <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> preserve trivial cofibrations, then every every fibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{inj}</annotation></semantics></math> is in particular a fibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{proj}</annotation></semantics></math>. Similarly, if <a class="existingWikiWord" href="/nlab/show/powers">powers</a> by the hom-objects of <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> preserve trivial fibrations, then every cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{proj}</annotation></semantics></math> is in particular a cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{inj}</annotation></semantics></math>. The hypotheses are satisfied if <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> is unenriched, or in the monoidal model category case if the hom-objects of <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> are cofibrant.</p> </div> <p>This is argued in the beginning of the proof of <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, lemma A.2.8.3</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>-enriched functors, this is (<a href="#Piacenza91">Piacenza 91, section 5</a>). For details see at <em><a href="classical+model+structure+on+topological+spaces#ModelStructureOnTopEnrichedFunctors">classical model structure on topological spaces – Model structure on enriched functors</a></em>.</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>If <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>B</mi></mrow><annotation encoding="application/x-tex">i:A\longrightarrow B</annotation></semantics></math> is a trivial cofibration 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ob</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\in ob(D)</annotation></semantics></math>, then the first assumption implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>F</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>F</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_x(i) : F_x(A) \longrightarrow F_x(B)</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊙</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">F_x(A) (y) = D(x,y) \odot A</annotation></semantics></math> the left adjoint of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mi>x</mi></msub><mo>:</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo>⟶</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">ev_x : [D,C] \longrightarrow C</annotation></semantics></math>, is a trivial cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{inj}</annotation></semantics></math>. Thus, any fibration <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{inj}</annotation></semantics></math> has the right lifting property with respect to it, which is to say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ev_x(p)</annotation></semantics></math> has the right lifting property with respect to <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>. Since this is true for any <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>, each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ev_x(p)</annotation></semantics></math> is a fibration, i.e. <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 in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">[D,C]_{inj}</annotation></semantics></math>. The other half is dual.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>The <a class="existingWikiWord" href="/nlab/show/identity+functors">identity functors</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><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mover><munder><mo>⟶</mo><mi>Id</mi></munder><mover><mo>⟵</mo><mi>Id</mi></mover></mover><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [D,C]_{inj} \stackrel{\overset{Id}{\longleftarrow}}{\underset{Id}{\longrightarrow}} [D,C]_{proj} </annotation></semantics></math></div> <p>form a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Id</mi><mo>:</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">Id : [D,C]_{proj} \longrightarrow [D,C]_{inj}</annotation></semantics></math> being the left Quillen functor).</p> <p>If <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> is a <a class="existingWikiWord" href="/nlab/show/Reedy+category">Reedy category</a> this factors through the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</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><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mover><munder><mo>⟶</mo><mi>Id</mi></munder><mover><mo>⟵</mo><mi>Id</mi></mover></mover><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub><mover><munder><mo>⟶</mo><mi>Id</mi></munder><mover><mo>⟵</mo><mi>Id</mi></mover></mover><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [D,C]_{inj} \stackrel{\overset{Id}{\longleftarrow}}{\underset{Id}{\longrightarrow}} [D,C]_{Reedy} \stackrel{\overset{Id}{\longleftarrow}}{\underset{Id}{\longrightarrow}} [D,C]_{proj} </annotation></semantics></math></div></div> <h3 id="functoriality_in_domain_and_codomain">Functoriality in domain and codomain</h3> <div class="num_prop" id="QuillenFunctorialityInCodomain"> <h6 id="proposition_4">Proposition</h6> <p>The functor model structures depend <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen-functorially</a> on their <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a>, in that for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>1</mn></msub><mover><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></mover><msub><mi>D</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> D_1 \stackrel {\overset{L}{\longleftarrow}} {\underset{R}{\longrightarrow}} D_2 </annotation></semantics></math></div> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a> between <a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+model+categories">enriched model categories</a>, postcomposition induces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunctions">enriched Quillen adjunctions</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><mi>C</mi><mo>,</mo><msub><mi>D</mi> <mn>1</mn></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mover><munder><mo>⟶</mo><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>R</mi><mo stretchy="false">]</mo></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>L</mi><mo stretchy="false">]</mo></mrow></mover></mover><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><msub><mi>D</mi> <mn>2</mn></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [C,D_1]_{proj} \stackrel {\overset{[C,L]}{\longleftarrow}} {\underset{[C,R]}{\longrightarrow}} [C,D_2]_{proj} </annotation></semantics></math></div> <p>and</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>C</mi><mo>,</mo><msub><mi>D</mi> <mn>1</mn></msub><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mover><munder><mo>⟶</mo><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>R</mi><mo stretchy="false">]</mo></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>L</mi><mo stretchy="false">]</mo></mrow></mover></mover><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><msub><mi>D</mi> <mn>2</mn></msub><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [C,D_1]_{inj} \stackrel {\overset{[C,L]}{\longleftarrow}} {\underset{[C,R]}{\longrightarrow}} [C,D_2]_{inj} \,. </annotation></semantics></math></div> <p>Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L \dashv R)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>, then so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>L</mi><mo stretchy="false">]</mo><mo>⊣</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>R</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">([C,L] \dashv [C,R])</annotation></semantics></math>.</p> </div> <p>For the case that <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 small category this is [<a href="#Lurie">Lurie, remark A.2.8.6</a>], for the enriched case this is [<a href="#Lurie">Lurie, prop. A.3.3.6</a>].</p> <p>The Quillen-functoriality on the <a class="existingWikiWord" href="/nlab/show/domain">domain</a> is more asymmetric.</p> <p> <div class='num_prop' id='QuillenFunctorialityInDomain'> <h6>Proposition</h6> <p><strong>(Quillen functoriality in the domain category)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p \colon C_1 \longrightarrow C_2</annotation></semantics></math> a functor between small categories or an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>, let</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><msub><mi>p</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>p</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>p</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mover><mover><munder><mo>⟵</mo><mrow><msub><mi>p</mi> <mo>*</mo></msub></mrow></munder><mover><mo>⟶</mo><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow></mover></mover><mover><mo>⟵</mo><mrow><msub><mi>p</mi> <mo>!</mo></msub></mrow></mover></mover><mo stretchy="false">[</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (p_! \dashv p^* \dashv p_*) \;\colon\; [C_2,D] \stackrel{\overset{p_!}{\longleftarrow}}{\stackrel{\overset{p^*}{\longrightarrow}}{\underset{p_*}{\longleftarrow}}} [C_1,D] </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math> is precomposition with <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> and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">p_!</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">p_*</annotation></semantics></math> are left and right <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> along <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>, respectively.</p> <p>Then we have <a class="existingWikiWord" href="/nlab/show/Quillen+adjunctions">Quillen adjunctions</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><msub><mi>p</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>p</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><mi>D</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mover><munder><mo>⟵</mo><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mo>!</mo></msub></mrow></mover></mover><mo stretchy="false">[</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><mi>D</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> (p_! \dashv p^*) \;\colon\; [C_1,D]_{proj} \stackrel{\overset{p_!}{\longrightarrow}}{\underset{p^*}{\longleftarrow}} [C_2,D]_{proj} </annotation></semantics></math></div> <p>and</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><msup><mi>p</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>p</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><mi>D</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mover><munder><mo>⟶</mo><mrow><msub><mi>p</mi> <mo>*</mo></msub></mrow></munder><mover><mo>⟵</mo><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow></mover></mover><mo stretchy="false">[</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><mi>D</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (p^* \dashv p_*) \;\colon\; [C_1,D]_{inj} \stackrel{\overset{p^*}{\longleftarrow}}{\underset{p_*}{\longrightarrow}} [C_2,D]_{inj} \,. </annotation></semantics></math></div> <p></p> </div> </p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> not enriched this appears as [<a href="#Lurie">Lurie, prop. A.2.8.7</a>], for the enriched case it appears as [<a href="#Lurie">Lurie, prop. A.3.3.7</a>].