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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 simplicial presheaves </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/1352/#Item_12" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="model_category_theory">Model category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></strong>, <a class="existingWikiWord" href="/nlab/show/model+%28infinity%2C1%29-category">model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-category</a></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/resolution">resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">of a model category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a></p> </li> </ul> <p><strong>Refinements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+category">locally cartesian closed model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> </ul> <p><strong>Producing new model structures</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a></p> </li> </ul> <p><strong>Presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a></p> </li> </ul> <p><strong>Model structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+semi-simplicial+sets">on semi-simplicial sets</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-groupoids">on strict ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/natural+model+structure+on+groupoids">on groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">on chain complexes</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a></p> </li> </ul> <p><em>for equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>)</p> </li> </ul> <p><em>for rational <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> <p><em>for rational equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-groupoids</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-types</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <p><em>general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">on simplicial T-algebras</a>, on <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> </ul> <p><em>specific <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">on dg-algebras</a> and on <a class="existingWikiWord" href="/nlab/show/simplicial+ring">on simplicial rings</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">on cosimplicial rings</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a></p> </li> </ul> <p><em>for stable/spectrum objects</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+quasi-categories">Joyal model for quasi-categories</a> (and its <a class="existingWikiWord" href="/nlab/show/model+structure+for+cubical+quasicategories">cubical version</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a></p> </li> </ul> <p><em>for stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-operads</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">on operads</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">for Segal operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">for dendroidal complete Segal spaces</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">for dendroidal Cartesian fibrations</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a>/<a class="existingWikiWord" href="/nlab/show/Cech+model+structure+on+simplicial+presheaves">Cech model structure</a>/<a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks</p> </li> </ul> </div></div> <h4 id="topos_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <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/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><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/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#the_different_model_structures_and_their_interrelation'>The different model structures and their interrelation</a></li> <ul> <li><a href='#injectiveprojective__localglobal__presheavessheaves'>Injective/projective - local/global - presheaves/sheaves</a></li> <li><a href='#reedy_and_intermediate_model_structures'>Reedy and intermediate model structures</a></li> <li><a href='#DependencyOnSite'>Dependency on the underlying site</a></li> </ul> <li><a href='#PresentationOfInfiniToposes'>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>-toposes</a></li> <li><a href='#FibAndCofibObjects'>Fibrant and cofibrant objects</a></li> <ul> <li><a href='#fibrant_objects'>Fibrant objects</a></li> <li><a href='#CofibrantObjects'>Cofibrant objects</a></li> <li><a href='#CofibrantReplacement'>Cofibrant replacement</a></li> </ul> <li><a href='#local_fibrations'>Local fibrations</a></li> <li><a href='#Descent'>Localization and descent</a></li> <ul> <li><a href='#ČechLocalization'>Čech localization at Grothendieck (pre)topologies</a></li> <ul> <li><a href='#defintion'>Defintion</a></li> </ul> <li><a href='#for_values_in_strict_and_abelian_groupoids'>For values in strict and abelian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</a></li> </ul> <li><a href='#Properness'>Properness</a></li> <li><a href='#MonoidalStructure'>Closed monoidal structure</a></li> <li><a href='#HomotopyLimits'>Homotopy (co)limits</a></li> <li><a href='#InclusionOfChainComplexes'>Inclusion of chain complexes of sheaves</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><a class="existingWikiWord" href="/nlab/show/model+category">Model structures</a> on <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaves</a> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">present</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-categories of (∞,1)-presheaves</a> and <a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localizations of these</a>, such as notably the left exact localizations that are <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-categories of (∞,1)-sheaves</a>: these model structures are <a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a>.</p> <p>Recall that</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> is a way to <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">present</a> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>;</p> </li> <li> <p>in the context of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> <a class="existingWikiWord" href="/nlab/show/presheaf">presheaves</a> on an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <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> are given by <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functors">(∞,1)-functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo></mrow><annotation encoding="application/x-tex">C^{op} \to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a>.</p> </li> </ul> <p>This suggests that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a> on some <a class="existingWikiWord" href="/nlab/show/site">site</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> can be <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presented</a> by a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on the ordinary <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</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><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>SSet</mi><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [C^{op},SSet] \simeq [\Delta^{op}, PSh(C)] </annotation></semantics></math></div> <p>– the category of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaves</a> .</p> <p>Various interrelated flavors of model structures on the category of simplicial presheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> have been introduced and studied since the 1970s, originally by K. Brown and <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a> and then developed in detail by <a class="existingWikiWord" href="/nlab/show/J.+F.+Jardine">J. F. Jardine</a>.</p> <p>Notice that when regarded as a presentation of an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, i.e. of an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, a simplicial presheaf – being an ordinary functor instead of a <a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a> – corresponds to a <a class="existingWikiWord" href="/nlab/show/rectified+%E2%88%9E-stack">rectified ∞-stack</a>. It might therefore seem that a model given by simplicial presheaves is too restrictive to capture the full expected flexibility of a notion of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>. But this is not so.</p> <p>In</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></li> </ul> <p>a fully general definition of a <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves">(∞,1)-category of ∞-stacks</a> is given it is shown – proposition 6.5.2.1 – that, indeed, the Brown–Joyal–Jardine model is a <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentation</a> of that.</p> <p>More precisely</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure on simplicial presheaves</a> on a category is a <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentation</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+model+structure+on+simplicial+presheaves">Čech model structure on simplicial presheaves</a> on a <a class="existingWikiWord" href="/nlab/show/site">site</a> is a <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentation</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a> on a <a class="existingWikiWord" href="/nlab/show/site">site</a> is a <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentation</a> of the <em>hypercompletion</em> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a> (see the discussion at <a class="existingWikiWord" href="/nlab/show/hypercover">hypercover</a>).</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Bousfield+localization">Bousfield localization</a> of the global <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure to the local one presents the corresponding <a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a> from presheaves to sheaves, mimicking the corresponding statement for a <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a>.</p> </li> </ul> <p>Originally K. Brown had considered in <a class="existingWikiWord" href="/nlab/show/BrownAHT">BrownAHT</a> not a model structure on simplicial presheaves but</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> structure on locally Kan simplicial sheaves (see there for details)</li> </ul> <p>and Joyal had originally considered a</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+sheaves">local model structure on simplicial sheaves</a>.</li> </ul> <p>Joyal’s <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+sheaves">local model structure on simplicial sheaves</a> is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to the injective <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a>.</p> <p>By repackaging <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es as <a class="existingWikiWord" href="/nlab/show/simplicial+groupoids">simplicial groupoids</a> one obtains a <a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">model structure on presheaves of simplicial groupoids</a> which is also Quillen equivalent to the above.</p> <p>If K. Brown’s <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> on locally Kan simplicial sheaves is restricted to globally Kan simplicial sheaves on a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> with <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough point</a> then it is the full subcategory on the fibrant objects in the projective <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+sheaves">local model structure on simplicial sheaves</a>.</p> <p>But since in all cases the weak equivalences are the same (where they apply, for Brown’s model if the topos has enough points), all these local <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical categories</a> define equivalent <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy categories</a>.</p> <p>By Lurie’s result these are in each case in turn equivalent to the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category of</a> the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a>. So in particular they serve as a home for general <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>.</p> <p>Various old results appear in a new light this way. For instance using the old result of <a class="existingWikiWord" href="/nlab/show/BrownAHT">BrownAHT</a> on the way ordinary <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> is embedded in the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of simplicial sheaves, one sees that the old right derived functor definition of <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> really computes the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stackification">∞-stackification</a> of a sheaf of <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es regarded under the <a class="existingWikiWord" href="/nlab/show/Dold%E2%80%93Kan+correspondence">Dold–Kan correspondence</a> as a simplicial sheaf.</p> <h2 id="the_different_model_structures_and_their_interrelation">The different model structures and their interrelation</h2> <p>It is the very point of <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures on a given <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> that there may be several of them: each <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presenting</a> the same <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> but also each suited for different computational purposes.