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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 sheaves </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/2425/#Item_3" 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='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Joyal</a>- <em><a class="existingWikiWord" href="/nlab/show/model+category">model structure</a> on <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial</a> <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a></em> over a <a class="existingWikiWord" href="/nlab/show/site">site</a> is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentation</a> of the <a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a> over that site.</p> <p>It is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to the Jardine-local <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>.</p> <p>There is also a <em><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+model+structure+on+simplicial+sheaves">Čech model structure on simplicial sheaves</a></em> (see there) modelling not the hypercompletion but just the <a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</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> <h2 id="definition">Definition</h2> <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/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/site">site</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Sh(C)^{\Delta^{op}}</annotation></semantics></math> for the category of <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C)</annotation></semantics></math>: the category of <em>simplicial sheaves</em> 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> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/left+proper+model+category">left proper</a> <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched</a> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Sh(C)^{\Delta^{op}}</annotation></semantics></math> such that</p> <ul> <li> <p>cofibrations are precisely the objectwise cofibrations (monomorphisms) of <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s;</p> </li> <li> <p>weak equivalences are the <em>local weak equivalences</em> of the underlying simplicial presheaves as defined at <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>.</p> </li> </ul> </div> <p>This is (<a href="#Jardine">Jardine, theorem 5</a>).</p> <p>Call this the <strong>local injective model structure</strong> on simplicial sheaves.</p> <h2 id="properties">Properties</h2> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> of a <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> into its <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</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>L</mi><mo>⊣</mo><mi>i</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mo>↪</mo><mover><mo>←</mo><mi>L</mi></mover></mover><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (L \dashv i) : Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C) </annotation></semantics></math></div> <p>with <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> given by <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> extends to 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><mi>L</mi><mo>⊣</mo><mi>i</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>loc</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msubsup><mover><mo>↪</mo><mover><mo>←</mo><mi>L</mi></mover></mover><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>loc</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msubsup></mrow><annotation encoding="application/x-tex"> (L \dashv i) : Sh(C)^{\Delta^{op}}_{loc} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)^{\Delta^{op}}_{loc} </annotation></semantics></math></div> <p>between the above local model structure on simplicial sheaves and the injective hyperlocal Jardine-<a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>.</p> </div> <p>This is (<a href="#Jardine">Jardine, theorem 5</a>).</p> <div class="num_prop" id="PresentationOfhypercompleteTopos"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/simplicial+combinatorial+model+category">simplicial combinatorial model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>loc</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msubsup></mrow><annotation encoding="application/x-tex">Sh(C)^{\Delta^{op}}_{loc}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentation</a> for the <a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>Sh</mi><mo stretchy="false">^</mo></mover> <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">\hat Sh_{(\infty,1)}(c)</annotation></semantics></math> 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 <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 class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>Sh</mi><mo stretchy="false">^</mo></mover> <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><mo>≃</mo><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>loc</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msubsup><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup><mo>.</mo></mrow><annotation encoding="application/x-tex"> \hat Sh_{(\infty,1)}(C) \simeq (Sh(C)^{\Delta^{op}}_{loc})^\circ . </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The proof is spelled out at <a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a>.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/dense+sub-site">dense sub-site</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> 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><msub><mover><mi>Sh</mi><mo stretchy="false">^</mo></mover> <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><mo>≃</mo><msub><mover><mi>Sh</mi><mo stretchy="false">^</mo></mover> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \hat Sh_{(\infty,1)}(C) \simeq \hat Sh_{(\infty,1)}(D) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By the <em>comparison lemma</em> at <a class="existingWikiWord" href="/nlab/show/dense+sub-site">dense sub-site</a> we have already an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sh(C) \simeq Sh(D) \,. </annotation></semantics></math></div> <p>This implies the claim with the <a href="#PresentationOfhypercompleteTopos">above proposition</a>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+model+structure+on+simplicial+sheaves">Čech model structure on simplicial sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+presheaves">model structure on cubical presheaves</a></p> </li> </ul> <h2 id="references">References</h2> <p>The local model structure on simplicial sheaves was proposed in</p> <ul> <li id="Joyal1984"><a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a>, Letter to <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, 11.4.1984, (<a href="http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf">pdf scan</a>).</li> </ul> <p>This is, with <a class="existingWikiWord" href="/nlab/show/BrownAHT">BrownAHT</a>, among the first proposals for models for <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> which eventually came to be used in the theory of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a>.</p> <p>Discussion of the model structure:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sjoerd+Crans">Sjoerd Crans</a>, <em>Quillen closed model structure for sheaves</em>, J. Pure Appl. Algebra 101 (1995), 35-57 (<a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/Crans.pdf">pdf</a>)</li> </ul> <p>Jardine’s lectures</p> <ul> <li id="Jardine"><a class="existingWikiWord" href="/nlab/show/John+Frederick+Jardine">John Frederick Jardine</a>, <em>Field Lectures: Simplicial presheaves</em> (<a href="https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf">pdf</a>)</li> </ul> <p>discuss the <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between the model structure on simplicial sheaves and the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>.</p> <p>Wendt discusses “the construction of classifying spaces of fibre sequences in model categories of simplicial sheaves” in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Matthias+Wendt">Matthias Wendt</a>, <em>Classifying spaces and fibrations of simplicial sheaves</em>, <a href="http://arxiv.org/abs/1009.2930">arxiv/1009.2930</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 19, 2022 at 13:55:02. 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