</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>-enriched case, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><msub><mi>D</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>D</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p : D_1 \longrightarrow D_2</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">model structure on sSet-categories</a>, then these two Quillen adjunctions are both <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>s.</p> </div> <h3 id="relation_to_homotopy_kan_extensionslimitscolimits">Relation to homotopy Kan extensions/limits/colimits</h3> <p>Often functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>⟶</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">D \longrightarrow C</annotation></semantics></math> are thought of as <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>s in the model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, and one is interested in obtaining their <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> or <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> or, generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>D</mi><mo>⟶</mo><mi>D</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p : D \longrightarrow D'</annotation></semantics></math> any functor, their left and right <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a>.</p> <p>These are the left and right <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>HoLan</mi><mo>:</mo><mo>=</mo><mi>𝕃</mi><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">HoLan := \mathbb{L} p_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>HoRan</mi><mo>:</mo><mo>=</mo><mi>ℝ</mi><msub><mi>p</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">HoRan := \mathbb{R} p_*</annotation></semantics></math> of</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>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mo>!</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>D</mi><mo>′</mo><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [D,C]_{proj} \stackrel{p_!}{\longrightarrow} [D',C]_{proj} </annotation></semantics></math></div> <p>and</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>D</mi><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mo>*</mo></msub></mrow></mover><mo stretchy="false">[</mo><mi>D</mi><mo>′</mo><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex"> [D,C]_{inj} \stackrel{p_*}{\longrightarrow} [D',C]_{inj} </annotation></semantics></math></div> <p>respectively.</p> <p>For more on this see <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a>. For the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>′</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">D' = *</annotation></semantics></math> this reduces to <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a>.</p> <h3 id="RelationToInfinityFunctorCategories">Relation to <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>-functor categories</h3> <p> <div class='num_prop'> <h6>Proposition</h6> <p>For</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/combinatorial+simplicial+model+category">combinatorial simplicial model category</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>D</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{D}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/sSet-enriched+category">sSet-enriched category</a></p> </li> </ul> <p>then both the projective and the injective model structure on the <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sFunc</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sFunc(\mathbf{D}, \mathbf{C})</annotation></semantics></math> (which exist by the above discussion) <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">present</a> the corresponding <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors"><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 of <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>-functors</a>. Concretely:</p> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a> of the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full</a> <a class="existingWikiWord" href="/nlab/show/sSet-enriched+category">sSet-enriched</a> <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of the functor model category on its <a class="existingWikiWord" href="/nlab/show/bifibrant+objects">bifibrant objects</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mstyle mathvariant="bold"><mi>sFunc</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">)</mo><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\big(\mathbf{sFunc}(\mathbf{D}, \mathbf{C})\big)^\circ</annotation></semantics></math>, is <a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-functor+%28infinity%2C1%29-category"><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>-functor</a> <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> between the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerves">homotopy coherent nerves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mstyle