</p> <p>So it is good that there are many model structures on simplicial (pre)sheaves, as there are.</p> <h3 id="injectiveprojective__localglobal__presheavessheaves">Injective/projective - local/global - presheaves/sheaves</h3> <p>The following diagram is a map for part of the territory:</p> <div style="overflow:auto"><div class="maruku-equation"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"> <semantics> <mrow> <mrow> <mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mi>presentation</mi></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mi>presentation</mi></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mi>presentation</mi></msup></mtd></mtr> <mtr><mtd><mi>SSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>inj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>→</mo><mi>embedding</mi></mover><mover><mo>←</mo><mi>sheafification</mi></mover></mover></mtd> <mtd><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>inj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mo>←</mo><mrow></mrow></mover><mo stretchy="false">|</mo></mtd> <mtd><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>inj</mi></msub></mtd> <mtd><mover><mover><mo>→</mo><mi>Id</mi></mover><mover><mo>←</mo><mi>Id</mi></mover></mover></mtd> <mtd><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mtd> <mtd><mover><mo>↦</mo><mrow></mrow></mover></mtd> <mtd><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>proj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>←</mo><mi>embedding</mi></mover><mover><mo>→</mo><mi>sheafification</mi></mover></mover></mtd> <mtd><mi>SSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>proj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd></mtr> <mtr><mtd><mi>Joyal</mi></mtd> <mtd><mover><mo>↔</mo><mrow><mi>Quillen</mi><mi>equivalence</mi></mrow></mover></mtd> <mtd><mi>Jardine</mi></mtd> <mtd><mover><mrow><mo>←</mo><mo stretchy="false">|</mo></mrow><mrow><mi>left</mi><mi>Bousf</mi><mo>.</mo><mi>localization</mi></mrow></mover></mtd> <mtd><mi>Heller</mi></mtd> <mtd><mover><mo>↔</mo><mrow><mi>Quillen</mi><mi>equivalence</mi></mrow></mover></mtd> <mtd><mi>Bousfield</mi><mo>−</mo><mi>Kan</mi></mtd> <mtd><mover><mo>↦</mo><mrow><mi>left</mi><mi>Bousf</mi><mo>.</mo><mi>localization</mi></mrow></mover></mtd> <mtd><mi>Blander</mi></mtd> <mtd><mover><mo>↔</mo><mrow><mi>Quillen</mi><mi>equivalence</mi></mrow></mover></mtd> <mtd><mi>Brown</mi><mo>−</mo><mi>Gersten</mi></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>everything</mi><mi>cofibrant</mi><mo>;</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>fibrant</mi><mo>=</mo><mi>global</mi><mi>injective</mi><mi>fib</mi><mo>.</mo><mo>.</mo><mo>.</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mtd> <mtd><mo>.</mo><mo>.</mo><mo>.</mo><mi>satisfying</mi><mi>descent</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>cofibrant</mi><mo>=</mo><mi>global</mi><mi>projective</mi><mi>cofib</mi><mo>;</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>fibrant</mi><mo>=</mo><mi>Kan</mi><mi>valued</mi><mi>and</mi><mo>.</mo><mo>.</mo><mo>.</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>.</mo><mo>.</mo><mo>.</mo><mi>satisfying</mi><mi>descent</mi></mtd></mtr></mtable> </mrow> </mrow> <annotation encoding="application/x-tex"> \array{ &amp;&amp; (\infty,1)Sh(C) &amp;&amp;&amp; (\infty,1)PSh(C) &amp;&amp;&amp; (\infty,1)Sh(C) \\ &amp;&amp; \uparrow^{presentation} &amp;&amp;&amp; \uparrow^{presentation} &amp;&amp;&amp; \uparrow^{presentation} \\ SSh(C)^{l loc}_{inj} &amp; \stackrel{\stackrel{sheafification}{\leftarrow}} {\stackrel{embedding}{\to}}&amp; SPSh(C)^{l loc}_{inj} &amp;\stackrel{}{\leftarrow}|&amp; SPSh(C)_{inj} &amp;\stackrel{\stackrel{Id}{\leftarrow}} {\stackrel{Id}{\rightarrow}}&amp; SPSh(C)_{proj} &amp;\stackrel{}{\mapsto}&amp; SPSh(C)_{proj}^{l loc} &amp; \stackrel{\stackrel{sheafification}{\to}} {\stackrel{embedding}{\leftarrow}}&amp; SSh(C)_{proj}^{l loc} \\ Joyal &amp;\stackrel{Quillen equivalence}{\leftrightarrow}&amp; Jardine &amp;\stackrel{left Bousf. localization}{\leftarrow|}&amp; Heller &amp;\stackrel{Quillen equivalence}{\leftrightarrow}&amp; Bousfield-Kan &amp;\stackrel{left Bousf. localization}{\mapsto}&amp; Blander &amp;\stackrel{Quillen equivalence}{\leftrightarrow}&amp; Brown-Gersten \\ \\ &amp; everything cofibrant; \\ &amp; fibrant = global injective fib... \\ \;\;\; &amp; ...satisfying descent &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp; cofibrant = global projective cofib; \\ &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp; fibrant = Kan valued and... \\ &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp; \;\;\; ...satisfying descent } </annotation> </semantics> </math> </div></div> <p>Here</p> <ul> <li> <p>“inj” denotes the injective model structure: cofibrations are objectwise cofibrations</p> </li> <li> <p>“proj” denotes the projective model structure: fibrations are objectwise fibrations</p> </li> <li> <p>no “loc” subscript means global model structure: weak equivalences are the objectwise weak equivalences:</p> </li> <li> <p>“l loc” denotes <strong>left</strong> <a class="existingWikiWord" href="/nlab/show/Bousfield+localization">Bousfield localization</a> at <a class="existingWikiWord" href="/nlab/show/hypercovers">hypercovers</a> (at <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise acyclic fibrations if the <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">topos has enough points</a>)</p> </li> </ul> <p>The identity functor on the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SPSh(C)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> for the projective and injective <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a> and this is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+sheaves">local model structures on simplicial sheaves</a> are just the restrictions of the <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">those on simplicial presheaves</a>. (For the injective structure this is in <a href="#JardineLecture">Jardine</a>, for the projective one in <a href="#Blander">Blander, theorem 2.1, 2.2</a>).</p> <p>These are related by a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> given by the usual <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> of the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> as a full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of that of presheaves – with <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> – and this is also <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>.</p> <p>The characteristic of the <em>left</em> Bousfield localizations is that for them the fibrant objects are those that satisfy <a class="existingWikiWord" href="/nlab/show/descent">descent</a>: see <a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a>.</p> <p>In either case</p> <ul> <li>the global model structures <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presents</a> the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></li> </ul> <p>while</p> <ul> <li>the local model structures <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presents</a> the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a> (i.e. <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a>).</li> </ul> <p>The following diagram collection <a class="existingWikiWord" href="/nlab/show/model+category">model categories</a> that are <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentations</a> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a>. All indicated morphism pairs are <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a>.</p> <div style="overflow:auto"><div class="maruku-equation"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"> <semantics> <mrow> <mrow> <mtable><mtr><mtd><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>SGrpd</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mover><mo>→</mo><mi>sheafification</mi></mover><mover><mo>←</mo><mi>embedding</mi></mover></mover></mtd> <mtd><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>SGrpd</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↔</mo><mrow></mrow></mover></mtd> <mtd><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>SSet</mi><msubsup><mo stretchy="false">)</mo> <mi>inj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>→</mo><mi>embedding</mi></mover><mover><mo>←</mo><mi>sheafification</mi></mover></mover></mtd> <mtd><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>SSet</mi><msubsup><mo stretchy="false">)</mo> <mi>inj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>→</mo><mi>Id</mi></mover><mover><mo>←</mo><mi>Id</mi></mover></mover></mtd> <mtd><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>SSet</mi><msubsup><mo stretchy="false">)</mo> <mi>proj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>→</mo><mi>sheafification</mi></mover><mover><mo>←</mo><mi>embedding</mi></mover></mover></mtd> <mtd><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>SSet</mi><msubsup><mo stretchy="false">)</mo> <mi>proj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mi>Luo</mi><mo>−</mo><mi>Bubenik</mi><mo>−</mo><mi>Kim</mi></mtd> <mtd></mtd> <mtd><mi>Joyal</mi><mo>−</mo><mi>Tierney</mi></mtd> <mtd></mtd> <mtd><mi>Joyal</mi></mtd> <mtd></mtd> <mtd><mi>Jardine</mi></mtd> <mtd></mtd> <mtd><mi>Blander</mi></mtd> <mtd></mtd> <mtd><mi>Brown</mi><mo>−</mo><mi>Gersten</mi></mtd></mtr></mtable> </mrow> </mrow> <annotation encoding="application/x-tex"> \array{ PSh(X, SGrpd) &amp;\stackrel{\stackrel{embedding}{\leftarrow}} {\stackrel{sheafification}{\to}}&amp; Sh(X,SGrpd) &amp;\stackrel{}{\leftrightarrow}&amp; Sh(X, SSet)^{l loc}_{inj} &amp;\stackrel{\stackrel{sheafification}{\leftarrow}} {\stackrel{embedding}{\to}}&amp; PSh(X, SSet)^{l loc}_{inj} &amp;\stackrel{\stackrel{Id}{\leftarrow}} {\stackrel{Id}{\to}}&amp; PSh(X, SSet)^{l loc}_{proj} &amp;\stackrel{\stackrel{embedding}{\leftarrow}} {\stackrel{sheafification}{\to}}&amp; Sh(X, SSet)^{l loc}_{proj} \\ \\ Luo-Bubenik-Kim &amp;&amp; Joyal-Tierney &amp;&amp; Joyal &amp;&amp; Jardine &amp;&amp; Blander &amp;&amp; Brown-Gersten } </annotation> </semantics> </math> </div></div> <p>On the right this lists the model structures on simplicial (pre)sheaves, here displayed as (pre)sheaves with values in <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>, using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>SSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SPSh(C) \simeq PSh(C,SSet)</annotation></semantics></math>.</p> <p>On the left we have the Joyal–Tierney and Luo–Bubenik–Tim <a class="existingWikiWord" href="/nlab/show/model+structures+on+presheaves+of+simplicial+groupoids">model structures on presheaves of simplicial groupoids</a>.</p> <p>(…have to check here the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>SGrpd</mi><mo stretchy="false">)</mo><mo>↔</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>SGrpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X,SGrpd)\leftrightarrow PSh(X, SGrpd)</annotation></semantics></math>)</p> <h3 id="reedy_and_intermediate_model_structures">Reedy and intermediate model structures</h3> <p>To some extent the injective and projective model structures on simplicial presheaves are the two extremes of a larger family of model structures on simplicial presheaves that all have the same weak equivalences but different classes of cofibrations.</p> <p>Notably if the domain <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> has the special property that it is a <a class="existingWikiWord" href="/nlab/show/Reedy+category">Reedy category</a> there is the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C, sSet]</annotation></semantics></math>. Its class of cofibrations is intermediate that of the projective and the injective <a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a> and we have <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>s</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><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mover><munder><mo>→</mo><mi>Id</mi></munder><mover><mo>←</mo><mi>Id</mi></mover></mover><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>sSet</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>C</mi><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [C,sSet]_{proj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [C,sSet]_{Reedy} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [C,sSet]_{inj} \,. </annotation></semantics></math></div> <p>For 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>, there is still a whole family of model structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]</annotation></semantics></math> that interpolates between the injective and the projective model structure. See <a class="existingWikiWord" href="/nlab/show/intermediate+model+structure">intermediate model structure</a>.