mathvariant="bold"><mi>sFunc</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mo>≃</mo><mi>qCat</mi></munder><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Func</mi> <mn>∞</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>N</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>N</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mo>∘</mo></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N \big( \mathbf{sFunc}(\mathbf{D}, \mathbf{C})^\circ \big) \;\; \underset{qCat}{\simeq} \;\; Func_\infty \Big( N(\mathbf{D}) ,\, N\big( \mathbf{C}^\circ \big) \Big) \,. </annotation></semantics></math></div> <p></p> </div> This is due to <a href="#Lurie">Lurie (2009), Prop. 4.2.4.4</a>. See also the discussion <em><a href="infinity1-category+of+infinity1-functors#ModelCategoryPresentation">here</a></em> at <em><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors"><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 of <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>-functors</a></em>.</p> <h2 id="examples">Examples</h2> <p>Examples of cofibrant objects in the projective model structure are discussed at <em><a class="existingWikiWord" href="/nlab/show/projectively+cofibrant+diagram">projectively cofibrant diagram</a></em>.</p> <p> <div class='num_remark'> <h6>Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>)</strong> <br /> Model structures on <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-valued <a class="existingWikiWord" href="/nlab/show/functors">functors</a>, hence on <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a>, play a central role in the theory of <a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a> and specifically in <a class="existingWikiWord" href="/nlab/show/model+topos">model topos</a>-theory presenting <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos"><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>-toposes</a>. See at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></em> for more.</p> </div> </p> <p>Specifically:</p> <p> <div class='num_remark' id='BorelModelStructureOnSimplicialGroupActions'> <h6>Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a> on <a class="existingWikiWord" href="/nlab/show/simplicial+group+actions">simplicial group actions</a>)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>Grp</mi><mo stretchy="false">(</mo><mi>sSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{G} \,\in\, Grp(sSet)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> with <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+groupoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/delooping+groupoid">delooping groupoid</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝒢</mi><mo>∈</mo><mi>sSet</mi><mtext>-</mtext><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathcal{G} \in sSet\text{-}Grpd</annotation></semantics></math>, an <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝒢</mi><mo>⟶</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathcal{G} \longrightarrow sSet</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/simplicial+group+action">simplicial group action</a> of <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">\mathcal{G}</annotation></semantics></math>.</p> <p>Under this identification, the projective model strcuture on <a class="existingWikiWord" href="/nlab/show/simplicial+functors">simplicial functors</a> (Prop. <a class="maruku-ref" href="#ExistenceOfProjectiveStructureOnEnrichedFunctors"></a>) is equivalently the <em><a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a></em> on <a class="existingWikiWord" href="/nlab/show/simplicial+group+actions">simplicial group actions</a>, a context of Borel-<a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sFunc</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝒢</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>sSet</mi><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>proj</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>𝒢</mi><mi>Act</mi><mo stretchy="false">(</mo><mi>sSet</mi><msub><mo stretchy="false">)</mo> <mi>Borel</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sFunc\big( \mathbf{B}\mathcal{G} ,\, sSet \big)_{proj} \;\; = \;\; \mathcal{G}Act(sSet)_{Borel} \,. </annotation></semantics></math></div> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi><mo>∈</mo><mi>sSet</mi><mtext>-</mtext><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathcal{X} \in sSet\text{-}Grpd</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+groupoid">enriched groupoid</a> (Dwyer-Kan <a class="existingWikiWord" href="/nlab/show/simplicial+groupoid">simplicial groupoid</a>) with a single <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>𝒳</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\pi_0(\mathcal{X}) \simeq \{[x]\}</annotation></semantics></math>, so that the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>𝒳</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>↪</mo><mphantom><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></mphantom></mover><mi>𝒳</mi></mrow><annotation encoding="application/x-tex"> \iota \,\colon\, \mathbf{B}(\mathcal{X}(x,x)) \xhookrightarrow{\phantom{---}} \mathcal{X} </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+adjoint+equivalence">enriched adjoint equivalence</a> (see discussion <a href="simplicial+groupoid#RelationToSimplicialGroups">there</a>) the projective model structure on <a class="existingWikiWord" href="/nlab/show/simplicial+functors">simplicial functors</a> from Prop. <a class="maruku-ref" href="#ExistenceOfProjectiveStructureOnEnrichedFunctors"></a> is <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred</a> under the induced <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a> of <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor+categories">enriched functor categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sFunc</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒳</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>sSet</mi><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>proj</mi></msub><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mo>⊥</mo> <mo>≃</mo></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mrow><msup><mi>ι</mi> <mo>*</mo></msup></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>ι</mi> <mo>!