</p> <h3 id="DependencyOnSite">Dependency on the underlying site</h3> <div class="num_prop" id="SiteDependence"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>,</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C,D</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/site">site</a>s and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">f : C \to D</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> that induces a morphism of <a class="existingWikiWord" href="/nlab/show/site">site</a>s in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_* : PSh(D) \to PSh(C)</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/sheaf">sheaves</a> and its <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><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^* : PSh(C) \to PSh(D)</annotation></semantics></math> (given by left <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a>) is left <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a> in that it preserves <a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a>s.</p> <p>Then the induced <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mi>SPSh</mi><mo stretchy="false">(</mo><mi>D</mi><msubsup><mo stretchy="false">)</mo> <mi>inj</mi> <mi>loc</mi></msubsup><mover><mo>→</mo><mo>←</mo></mover><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>inj</mi> <mi>loc</mi></msubsup><mo>:</mo><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> f_* : SPSh(D)_{inj}^{loc} \stackrel{\leftarrow}{\to} SPSh(C)_{inj}^{loc} : f^* </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> for the local injective model structure on presheaves on both sides.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This is “little fact 5)” on page 10, 11 of (<a href="#JardineLecture">JardineLectures</a>).</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>D</mi><mo>↪</mo></mrow><annotation encoding="application/x-tex">f : D \hookrightarrow</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full</a> <a class="existingWikiWord" href="/nlab/show/dense+sub-site">dense sub-site</a>. Then right <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f_* : [D^{op}, sSet] \to [C^{op}, sSet]</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> yields a <a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</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><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>inj</mi><mo>,</mo><mi>loc</mi></mrow></msub><mover><munder><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mover><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>inj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> (f^* \dashv f_*) : [D^{op}, sSet]_{inj,loc} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} [C^{op}, sSet]_{inj,loc} </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">left Bousfield localizations</a> of the projective model structures at the <a class="existingWikiWord" href="/nlab/show/sieve">sieve</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">S(\{U_i\}) \to U</annotation></semantics></math> for each <a class="existingWikiWord" href="/nlab/show/covering">covering</a> family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to U\}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>It is immediate that we have a simplicial Quillen adjunction on the global injective model structure: by definition of right <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> we have an <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> and the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> restriction functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> trivially preserves injective cofibrations and acyclic cofibrations.</p> <p>Since we have <a class="existingWikiWord" href="/nlab/show/left+proper+model+categories">left proper model categories</a> it is sufficient (by the discussion at <a href="http://nlab.mathforge.org/nlab/show/simplicial%20Quillen%20adjunction#Recognition">recognition of simplicial Quillen adjunctions</a>) for deducing that the Quillen adjunction descends to the local strucuture to check that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_*</annotation></semantics></math> preserves locally fibrant objects, which in turn by properties of <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">left Bousfield localization</a> is equivalent to checking that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> sends covering sieve inclusions to weak equivalences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[D^{op}, sSet]_{proj,loc}</annotation></semantics></math>.</p> <p>By the <a href="#GeneralizedCover">result on generalized covers</a>, for this it is sufficient to check that for every covering sieve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S(\{U_i\}) \to X</annotation></semantics></math> and every representable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">K \in D</annotation></semantics></math> and morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">K \to f^* X</annotation></semantics></math>, there is a <a class="existingWikiWord" href="/nlab/show/covering">covering</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>K</mi> <mi>j</mi></msub><mo>→</mo><mi>K</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{K_j \to K\}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> and local lifts</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>K</mi> <mi>j</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>K</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>f</mi> <mo>*</mo></msup><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ K_j &\to& f^*(S(\{U_i\})) \\ \downarrow && \downarrow \\ K &\to& f^* X } \,. </annotation></semantics></math></div> <p>This follows directly from the single defining condition on a <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </div> <h2 id="PresentationOfInfiniToposes">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>-toposes</h2> <div class="num_def"> <h6 id="definition">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj}</annotation></semantics></math> be the projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> over <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> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>=</mo><mo stretchy="false">{</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">W = \{C(\{U_i\}) \to U\}</annotation></semantics></math> be the set of <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> projections in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C, sSet]</annotation></semantics></math> for each <a class="existingWikiWord" href="/nlab/show/covering">covering</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to U\}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>Write</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>Id</mi><mo>⊣</mo><mi>Id</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><mover><munder><mo>→</mo><mi>Id</mi></munder><mover><mo>←</mo><mi>Id</mi></mover></mover><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> (Id \dashv Id) : [C^{op}, sSet]_{proj,loc} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [C^{op}, sSet]_{proj} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">left Bousfield localization</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">([C^{op}, sSet]_{proj})^\circ</annotation></semantics></math> for the full sub-<a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a> on the fibrant-cofibrant objects, similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">([CartSp^{op}, sSet]_{proj,loc})^\circ</annotation></semantics></math>.</p> </div> <div class="num_prop" id="PresentationOfTheInfinTopos"> <h6 id="proposition_3">Proposition</h6> <p>We have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↪</mo><mover><mo>←</mo><mi>L</mi></mover></mover></mtd> <mtd><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mtd> <mtd><mover><munder><mo>→</mo><mrow><mi>ℝ</mi><mi>Id</mi></mrow></munder><mover><mo>←</mo><mrow><mi>𝕃</mi><mi>Id</mi></mrow></mover></mover></mtd> <mtd><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ Sh_{(\infty,1)}(C) &\stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}& PSh_{(\infty,1)}(C) \\ \uparrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}} \\ ([C^{op}, sSet]_{proj,loc})^\circ & \stackrel { \overset{\mathbb{L} Id}{\leftarrow} } { \underset{\mathbb{R}Id}{\to} } & ([C^{op}, sSet]_{proj})^\circ } \,, </annotation></semantics></math></div> <p>where at the bottom we have the left and right <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>s of the identity functors, as discussed at <a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This follows using the arguments in the proof of <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, 6.5.2.14</a> and <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. A.3.7.6</a>.</p> </div> <h2 id="FibAndCofibObjects">Fibrant and cofibrant objects</h2> <h3 id="fibrant_objects">Fibrant objects</h3> <p>The fibrant objects in the <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a> are those that</p> <ul> <li> <p>are fibrant with respect to the respective <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a></p> </li> <li> <p>and satisfy <a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a>. See there for more details.</p> </li> </ul> <p>This descent condition is the analog in this model of the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>-condition and the <a class="existingWikiWord" href="/nlab/show/stack">stack</a>-condition. In fact, it reduces to these for truncated simplicial presheaves.</p> <p>Since the fibrancy condition in the global projective model structure is simple – it just requires that the <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a> is in fact a presheaf of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es – the local projective model structure has slightly more immediate characterization of fibrant objects than the local injective model structures. (In fact, for suitable choices of <a class="existingWikiWord" href="/nlab/show/site">site</a>s it may become very simple, as the above discussion of site dependence of the model structure shows).</p> <p>On the other hand the cofibrancy condition on objects is entirely <em>trivial</em> in the global and local injective model structure: since a cofibration there is just an objectwise cofibration, and since every <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is cofibrant, every object is injective cofibrant.</p> <p>But the cofibrant objects in the projective structure are not too nasty either: every object that is degreewise a coproduct of representables is cofibrant. In particular the <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a>s of any <em><a class="existingWikiWord" href="/nlab/show/good+cover">good cover</a></em> (see below for more details) is a projectively cofibrant object.</p> <p>A <strong>cofibrant replacement</strong> functor in the local projective structure is discussed in <a href="#Dugger01">Dugger 01</a>.</p> <p>Something related to a <strong>fibrant replacement</strong> functor (“<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>-stackification”) is discussed in section 6.5.3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></li> </ul> <h3 id="CofibrantObjects">Cofibrant objects</h3> <p>In the injective <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a> all objects are cofibrant. For the projective local structure</p> <p>necessary and sufficient conditions are given here:</p> <ul> <li id="Garner13"><a class="existingWikiWord" href="/nlab/show/Richard+Garner">Richard Garner</a>, <a href="https://mathoverflow.net/a/127187/381">MO comment</a></li> </ul> <p>More specifically, there is the following useful statement (see also <em><a class="existingWikiWord" href="/nlab/show/projectively+cofibrant+diagram">projectively cofibrant diagram</a></em>) (see also <a href="#Low">Low, remark 8.2.3</a>).</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A simplicial presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in sPSh(C)</annotation></semantics></math> is said to have <strong>free degeneracies</strong> or the <strong>degenerate cells split off</strong> if in each degree there is a <a class="existingWikiWord" href="/nlab/show/subobject">sub</a>-presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>k</mi></msub><mo>↪</mo><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">N_k \hookrightarrow X_k</annotation></semantics></math> such that the canonical mophism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <munder><mrow><mi>σ</mi><mo>:</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><mrow><mi>surj</mi><mo>.</mo></mrow></munder></munder><msub><mi>N</mi> <mi>n</mi></msub><mover><mo>→</mo><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <munder><mrow><mi>σ</mi><mo>:</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><mrow><mi>surj</mi><mo>.</mo></mrow></munder></munder><msup><mi>σ</mi> <mo>*</mo></msup></mrow></mover><msub><mi>F</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> \coprod_{\underset{surj.}{\sigma : [k] \to [n]}} N_n \stackrel{\coprod_{\underset{surj.}{\sigma : [k] \to [n]}} \sigma^*}{\to} F_k </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <p>So if degenerate cells split off we have in particular that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>k</mi></msub><mo>=</mo><msubsup><mi>X</mi> <mi>k</mi> <mi>nd</mi></msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msubsup><mi>X</mi> <mi>k</mi> <mi>dg</mi></msubsup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X_k = X_k^{nd} \coprod X_k^{dg} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>X</mi> <mi>k</mi> <mi>nd</mi></msubsup></mrow><annotation encoding="application/x-tex">X_k^{nd}</annotation></semantics></math> is the presheaf of non-degenerate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-cells and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>X</mi> <mi>k</mi> <mi>dg</mi></msubsup></mrow><annotation encoding="application/x-tex">X_k^{dg}</annotation></semantics></math> is a separate presheaf containing all the degenerate cells (and itself a coproduct over separate presheaves for each degree and order of degeneracy).</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>In the <em>projective</em> <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a> all objects that are</p> <ol> <li> <p>degreewise <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> of <a class="existingWikiWord" href="/nlab/show/representable+functor">representables</a></p> </li> <li> <p>and whose degenerate cells split off</p> </li> </ol> <p>are cofibrant.</p> </div> <p>This is in <a href="#Dugger01">Dugger 01, Cor. 9.4</a>.</p> <div class="num_example"> <h6 id="example">Example</h6> <p><strong>(split hypercovers)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> is an acyclic fibration in the local projective model structure with <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> a representable and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> cofibration in the above way, it is called a <strong><a class="existingWikiWord" href="/nlab/show/split+hypercover">split hypercover</a></strong> .</p> <p>All <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\{U_i\})</annotation></semantics></math> coming from an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> have split degeneracies. The condition that the Čech nerve be degreewise a coproduct of representables is a condition akin to that of <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>s (which is precisely the special case for <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/CartSp">CartSp</a>). This is then a split hypercover of <em>height</em> 0.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p><strong>(good cover)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> with a weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">U \stackrel{\simeq}{\to} X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mi>loc</mi></msup></mrow><annotation encoding="application/x-tex">SPSh(C)^{loc}</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/good+cover">good cover</a></strong> if it is degreewise a coproduct of <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a>s.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This reduces to the ordinary notion of <a class="existingWikiWord" href="/nlab/show/good+cover">good cover</a> as an open cover by contractible spaces such that all finite intersections of these are again contractibe when using a <a class="existingWikiWord" href="/nlab/show/site">site</a> like <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/CartSp">CartSp</a>.</p> </div> <h3 id="CofibrantReplacement">Cofibrant replacement</h3> <p>In</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <em><a class="existingWikiWord" href="/nlab/files/DuggerUniv.pdf" title="Universal homotopy theories">Universal homotopy theories</a></em></li> </ul> <p>a useful cofibrant replacement functor for the projective local model structure is discussed.</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in PSh(C) \hookrightarrow SPSh(C)</annotation></semantics></math> an ordinary presheaf (simplicially discrete simplicial presheaf) let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Q</mi><mo stretchy="false">˜</mo></mover><mi>A</mi></mrow><annotation encoding="application/x-tex">\tilde Q A</annotation></semantics></math> be the simplicial presheaf which in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is</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><mover><mi>Q</mi><mo stretchy="false">˜</mo></mover><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><mo>:</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><msub><mi>U</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>U</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo>→</mo><msub><mi>U</mi> <mn>0</mn></msub><mo>→</mo><mi>A</mi></mrow></munder><msub><mi>U</mi> <mi>k</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (\tilde Q A)_k := \coprod_{U_k \to U_{k-1} \to \cdots \to U_0 \to A} U_k \,, </annotation></semantics></math></div> <p>where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_k</annotation></semantics></math> range over the representables, i.e. the objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↪</mo><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \hookrightarrow SPSh(C)</annotation></semantics></math>. The face and degeneracy maps are the obvious ones coming from composing maps and inserting identity maps in the labels over which the coproduct ranges.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in SPSh(C)</annotation></semantics></math> an arbitrary simplicial presheaf let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">Q A</annotation></semantics></math> be the diagonal of the bisimplicial presheaf obtained by applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Q</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde Q</annotation></semantics></math> degreewise</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>A</mi><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>U</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow></munder><msub><mi>U</mi> <mn>1</mn></msub><mover><mo>→</mo><mo>→</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><msub><mi>U</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>0</mn></msub></mrow></munder><msub><mi>U</mi> <mn>0</mn></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Q A = \left( \cdots \coprod_{U_1 \to U_0 \to A_1} U_1 \stackrel{\to}{\to}\coprod_{U_0 \to A_0} U_0 \right) \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in SPSh(C)</annotation></semantics></math> the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">Q A</annotation></semantics></math> is cofibrant and is weakly equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>proj</mi> <mi>loc</mi></msubsup></mrow><annotation encoding="application/x-tex">SPSh(C)_{proj}^{loc}</annotation></semantics></math>.</p> </div> <p>This is in prop 2.8 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <em><a class="existingWikiWord" href="/nlab/show/Universal+Homotopy+Theories">Universal Homotopy Theories</a></em></li> </ul> <h2 id="local_fibrations">Local fibrations</h2> <p>A <em><a class="existingWikiWord" href="/nlab/show/local+fibration">local fibration</a></em> or <em>local weak equivalence</em> of simplicial (pre)sheaves is defined to be one whose lifting property is satisfied after refining to some cover.</p> <p><strong>Warning</strong>. Notice that this is a priori unrelated to equivalences and fibrations with respect to any local model structure.</p> <p>If the <a class="existingWikiWord" href="/nlab/show/site">site</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> has <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a>, then local fibrations of simplicial presheaves are equivalently those that are <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise fibrations of simplicial sets.</p> <p>This is discussed in (<a href="#Jardine96">Jardine 96</a>).</p> <h2 id="Descent">Localization and descent</h2> <h3 id="ČechLocalization">Čech localization at Grothendieck (pre)topologies</h3> <p>We discuss some aspects of the <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">left Bousfield localization</a> of the projective global model structure on simplicial presheaves at <a class="existingWikiWord" href="/nlab/show/Grothendieck+topologies">Grothendieck topologies</a> and <a class="existingWikiWord" href="/nlab/show/covering">covering</a> families. By the discussion at <a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a> these are models for <a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a>s leading to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-categories of (∞,1)-sheaves</a>.</p> <p>The central reference is (<a href="#DuggerHollanderIsaksen">DuggerHollanderIsaksen</a>) with the central theorem being this one:</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a> given by a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>. The left <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">sPSh(C)_{proj}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>inj</mi></msub></mrow><annotation encoding="application/x-tex">sPSh(C)_{inj}</annotation></semantics></math>, respectively, at the following classes of morphisms exist and coincide:</p> <ol> <li> <p>the set of all <a class="existingWikiWord" href="/nlab/show/covering">covering</a> <a class="existingWikiWord" href="/nlab/show/sieve">sieve</a> <a class="existingWikiWord" href="/nlab/show/subfunctor">subfunctor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>↪</mo><mi>j</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R \hookrightarrow j(X)</annotation></semantics></math>;</p> </li> <li> <p>the set of all morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>hocolim</mi> <mi>R</mi></msub><mo>→</mo><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">hocolim_R \to U \to X</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a covering sieve of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>the set of all <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C(\{U_i\}) \to X</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> a covering sieve;</p> </li> <li> <p>the class of all bounded <a class="existingWikiWord" href="/nlab/show/hypercover">hypercover</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \to X</annotation></semantics></math>;</p> </li> <li> <p>the class of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><mover><mi>F</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">F \to \bar F</annotation></semantics></math> from a <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> to the simplicial <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> obtained by degreewise <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a>.</p> </li> <li> <p>if the topology is generated from a <a class="existingWikiWord" href="/nlab/show/basis+for+a+topology">basis</a>, then: the set of covering sieve subfunctors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>U</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">R_U \to X</annotation></semantics></math> for each covering family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> in the basis.</p> </li> </ol> </div> <p>This is theorem A5 in <a href="http://front.math.ucdavis.edu/0205.5027">DugHolIsak</a>.</p> <p>This localization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">sPSh(C)_{proj,cov}</annotation></semantics></math> is the <strong>Čech localization</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sPSh(C)</annotation></semantics></math> with respect to the given <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>. It is a presentation of <a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a> of an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> to an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↪</mo><mover><mo>→</mo><mi>L</mi></mover></mover></mtd> <mtd><msub><mi>Psh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mtd> <mtd><mover><munder><mo>→</mo><mrow></mrow></munder><mover><mo>←</mo><mrow><mi>left</mi><mo>.</mo><mi>Bousf</mi><mo>.</mo></mrow></mover></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Sh_{(\infty,1)}(C) &\stackrel{\overset{L}{\to}}{\hookrightarrow}& Psh_{(\infty,1)}(C) \\ \uparrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}} \\ (sPSh(C)_{proj,cov})^\circ &\stackrel{\overset{left. Bousf.}{\leftarrow}}{\underset{}{\to}}& (sPSh(C)_{proj})^\circ } \,. </annotation></semantics></math></div> <p>The following definition and proposition provides information on what the general morphisms are which become weak equivalences after localization at</p> <div class="num_def" id="GeneralizedCover"> <h6 id="defintion">Defintion</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/generalized+cover">generalized cover</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a>. A <strong><a class="existingWikiWord" href="/nlab/show/local+epimorphism">local epimorphism</a></strong> (or <strong><a class="existingWikiWord" href="/nlab/show/generalized+cover">generalized cover</a></strong>) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sPSh(C)</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : E \to B</annotation></semantics></math> of simplicial presheaves with the property that for every <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> and every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">j(U) \to B</annotation></semantics></math> there exists a <a class="existingWikiWord" href="/nlab/show/covering">covering</a> <a class="existingWikiWord" href="/nlab/show/sieve">sieve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to U\}</annotation></semantics></math> such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U_i \to U</annotation></semantics></math> the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">U_i \to U \to B</annotation></semantics></math> has a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>∃</mo><mi>σ</mi></mrow></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mo>∀</mo></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ j(U_i) &\stackrel{\exists \sigma}{\to}& E \\ \downarrow && \downarrow \\ j(U) &\stackrel{\forall}{\to} & B } \,. </annotation></semantics></math></div></div> <p>(<a href="#DuggerHollanderIsaksen">Dugger-Hollander-Isaksen, corollary A.3</a>)</p> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : E \to B</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/local+epimorphism">local epimorphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sPSh(C)</annotation></semantics></math> in the above sense, its <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> projection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex"> C(E) \to B </annotation></semantics></math></div> <p>is a weak equivalence in the projective local model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">sPSh(C)_{proj, loc}</annotation></semantics></math>.