</mo></msub></mrow></mover></munderover><mi>sFunc</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒳</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>sSet</mi><msub><mo maxsize="1.8em" minsize="1.8em">)</mo> <mi>proj</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒳</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mi>Act</mi><mo stretchy="false">(</mo><mi>sSet</mi><msub><mo stretchy="false">)</mo> <mi>Borel</mi></msub></mrow><annotation encoding="application/x-tex"> sFunc\big( \mathcal{X} ,\, sSet \big)_{proj} \underoverset {\underset{\iota^\ast}{\longrightarrow}} {\overset{\iota_!}{\longleftarrow}} {\;\; \bot_{\simeq} \;\;} sFunc\Big( \mathbf{B}\big(\mathcal{X}(x,x)\big) ,\, sSet \Big)_{proj} \;=\; \big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel} </annotation></semantics></math></div> <p>By <a href="transferred+model+structure#RightTransferAlongAdjointEquivalence">this example</a> it follows that morphisms in all three classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">W</mi><mo>,</mo><mi>Fib</mi><mo>,</mo><mi>Cof</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathrm{W}, Fib, Cof)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sFunc</mi><mo stretchy="false">(</mo><mi>𝒳</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>sSet</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">sFunc(\mathcal{X}, \, sSet)_{proj}</annotation></semantics></math> are those which restrict on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒳</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in Obj(\mathcal{X})</annotation></semantics></math> to the respective class in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒳</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mi>Act</mi><mo stretchy="false">(</mo><mi>sSet</mi><msub><mo stretchy="false">)</mo> <mi>Borel</mi></msub></mrow><annotation encoding="application/x-tex">\big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel}</annotation></semantics></math>.</p> <p>It follows that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>sSet</mi><mtext>-</mtext><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathcal{X} \,\in\, sSet\text{-}Grpd</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+groupoid">simplicial groupoid</a> with any set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>𝒳</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\mathcal{X})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a>, the projective model structure of simplicial functors over it is the <a href="model+category#ProductModelStructure">product model structure</a> of the <a class="existingWikiWord" href="/nlab/show/Borel+model+structures">Borel model structures</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+group+actions">simplicial group actions</a>, one for each connected component:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sFunc</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒳</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>sSet</mi><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>proj</mi></msub><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mo>⊥</mo> <mo>≃</mo></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mrow><msup><mi>ι</mi> <mo>*</mo></msup></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>ι</mi> <mo>!</mo></msub></mrow></mover></munderover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>𝒳</mi><mo stretchy="false">)</mo></mrow></munder><mi>sFunc</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒳</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>sSet</mi><msub><mo maxsize="1.8em" minsize="1.8em">)</mo> <mi>proj</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>𝒳</mi><mo stretchy="false">)</mo></mrow></munder><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝒳</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mi>Act</mi><mo stretchy="false">(</mo><mi>sSet</mi><msub><mo stretchy="false">)</mo> <mi>Borel</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sFunc\big( \mathcal{X} ,\, sSet \big)_{proj} \underoverset {\underset{\iota^\ast}{\longrightarrow}} {\overset{\iota_!