</p> </div> <p>This is <a href="#DuggerHollanderIsaksen">Dugger-Hollander-Isaksen, corollary A.3</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="for_values_in_strict_and_abelian_groupoids">For values in strict and abelian <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</h3> <p>Many simplicial presheaves appearing in practice are (equivalent) to objects in <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-categories">sub-(∞,1)-categories</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(C)</annotation></semantics></math> of abelian or at least <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a>s. These subcategories typically offer convenient and desireable contexts for formulating and proving statements about special cases of general simplicial presheaves.</p> <p>One well-known such notion is given by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a>. This identifies <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s with strict and strictly <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28infinity%2C1%29-category">symmetric monoidal</a> <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.</p> <p>Dropping the condition on symmetric monoidalness we obtain a more general such inclusion, a kind of non-abelian Dold-Kan correspondence:</p> <p>the identification of <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a>es of groupoids as precisely the <a class="existingWikiWord" href="/nlab/show/strict+omega-groupoid">strict ∞-groupoid</a>s. This has been studied in particular in <a class="existingWikiWord" href="/nlab/show/nonabelian+algebraic+topology">nonabelian algebraic topology</a>.</p> <p>So we have a sequence of inclusions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ChainCplx</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>CrsCpl</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>KanCplx</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>simeq</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>simeq</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>simeq</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>StrAb</mi><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mn>∞</mn><mi>Grpd</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ ChainCplx &\hookrightarrow& CrsCpl &\hookrightarrow& KanCplx \\ \downarrow^{\mathrlap{simeq}} && \downarrow^{\mathrlap{simeq}} && \downarrow^{\mathrlap{simeq}} \\ StrAb Str\infty Grpd &\hookrightarrow& Str \infty Grpd &\hookrightarrow& \infty Grpd } </annotation></semantics></math></div> <p>of strict <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 into all <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. See also the <a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>.</p> <p>Among the special tools for handling <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 on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> that factor at some point through the above inclusion are the following:</p> <ul> <li> <p><strong>restriction to abelian sheaf cohomology</strong> – Using the fact that the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(C)</annotation></semantics></math> are modeled by <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> symmetric monoidal <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>-Lie groupoids are identified under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> with <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">\mathbb{N}</annotation></semantics></math>-graded <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complexes</a> of sheaves. To these the rich set of tools for <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> apply.</p> </li> <li> <p><strong>descent for strict <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>-groupoid valued sheaves</strong> – There is a good theory of <a class="existingWikiWord" href="/nlab/show/descent">descent</a> for (presheaves) with values in strict <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 (more restrictive than the fully general theory but more general than <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>). This goes back to <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a> and its relation to the full theory has been clarified by <a class="existingWikiWord" href="/nlab/show/Dominic+Verity">Dominic Verity</a> in <a href="#Verity">Verity09</a>.</p> </li> </ul> <p>We state a useful theorem for the computation of <a class="existingWikiWord" href="/nlab/show/descent">descent</a> for presheaves with values in <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a>s. Recall the standard terminology for <a class="existingWikiWord" href="/nlab/show/descent">descent</a>, i.e. for 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>-categorical <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>-condition:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">U \in C</annotation></semantics></math> a representable, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Y,A \in [C^{op}, sSet]</annotation></semantics></math> simplicial presheaves and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">p : Y \to U</annotation></semantics></math> a morphism, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> <em>satisfies <a class="existingWikiWord" href="/nlab/show/descent">descent</a></em> 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> or equivalently that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <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>-<a class="existingWikiWord" href="/nlab/show/local+object">local object</a> if the canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>=</mo></mover><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A(U) \stackrel{=}{\to} [C^{op}, sSet](U,A) \to [C^{op}, sSet](Y,A) </annotation></semantics></math></div> <p>is a weak equivalence. Here the first equality is the enriched <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>. By the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a> we may decompose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> into its cells as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>Y</mi> <mi>n</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Y = \int^{[n] \in \Delta} \Delta[n] \cdot Y_n \,, </annotation></semantics></math></div> <p>where in the integrand we have the <a class="existingWikiWord" href="/nlab/show/copower">tensoring</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>. Using that the enriched <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> sends coends to ends, the enriched <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> on the right we may equivalently write out as an <a class="existingWikiWord" href="/nlab/show/end">end</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mi>sSet</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mi>sSet</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo>:</mo><mi>Desc</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [C^{op}, sSet](Y,A) & = [C^{op}, sSet](\int^{[n] \in \Delta} \Delta[n] \cdot Y_n ,A) \\ & = \int_{[n] \in \Delta}[C^{op}, sSet](\Delta[n] \cdot Y_n ,A) \\ & = \int_{[n] \in \Delta} sSet(\Delta[n], [C^{op}, sSet](Y_n, A)) \\ & = \int_{[n] \in \Delta} sSet(\Delta[n], A(Y_n)) \\ & =:Desc(Y,A) \end{aligned} </annotation></semantics></math></div> <p>(equality signs denote <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>s), where in the second but last line we again used the <a class="existingWikiWord" href="/nlab/show/copower">tensoring</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>.</p> <p>In the last line we have the <em><a class="existingWikiWord" href="/nlab/show/totalization">totalization</a></em> of the cosimplicial <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> A(Y_\bullet) : \Delta \to sSet \,, </annotation></semantics></math></div> <p>sometimes called the <em>descent object</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, even though in this case it is really nothing but the hom-object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is fibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> cofibrant, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Desc</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Desc(Y,A)</annotation></semantics></math> is a Kan complex: the <em>descent <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>-groupoid</em> .</p> <p>Now suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} : C^{op} \to Str \infty Grpd</annotation></semantics></math> is a presheaf with values in <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a>s. In the context of strict <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 the standard <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>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> is given by the <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>th <a class="existingWikiWord" href="/nlab/show/oriental">oriental</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>. This allows to perform a construction that looks like a descent object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">Str\infty Grpd</annotation></semantics></math>:</p> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p><strong>(Ross Street)</strong></p> <p>The descent object for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{A} \in [C^{op}, Str \infty Grpd]</annotation></semantics></math> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Y \in [C^{op}, sSet]</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Desc</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mi>Str</mi><mn>∞</mn><mi>Cat</mi><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mi>𝒜</mi><mo stretchy="false">(</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Desc(Y,\mathcal{A}) := \int_{[n] \in \Delta} Str\infty Cat(O(n), \mathcal{A}(Y_n)) \;\in Str \infty Grpd \,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/end">end</a> is taken in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">Str \infty Grpd</annotation></semantics></math>.</p> </div> <p>This objects had been suggested by <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a> to be the right descent object for strict <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-valued presheaves in <a href="#Street03">Street03</a></p> <p>Under the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-nerve">∞-nerve</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mo>:</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">N_O : Str\infty Grpd \to sSet</annotation></semantics></math> this yields a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>0</mn></msub><mi>Desc</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_0 Desc(Y,\mathcal{A})</annotation></semantics></math>. On the other hand, applying the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-nerve directly to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> yields a simplicial presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">N_O\mathcal{A}</annotation></semantics></math> to which the above simplicial descent applies.</p> <p>The following theorem asserts that under certain conditions both notions coincide.</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p><strong>(Dominic Verity)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} : C^{op}, Str \infty Grpd</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">Y : C^{op} \to sSet</annotation></semantics></math> are such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy="false">(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">N_O \mathcal{A}(Y_\bullet) : \Delta \to sSet</annotation></semantics></math> is fibrant in the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta, sSet_{Quillen}]_{Reedy}</annotation></semantics></math>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>Desc</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Desc</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A}) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es.</p> </div> <p>This is proven in <a href="#Verity">Verity09</a>.</p> <div class="num_corollary"> <h6 id="corollary">Corollary</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Y \in [C^{op}, sSet]</annotation></semantics></math> is such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mo>•</mo></msub><mo>:</mo><mi>Δ</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>↪</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Y_\bullet : \Delta \to [C^{op}, Set] \hookrightarrow [C^{op}, sSet]</annotation></semantics></math> is cofibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta, [C^{op}, sSet]_{proj}]_{Reedy}</annotation></semantics></math> then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} : C^{op} \to Str \infty Grpd</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>Desc</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Desc</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A}) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">Y_\bullet</annotation></semantics></math> is Reedy cofibrant, then by definition the canonical morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mover><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><msub><mi>Y</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> \lim_{\to}( ([n] \stackrel{+}{\to} [k]) \mapsto Y_k ) \to Y_n </annotation></semantics></math></div> <p>are cofibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj}</annotation></semantics></math>. Since the latter is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">N_O \mathcal{A}</annotation></semantics></math> is fibrant, it follows that the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[C^{op}, sSet](-, N_O \mathcal{A})</annotation></semantics></math> sends cofibrations to fibrations, so that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy="false">(</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>→</mo><munder><mi>lim</mi> <mo>←</mo></munder><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mover><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy="false">(</mo><msub><mi>Y</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> N_O\mathcal{A}(Y_n) \to \lim_{\leftarrow}( [n]\stackrel{+}{\to} [k] \mapsto N_O\mathcal{A}(Y_k)) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>. But this says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy="false">(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_O \mathcal{A}(Y_\bullet)</annotation></semantics></math> is Reedy fibrant, so that the assumption of Verity’s theorem is met.</p> </div> <div class="num_corollary"> <h6 id="corollary_2">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> of a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>:</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>→</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} : CartSp^{op} \to Str \infty Grpd</annotation></semantics></math> we have that</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>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>Desc</mi><mo stretchy="false">(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [C^{op}, sSet](Y,N_O \mathcal{A}) \simeq N_O Desc(Y_\bullet, \mathcal{A}) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>By the above is sufices to note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">Y_\bullet</annotation></semantics></math> is cofibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, [C^{op}, sSet]_{proj}]_{Reedy}</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> of a good open cover. By the assumption of good open cover we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is degreewise a coproduct of representables and that the inclusion of all degenerate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cells into all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cells is a full inclusion into a coproduct, i.e. an inlusion of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>j</mi></munder><msub><mi>U</mi> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \coprod_{i \in I} U_i \to \coprod_j U_{j \in J} </annotation></semantics></math></div> <p>induced from an inclusion of subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>↪</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">I \hookrightarrow J</annotation></semantics></math>. Since all representables are cofibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj}</annotation></semantics></math> such an inclusion is a cofibration.</p> </div> <p>In conclusion we find that for determining the <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>-stack condition for <em>strict</em> <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>-Lie groupoids we may equivalently use Street’s formula for strict <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>-groupid valued presheaves. This is sometimes useful for computations in low categorical degree.</p> <h2 id="Properness">Properness</h2> <p>The global model structures on simplicial presheaves are all <a class="existingWikiWord" href="/nlab/show/left+proper+model+categories">left</a> and <a class="existingWikiWord" href="/nlab/show/right+proper+model+categories">right proper model categories</a>. Since left <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization of model categories</a> preserves left properness (as discussed <a href="Bousfield+localization+of+model+categories#ExistenceForLeftProperCombinatorialSimplicialModelCategories">there</a>), all local model structures on simplicial presheaves are also left proper.</p> <p>But the local model structures are not in general right proper anymore. A sufficient condition for them to be right proper is given in the following Prop. <a class="maruku-ref" href="#StalkwiseWeakEquivalencesImpliesRightProperness"></a>.</p> <p>The injective local model structures which are right proper are <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+categories">locally cartesian closed model categories</a> and <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">present</a> <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+%28infinity%2C1%29-categories">locally cartesian closed (infinity,1)-categories</a> (by the discussion <a href="locally+cartesian+closed+infinity,1-category#PresentationTheorem">there</a>).</p> <div class="num_prop" id="OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks"> <h6 id="proposition_7">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a> with <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a>. Then the weak equivalences in the local model structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sPSh(C)</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>-wise <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> of simplicial sets.</p> </div> <p>(p. 12 <a href="http://www.math.uiuc.edu/K-theory/0175/">here</a>)</p> <div class="num_prop" id="StalkwiseWeakEquivalencesImpliesRightProperness"> <h6 id="proposition_8">Proposition</h6> <p>A sufficient condition for an injective or projective local model structure of simplicial presheaves over a <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to be right proper is that the weak equivalences are precisely the <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>-wise <a class="existingWikiWord" href="/nlab/show/simplicial+weak+equivalence">weak equivalences of simplicial sets</a>.</p> </div> <p>This is mentioned for instance in (<a href="#Olsson">Olsson, remark 4.3</a>).</p> <p>By Prop. <a class="maruku-ref" href="#OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks"></a> this sufficient condition holds, for instance, for the injective Jardine model structure 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> has <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a>.</p> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>The key is that forming <a class="existingWikiWord" href="/nlab/show/stalks">stalks</a> is, being the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> of a <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">geometric morphism</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><msup><mi>x</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>x</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Set</mi><mover><munder><mo>→</mo><mrow><msub><mi>x</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>x</mi> <mo>*</mo></msup></mrow></mover></mover><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (x^* \dashv x_*) := Set \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} Sh(C) </annotation></semantics></math></div> <p>an operation that preserves <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>.</p> <p>Let therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f : X \to A</annotation></semantics></math> be a stalkwise weak equivalence of simplicial presheaves and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">g : A \to B</annotation></semantics></math> be a fibration. Notice that in all the model structures (injective, projective, global, local) the fibrations are always <em>in particular</em> objectwise fibrations.</p> <p>Then the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mo>*</mo></msup><mi>f</mi></mrow><annotation encoding="application/x-tex">g^* f</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>g</mi> <mo>*</mo></msup><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msup><mi>g</mi> <mo>*</mo></msup><mi>f</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ g^* X &\to& X \\ \big\downarrow {}^{\mathrlap{g^* f}} && \big\downarrow {}^{\mathrlap{f}} \\ A &\stackrel{g}{\to}& B } </annotation></semantics></math></div> <p>is a weak equivalence if for all topos points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> the stalk <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>g</mi> <mo>*</mo></msup><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x^* (g^* f)</annotation></semantics></math> is a weak equivalence of simplicial sets. But since stalks preserve <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>, we have a pullback diagram of simplicial sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>g</mi> <mo>*</mo></msup><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>g</mi> <mo>*</mo></msup><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ x^*(g^* X) &\to& x^*( X) \\ \downarrow^{\mathrlap{x^*(g^* f)}} && \downarrow^{\mathrlap{x^*(f)}} \\ x^*(A) &\stackrel{x^*(g)}{\to}& x^*(B) } \,. </annotation></semantics></math></div> <p>It is now sufficient to observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup><mi>g</mi></mrow><annotation encoding="application/x-tex">x^* g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, which implies the result by the fact that the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a> is right proper.</p> <p>To see this, notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x^*(g)</annotation></semantics></math> is a Kan fibration precisely if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">1 \leq k </annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">0 \leq i \leq k</annotation></semantics></math> the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>*</mo></msup><mi>A</mi><msup><mo stretchy="false">)</mo> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mo>→</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>*</mo></msup><mi>A</mi><msup><mo stretchy="false">)</mo> <mrow><mi>Λ</mi><mo stretchy="false">[</mo><mi>k</mi><msup><mo stretchy="false">]</mo> <mi>i</mi></msup></mrow></msup><msub><mo>×</mo> <mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>*</mo></msup><mi>B</mi><msup><mo stretchy="false">)</mo> <mrow><mi>Λ</mi><mo stretchy="false">[</mo><mi>k</mi><msup><mo stretchy="false">]</mo> <mi>i</mi></msup></mrow></msup></mrow></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>*</mo></msup><mi>B</mi><msup><mo stretchy="false">)</mo> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> (x^* A)^{\Delta[k]} \to (x^* A)^{\Lambda[k]^i} \times_{(x^* B)^{\Lambda[k]^i} } (x^* B)^{\Delta[k]} </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> of sets. Since stalks commute with finite limits, this is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup><mrow><mo>(</mo><msup><mi>A</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mo>→</mo><msup><mi>A</mi> <mrow><mi>Λ</mi><mo stretchy="false">[</mo><mi>k</mi><msup><mo stretchy="false">]</mo> <mi>i</mi></msup></mrow></msup><msub><mo>×</mo> <mrow><msup><mi>B</mi> <mrow><mi>Λ</mi><mo stretchy="false">[</mo><mi>k</mi><msup><mo stretchy="false">]</mo> <mi>i</mi></msup></mrow></msup></mrow></msub><msup><mi>B</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> x^* \left( A^{\Delta[k]} \to A^{\Lambda[k]^i} \times_{ B^{\Lambda[k]^i} } B^{\Delta[k]} \right) </annotation></semantics></math></div> <p>being an epimorphism. Now the morphism in parenthesis is an epimorphism since the fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is in particular an objectwise Kan fibration, and <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> functors such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math> preserve epimorphisms.</p> </div> <h2 id="MonoidalStructure">Closed monoidal structure</h2> <p>If the underlying site has <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a>, then both the injective and the projective, the global and the local model structure on simplicial presheaves becomes a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> with respect to the standard <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a>.