}{\longleftarrow}} {\;\; \bot_{\simeq} \;\;} \underset{i \in \pi_0(\mathcal{X})}{\prod} sFunc\Big( \mathbf{B}\big(\mathcal{X}(x_i,x_i)\big) ,\, sSet \Big)_{proj} \;=\; \underset{i \in \pi_0(\mathcal{X})}{\prod} \big(\mathcal{X}(x_i,x_i)\big) Act(sSet)_{Borel} \,. </annotation></semantics></math></div> <p></p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a>, <a class="existingWikiWord" href="/nlab/show/generalized+Reedy+model+structure">generalized Reedy model structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sections">model structure on sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">model structure on homotopical presheaves</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+n-excisive+functors">model structure for n-excisive functors</a></p> </li> </ul> <h2 id="references">References</h2> <p>The projective model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Top_{Quillen}</annotation></semantics></math>-enriched functors is discussed in</p> <ul> <li id="Piacenza91"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Piacenza">Robert Piacenza</a> section 5 of <em>Homotopy theory of diagrams and CW-complexes over a category</em>, Can. J. Math. Vol 43 (4), 1991 (<a class="existingWikiWord" href="/nlab/files/Piazenza91.pdf" title="pdf">pdf</a>)</p> <p>also chapter VI of <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a> et al., <em>Equivariant homotopy and cohomology theory</em>, 1996 (<a href="http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf">pdf</a>)</p> </li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a>, <em>Homotopy in functor categories</em>, Transactions of the AMS, vol 272, Number 1, July 1982 (<a href="http://www.jstor.org/stable/1998955">JSTOR</a>)</li> </ul> <p>Textbook account of the projective model structure</p> <ul> <li id="Hirschhorn02"><a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, §6.11 in: <em><a class="existingWikiWord" href="/nlab/show/Model+Categories+and+Their+Localizations">Model Categories and Their Localizations</a></em>, AMS Math. Survey and Monographs <strong>99</strong> (2002) [<a href="https://bookstore.ams.org/surv-99-s/">ISBN:978-0-8218-4917-0</a>, <a href="http://www.gbv.de/dms/goettingen/360115845.pdf">pdf toc</a>, <a href="https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf">pdf</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf">pdf</a>]</li> </ul> <p>The injective model structure for unenriched diagrams of simplicial sets was first constructed by</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a>, <em>Homotopy theories</em></li> </ul> <p>Probably the first general construction of injective model structures for enriched diagrams in combinatorial model categories was in</p> <ul> <li id="Lurie"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, sections A.2.8 (unenriched) and section A.3.3 (enriched) of <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em>, 2009</li> </ul> <p>The projective model structure for functors to <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> on a <em><a class="existingWikiWord" href="/nlab/show/large+category">large</a></em> domain is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Boris+Chorny">Boris Chorny</a>, <a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <em>Homotopy theory of small diagrams over large categories</em>, <a href="http://arxiv.org/abs/math/0607117">arXiv:math/0607117</a></li> </ul> <p>The case of diagrams in a 2-category is a special case of</p> <ul> <li id="Lack06"><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>Homotopy-theoretic aspects of 2-monads</em>, <a href="http://arxiv.org/abs/math.CT/0607646">arXiv</a></li> </ul> <p>The use of accessible model structures to construct both projective and injective model structures on unenriched diagrams was introduced in</p> <ul> <li> <p>Marzieh Bayeh, <a class="existingWikiWord" href="/nlab/show/Kathryn+Hess">Kathryn Hess</a>, <a class="existingWikiWord" href="/nlab/show/Varvara+Karpova">Varvara Karpova</a>, Magdalena Kędziorek, <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em>Left-induced model structures and diagram categories</em> (<a href="http://arxiv.org/abs/1401.3651">arXiv:1401.3651</a>)</p> </li> <li id="HKRS15"> <p><a class="existingWikiWord" href="/nlab/show/Kathryn+Hess">Kathryn Hess</a>, Magdalena Kędziorek, <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em>A necessary and sufficient condition for induced model structures</em> (<a href="http://arxiv.org/abs/1509.08154">arXiv:1509.08154</a>). This paper contains an error, corrected by:</p> </li> <li id="GKR18"> <p><a class="existingWikiWord" href="/nlab/show/Richard+Garner">Richard Garner</a>, Magdalena Kedziorek, <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em>Lifting accessible model structures</em>, <a href="https://arxiv.org/abs/1802.09889">arXiv:1802.09889</a></p> </li> </ul> <p>It was generalized to enriched diagrams in</p> <ul> <li id="Moser">Lyne Moser, <em>Injective and Projective Model Structures on Enriched Diagram Categories</em>, <a href="https://arxiv.org/abs/1710.11388">arXiv:1710.11388</a></li> </ul> <p>The more general result above on combinatoriality of injective model structures follows from Remark 3.8 of</p> <ul> <li id="MakkaiRosický14"><a class="existingWikiWord" href="/nlab/show/M.+Makkai">M. Makkai</a>, <a class="existingWikiWord" href="/nlab/show/J.+Rosick%C3%BD">J. Rosický</a>, <em>Cellular categories</em>, J. Pure Appl. Alg. <strong>218</strong> (2014) 1652-1664 [<a href="https://arxiv.org/abs/1304.7572">arXiv:1304.7572</a>, <a href="https://doi.org/10.1016/j.jpaa.2014.01.005">doi:10.1016/j.jpaa.2014.01.005</a>]</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+White">David White</a>, <em>Modified projective model structure</em> (<a href="http://mathoverflow.net/questions/76160/acyclic-models-via-model-categories/104423#104423">MO comment</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 8, 2024 at 14:33:43. 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