</p> <p>See for instance <a href="http://www.math.univ-toulouse.fr/~toen/crm-2008.pdf#page=24">here</a>.</p> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a category with <a class="existingWikiWord" href="/nlab/show/products">products</a>. Then the <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a> makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a>.</p> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>It is sufficient to check that the Cartesian product of presheaves</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⊗</mo><mo>:</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub><mo>×</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub><mo>→</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> \otimes : sPSh(C)_{proj} \times sPSh(C)_{proj} \to sPSh(C)_{proj} </annotation></semantics></math></div> <p>is a left <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a>. As discussed there, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sPSh(C)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> for that it is sufficient to check that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> satisfies the <span class="newWikiWord">pushout-prodct axiom<a href="/nlab/new/pushout-prodct+axiom">?</a></span> on <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">generating (acyclic) cofibrations</a>.</p> <p>As discussed at <a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a>, these are those morphisms of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Id</mi><mo>×</mo><mi>i</mi><mo>:</mo><mi>U</mi><mo>⋅</mo><mi>S</mi><mo>→</mo><mi>U</mi><mo>⋅</mo><mi>T</mi></mrow><annotation encoding="application/x-tex"> Id \times i : U \cdot S \to U \cdot T </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">U \in C</annotation></semantics></math> representable and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">i : S \to T</annotation></semantics></math> an (acylic) cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math>. For these morphisms checking the pushout-product axiom amounts to checking it in <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>, where it is evident.</p> </div> <div class="num_lemma"> <h6 id="lemma_2">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a> with <a class="existingWikiWord" href="/nlab/show/product">product</a>s and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj,cov}</annotation></semantics></math> be the left <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a> at the <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> projections.</p> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any cofibrant object, the <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a>-adjunction</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>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊣</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> (X \times (-) \dashv [X,-]) : [C^{op}, sSet]_{proj,cov} \to [C^{op}, sSet]_{proj,cov} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>.</p> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>The above lemma implies that the <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><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \times (-)</annotation></semantics></math> preserves cofibrations. As discussed in the <a href="http://ncatlab.org/nlab/show/Quillen+adjunction#sSet">section on sSet-enriched adjunctions</a> at <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> since the adjunction is <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 and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj,cov}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/proper+model+category">left proper</a> <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> it suffices to check that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,-]</annotation></semantics></math> preserves fibrant objects.</p> <p>For that let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to U\}</annotation></semantics></math> be a covering family and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\{U_i\})</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a>. We need to check that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A \in [C^{op}, sSet]_{proj,cov}</annotation></semantics></math> is fibrant, then</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>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [C^{op}, sSet](U, [X,A]) \to [C^{op},sSet](C(\{U_i\}), [X,A]) </annotation></semantics></math></div> <p>is an equivalence of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es.</p> <p>Writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">C(\{U_i\}) = \int^{[n]} \Delta[n] \cdot \coprod U_{i_0, \cdots, i_n}</annotation></semantics></math> and using that the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> preserves <a class="existingWikiWord" href="/nlab/show/end">end</a>s, this is eqivalent to</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>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [C^{op},sSet]( X \times C(\{U_i\}) \to X \times U , A) </annotation></semantics></math></div> <p>being an equivalence. Now we observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">X \times C(\{U_i\}) \to X\times U</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/local+epimorphism">local epimorphism</a> in the above sense, namely a morphism such that for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V \to X \times U</annotation></semantics></math> out of a representable, there is a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>U</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X \times C(\{U_i\}) \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow \\ V &\to& X \times U } \,. </annotation></semantics></math></div> <p>By the above discussion of the Čech-localization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj}</annotation></semantics></math>, this is a local morphism, hence does produce an equivalence when hommed into the fibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> </div> <h2 id="HomotopyLimits">Homotopy (co)limits</h2> <p>Properties of <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>s and <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a>s of simplicial presheaves are discussed at</p> <ul> <li><a href="http://ncatlab.org/nlab/show/homotopy+limit#SimpSheaves">Homotopy (co)limits of simplicial (pre)sheaves</a></li> </ul> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a>.</p> <div class="num_prop"> <h6 id="proposition_9">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F : D \to [C^{op}, sSet]</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/finite+limit">finite</a> diagram.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℝ</mi> <mi>glob</mi></msub><msub><mi>lim</mi> <mo>←</mo></msub><mi>F</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}_{glob}\lim_{\leftarrow} F \in [C^{op}, sSet]</annotation></semantics></math> for any representative of the <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> computed in the global model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj}</annotation></semantics></math>, well defined up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a>.</p> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℝ</mi> <mi>glob</mi></msub><msub><mi>lim</mi> <mo>←</mo></msub><mi>F</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}_{glob}\lim_{\leftarrow} F \in [C^{op},sSet]</annotation></semantics></math> presents also the <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> computed in the local model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj,loc}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>By the discussion at <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> the <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><msub><mi>lim</mi> <mo>←</mo></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}\lim_{\leftarrow}</annotation></semantics></math> computes the corresponding <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> and <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheafification">(∞,1)-sheafification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L </annotation></semantics></math> is a left <a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a> and preserves these finite <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a>s:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>L</mi> <mo>*</mo></msub></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>ℝ</mi><munder><mi>lim</mi> <mo>←</mo></munder></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>ℝ</mi><munder><mi>lim</mi> <mo>←</mo></munder></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mtd> <mtd><mover><mo>←</mo><mrow><mi>L</mi><mo>≃</mo><mi>𝕃</mi><mi>Id</mi></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ ([D, [C^{op}, sSet]_{proj, loc}]_{inj})^\circ &\stackrel{L_*}{\leftarrow}& ([D, [C^{op}, sSet]_{proj}]_{inj})^\circ \\ \downarrow^{\mathrlap{\mathbb{R} \lim_\leftarrow}} && \downarrow^{\mathrlap{\mathbb{R} \lim_\leftarrow}} \\ ([C^{op}, sSet]_{proj,loc})^\circ &\stackrel{L \simeq \mathbb{L} Id}{\leftarrow}& ([C^{op}, sSet]_{proj})^\circ } \,. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>≃</mo><mi>𝕃</mi><mi>Id</mi></mrow><annotation encoding="application/x-tex">L \simeq \mathbb{L} Id</annotation></semantics></math> is the left <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of the identity for the <a href="#PresentationOfTheInfinTopos">above</a> left Bousfield localization. Since left Bousfield localization does not change the cofibrations and includes the global weak equivalences into the local weak equivalences, the postcomposition of the diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>Id</mi></mrow><annotation encoding="application/x-tex">\mathbb{L} Id</annotation></semantics></math> is given by cofibrant replacement in the local structure, too. But the <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> of the diagram is invariant, up to equivalence, under cofibrant replacement, and hence a finite homotopy limit diagram in the global structure is also one in the local structure.</p> </div> <h2 id="InclusionOfChainComplexes">Inclusion of chain complexes of sheaves</h2> <p>We discuss how <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es of presheaves of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s embed into the model structure on simplicial presheaves. Under passing to the intrinsic <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a href="#PresentationOfInfiniToposes">presented by</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>loc</mi></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{loc}</annotation></semantics></math>, this realizes traditional <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> over <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 generalizes it to general base objects.</p> <p>Observe from the discussion at <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+abelian+groups">model structure on simplicial abelian groups</a> that the degreewise <a class="existingWikiWord" href="/nlab/show/free+functor">free functor</a>-<a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Ab</mi><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><mi>Set</mi></mrow><annotation encoding="application/x-tex">(F \dashv U) : Ab \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Set</annotation></semantics></math> (see <a class="existingWikiWord" href="/nlab/show/algebra+over+a+Lawvere+theory">algebra over a Lawvere theory</a> for details) induces a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>sAb</mi> <mi>Quillen</mi></msub><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> (F \dashv U) : sAb_{Quillen } \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet_{Quillen} </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+abelian+groups">model structure on simplicial abelian groups</a> and the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>, which exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sAb</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sAb_{Quillen}</annotation></semantics></math> as the corresponding <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a>.</p> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> constitutes in particular a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</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>N</mi> <mo>•</mo></msub><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><msub><mo></mo><mi>proj</mi></msub><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mrow><msub><mi>N</mi> <mo>•</mo></msub></mrow></mover></mover><msub><mi>sAb</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> (N_\bullet \dashv \Gamma) : Ch_\bullet^+_{proj} \stackrel{\overset{N_\bullet}{\leftarrow}}{\underset{\Gamma}{\to}} sAb_{Quillen} </annotation></semantics></math></div> <p>between the projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a> of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s in non-negative degree and simplicial abelian groups.</p> <p>We write</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>N</mi> <mo>•</mo></msub><mi>F</mi><mo>⊣</mo><mi>Ξ</mi><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><msub><mo></mo><mi>proj</mi></msub><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mrow><msub><mi>N</mi> <mo>•</mo></msub></mrow></mover></mover><msub><mi>sAb</mi> <mi>Quillen</mi></msub><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> (N_\bullet F \dashv \Xi) : Ch_\bullet^+_{proj} \stackrel{\overset{N_\bullet}{\leftarrow}}{\underset{\Gamma}{\to}} sAb_{Quillen} \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet_{Quillen} </annotation></semantics></math></div> <p>for the composite <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>. 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> any <a class="existingWikiWord" href="/nlab/show/category">category</a>, postcomposition with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi></mrow><annotation encoding="application/x-tex">\Xi</annotation></semantics></math> induces a Quillen adjunction</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>N</mi> <mo>•</mo></msub><mi>F</mi><mo>⊣</mo><mi>Ξ</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><msub><mo></mo><mi>proj</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mover><munder><mo>→</mo><mi>Ξ</mi></munder><mover><mo>←</mo><mrow><msub><mi>N</mi> <mo>•</mo></msub><mi>F</mi></mrow></mover></mover><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> (N_\bullet F \dashv \Xi) : [C^{op}, Ch_\bullet^+_{proj}]_{proj} \stackrel{\overset{N_\bullet F}{\leftarrow}}{\underset{\Xi}{\to}} [C^{op}, sSet]_{proj} </annotation></semantics></math></div> <p>between the projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><msub><mo></mo><mi>proj</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[C^{op}, Ch_\bullet^+_{proj}]_{proj}</annotation></semantics></math> and the global projective model structure on simplicial presheaves, which by convenient abuse of notation we denote by the same symbols.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+fibration">local fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Gersten+property">Brown-Gersten property</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/model+topos">model topos</a></p> <ul> <li> <p><strong>model structure on simplicial presheaves</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">model structure on simplicial sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">model structure on sSet-enriched presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+presheaves">model structure on cubical presheaves</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> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a></p> </li> </ul> <div> <p><strong>Locally presentable categories:</strong> <a class="existingWikiWord" href="/nlab/show/cocomplete+category">Cocomplete</a> possibly-<a class="existingWikiWord" href="/nlab/show/large+categories">large categories</a> generated under <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a> by <a class="existingWikiWord" href="/nlab/show/small+object">small</a> <a class="existingWikiWord" href="/nlab/show/generators">generators</a> under <a class="existingWikiWord" href="/nlab/show/small+colimit">small</a> <a class="existingWikiWord" href="/nlab/show/relations">relations</a>. Equivalently, <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible</a> <a class="existingWikiWord" href="/nlab/show/reflective+localizations">reflective localizations</a> of <a class="existingWikiWord" href="/nlab/show/free+cocompletions">free cocompletions</a>. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a> localization.</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/toposes">toposes</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th>locally presentable</th><th>loc finitely pres</th><th>localization theorem</th><th><a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a></th><th>accessible</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locales">locales</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/suplattice">suplattice</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+lattices">algebraic lattices</a></td><td style="text-align: left;"><a href="algebraic+lattice#RelationToLocallyFinitelyPresentableCategories">Porst’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/powerset">powerset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/poset">poset</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+finitely+presentable+categories">locally finitely presentable categories</a></td><td style="text-align: left;"><a href="locally+presentable+category#AsLocalizationsOfPresheafCategories">Gabriel–Ulmer’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+toposes">model toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dugger%27s+theorem">Dugger's theorem</a></td><td style="text-align: left;">global <a class="existingWikiWord" href="/nlab/show/model+structures+on+simplicial+presheaves">model structures on simplicial presheaves</a></td><td style="text-align: left;">n/a</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-categories">locally presentable (∞,1)-categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a href="locally+presentable+infinity-category#Definition">Simpson’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf+%28%E2%88%9E%2C1%29-categories">(∞,1)-presheaf (∞,1)-categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-categories">accessible (∞,1)-categories</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Introductions:</p> <ul> <li id="Dugger98"> <p><a class="existingWikiWord" href="/nlab/show/Dan+Dugger">Dan Dugger</a>, <em>Sheaves and homotopy theory</em>, (1998) [<a href="http://www.uoregon.edu/~ddugger/cech.html">web</a>, <a href="http://www.uoregon.edu/~ddugger/cech.dvi">dvi</a>, <a href="http://ncatlab.org/nlab/files/cech.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DuggerSheavesAndHomotopyTheory.pdf" title="pdf">pdf</a>]</p> </li> <li id="Jardine07"> <p><a class="existingWikiWord" href="/nlab/show/John+F.+Jardine">John F. Jardine</a>, <em>Simplicial Presheaves</em>, lecture notes, Fields Institute (2007) [<a href="https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf">pdf</a>, <a href="https://www.uwo.ca/math/faculty/jardine/courses/fields/fields_lectures.html">webpage</a>]</p> </li> </ul> <p>Detailed discussion of the injective model structures on simplicial presheaves:</p> <ul> <li id="JardineLecture"> <p><a class="existingWikiWord" href="/nlab/show/John+F.+Jardine">John F. Jardine</a>, <em>Simplicial presheaves</em>, Journal of Pure and Applied Algebra <strong>47</strong> (1987) 35-87 [<a href="https://doi.org/10.1016/0022-4049(87)90100-9">doi:10.1016/0022-4049(87)90100-9</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+F.+Jardine">John F. Jardine</a>, <em>Stacks and the homotopy theory of simplicial sheaves</em>, Homology, homotopy and applications, vol. 3 (2), 2001, pp.361–384 (<a href="https://projecteuclid.org/euclid.hha/1139840259">euclid:hha/1139840259</a>)</p> </li> <li id="Jardine96"> <p><a class="existingWikiWord" href="/nlab/show/John+F.+Jardine">John F. Jardine</a>, <em>Boolean localization, in practice</em>, Documenta Mathematica <strong>1</strong> (1996), 245-275 (<a href="https://www.math.uni-bielefeld.de/documenta/vol-01/13.html">documenta:vol-01/13</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+F.+Jardine">John F. Jardine</a>, <em><a class="existingWikiWord" href="/nlab/show/Local+homotopy+theory">Local homotopy theory</a></em>, Springer Monographs in Mathematics (2015) [<a href="https://doi.org/10.1007/978-1-4939-2300-7">doi:10.1007/978-1-4939-2300-7</a>]</p> </li> </ul> <p>The projective model structure is discussed in</p> <ul> <li id="Dugger01"><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <em>Universal homotopy theories</em>, Advances in Mathematics Volume 164, Issue 1, (2001) Pages 144-176 (<a href="http://hopf.math.purdue.edu/Dugger/dduniv.pdf">pdf</a>, <a href="https://arxiv.org/abs/math/0007070">arXiv:math/0007070</a>, <a href="https://doi.org/10.1006/aima.2001.2014">doi:10.1006/aima.2001.2014</a>)</li> </ul> <p>See also</p> <ul> <li>Benjamin Blander, <em>Local projective model structures on simplicial presheaves</em>, K-Theory, Volume 24, Number 3, (2001), 283–301, <a href="http://dx.doi.org/10.1023/a:1013302313123">doi</a>.</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/derived+hom-spaces">derived hom-spaces</a> (<a class="existingWikiWord" href="/nlab/show/function+complexes">function complexes</a>) in the projective model structure on simplicial presheaves:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <em>Function complexes for diagrams of simplicial sets</em>, Indagationes Mathematicae (Proceedings) <strong>86</strong> 2 (1983) 139-147 [<a href="https://doi.org/10.1016/1385-7258(83)90051-3">doi:10.1016/1385-7258(83)90051-3</a>, <a href="https://core.ac.uk/download/pdf/82652265.pdf">pdf</a>]</li> </ul> <p>A brief review in the context of <a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a> is in section 4 of</p> <ul> <li id="Olsson">Martin Olsson, <em>Towards non-abelian <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>-adic Hodge theory in the good reduction case</em> (<a href="http://math.berkeley.edu/~molsson/PHT3-24-08.pdf">pdf</a>)</li> </ul> <p>A detailed study of <a class="existingWikiWord" href="/nlab/show/descent">descent</a> for simplicial presheaves is given in</p> <ul> <li id="DuggerHollanderIsaksen"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <a class="existingWikiWord" href="/nlab/show/Sharon+Hollander">Sharon Hollander</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Isaksen">Daniel Isaksen</a>, <em>Hypercovers and simplicial presheaves</em>, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9–51 (<a href="http://www.math.uiuc.edu/K-theory/0563/">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Isaksen">Daniel Isaksen</a>, <em>Weak equivalences of simplicial presheaves</em> (<a href="http://arxiv.org/abs/math/0205025">arXiv</a>)</p> </li> </ul> <p>A survey of many of the model structures together with a treatment of the left local projective one is in</p> <ul> <li id="Blander"><a class="existingWikiWord" href="/nlab/show/Benjamin+Blander">Benjamin Blander</a>, <em>Local projective model structure on simplicial presheaves</em> (<a href="http://www.math.uiuc.edu/K-theory/0462/combination2.pdf">pdf</a>)</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Isaksen">Daniel Isaksen</a>, <em>Flasque model structure for simplicial presheaves</em> (<a href="http://www.math.uiuc.edu/K-theory/0679/">web</a>, <a href="http://www.math.uiuc.edu/K-theory/0679/flasque.pdf">pdf</a>)</li> </ul> <p>The characterization of the model category of simplicial presheaves as the canonical <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentation</a> of the (hypercompletion of) the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a> on a site is in</p> <ul> <li> <p><a href="http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=528">proposition 6.5.2.1</a> of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> </li> </ul> <p>A set of lecture notes on simplicial presheaves with an eye towards algebraic sites and <a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a> is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bertrand+Toen">Bertrand Toen</a>, <em>Simplicial presheaves and derived algebraic geometry</em> , lecture at <a href="http://www.crm.es/HigherCategories/">Simplicial methofs in higher categories</a> (<a href="http://www.crm.cat/HigherCategories/hc1.pdf">pdf</a>)</li> </ul> <p>Last not least, it is noteworthy that the idea of localizing simplicial sheaves at stalkwise weak equivalences is already described and applied in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kenneth+Brown">Kenneth Brown</a>, <em><a class="existingWikiWord" href="/nlab/show/BrownAHT">Abstract Homotopy Theory and Generalized Sheaf cohomology</a></em> ,</li> </ul> <p>using instead of a full <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure the more lightweight one of a Brown <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>.</p> <p>A comparison between Brown-Gersten and Joyal-Jardine approach:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Vladimir+Voevodsky">Vladimir Voevodsky</a>, <em>Homotopy theory of simplicial presheaves in completely decomposable topologies</em>, <a href="http://arxiv.org/abs/0805.4578">arxiv/0805.4578</a></li> </ul> <p>The proposal for descent objects for strict <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>-groupoid-valued presheaves discussed in <a href="#DescentForStrictInf">Descent for strict infinity-groupoids</a> appeared in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Categorical and combinatorial aspects of descent theory</em> (<a href="http://arxiv.org/abs/math/0303175">arXiv</a>)</li> </ul> <p>The relation to the general descent conditionF is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dominic+Verity">Dominic Verity</a>, <em><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">Relating descent notions</a></em></li> </ul> <p>A useful collection of facts is in</p> <ul> <li id="Low"><a class="existingWikiWord" href="/nlab/show/Zhen+Lin+Low">Zhen Lin Low</a>, <em><a class="existingWikiWord" href="/nlab/show/Notes+on+homotopical+algebra">Notes on homotopical algebra</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 15, 2023 at 12:03:45. 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