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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="locality_and_descent">Locality and descent</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/localization">localization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+object">local object</a>, <a class="existingWikiWord" href="/nlab/show/local+morphism">local morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+localization">reflective localization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a>, <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></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/descent">descent</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cover">cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+object">descent object</a>, <a class="existingWikiWord" href="/nlab/show/descent+morphism">descent morphism</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/matching+family">matching family</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/stack">stack</a>, <a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>,<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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+descent">cohomological descent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadic+descent">monadic descent</a>, <a class="existingWikiWord" href="/nlab/show/higher+monadic+descent">higher monadic descent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sweedler+coring">Sweedler coring</a>, <a class="existingWikiWord" href="/nlab/show/descent+in+noncommutative+algebraic+geometry">descent in noncommutative algebraic geometry</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/descent+and+locality+-+contents">Edit this sidebar</a> </p> </div></div></div> <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='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#existence_and_refinements'>Existence and refinements</a></li> <ul> <li><a href='#ech_nerves'>Čech nerves</a></li> <li><a href='#OverGeneralSites'>Over general sites</a></li> <li><a href='#OverVerdierSites'>Over Verdier sites</a></li> </ul> <li><a href='#HypercoverHomology'>Hypercover homology</a></li> <li><a href='#DescentAndCohomology'>Descent and cohomology</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#reference'>Reference</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>hypercover</em> is the generalization of a <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> of a <a class="existingWikiWord" href="/nlab/show/cover">cover</a>: it is a simplicial <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> of an object obtained by iteratively applying covering families.</p> <h2 id="definition">Definition</h2> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><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>be the <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> defining a <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</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><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C)</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(C)</annotation></semantics></math>.</p> <div class="num_defn" id="CoskeletonDefinition"> <h6 id="definition_2">Definition</h6> <p>A morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mover><mo>→</mo><mi>f</mi></mover><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>PSh</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"> (Y \stackrel{f}{\to} X) \in PSh(C)^{\Delta^{op}} </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/category+of+simplicial+objects">category of simplicial objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(C)</annotation></semantics></math>, hence the category of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a>, is called a <strong>hypercover</strong> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><msub><mo>×</mo> <mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow></msub><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> Y_{n} \to (\mathbf{cosk}_{n-1} Y)_n \times_{(\mathbf{cosk}_{n-1} X)_n} X_n </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(C)</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/local+epimorphisms">local epimorphisms</a> (in other words, <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 a “<a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy</a> local-epimorphism”).</p> <p>Here this morphism into the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> is that induced from the <a class="existingWikiWord" href="/nlab/show/naturality+square">naturality square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y &\longrightarrow& X \\ \downarrow && \downarrow \\ \mathbf{cosk}_{n-1} Y &\longrightarrow& \mathbf{cosk}_{n-1} X } </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> of the <a class="existingWikiWord" href="/nlab/show/coskeleton">coskeleton</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mi>n</mi></msub><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>PSh</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">\mathbf{cosk}_n : PSh(C)^{\Delta^{op}} \to PSh(C)^{\Delta^{op}}</annotation></semantics></math>.</p> <p>A hypercover is called <strong>bounded</strong> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \geq n</annotation></semantics></math> the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mi>k</mi></msub><mo>→</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><msub><mo>×</mo> <mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mrow></msub><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Y_{k} \to (\mathbf{cosk}_{k-1} Y)_k \times_{(\mathbf{cosk}_{k-1} X)_k} X_k</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>s.</p> <p>The smallest <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> for which this holds is called the <strong>height</strong> of the hypercover.</p> </div> <div class="num_defn" id="SplitHypercover"> <h6 id="definition_3">Definition</h6> <p>A hypercover that also satisfies the <a href="model+structure+on+simplicial+presheaves#CofibrantObjects">cofibrancy condition</a> in the projective <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a> in that</p> <ol> <li> <p>it is simplicial-degree wise a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <a class="existingWikiWord" href="/nlab/show/representable+functor">representables</a>;</p> </li> <li> <p>degenerate cells split off as a <a class="existingWikiWord" href="/nlab/show/coproduct">direct summand</a>)</p> </li> </ol> <p>is called a <strong><a class="existingWikiWord" href="/nlab/show/split+hypercover">split hypercover</a></strong>.</p> </div> <p>(<a href="#DuggerHollanderIsaksen02">Dugger-Hollander-Isaksen 02, def. 4.13</a>) see also (<a href="#Low">Low 14-05-26, 8.2.15</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Definition <a class="maruku-ref" href="#CoskeletonDefinition"></a> is equivalent to saying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : Y \to X</annotation></semantics></math> is a <em>local</em> acyclic fibration: for all <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> every lifting problem</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ \partial \Delta[n] \cdot U &\to& Y \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \Delta[n]\cdot U &\to& X } \right) \;\;\simeq \;\; \left( \array{ \partial \Delta[n] &\to& Y(U) \\ \downarrow && \downarrow^{\mathrlap{f(U)}} \\ \Delta[n] &\to& X(U) } \right) </annotation></semantics></math></div> <p>has a solution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>σ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\sigma_i)</annotation></semantics></math> after refining to some <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>i</mi><mo>:</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∃</mo><msub><mi>σ</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \forall i : \left( \array{ \partial \Delta[n] &\to& Y(U_i) \\ \downarrow &{}^{\mathllap{\exists \sigma_i}}\nearrow& \downarrow^{\mathrlap{f(U_i)}} \\ \Delta[n] &\to& X(U_i) } \right) \,, </annotation></semantics></math></div></div> <p>(<a href="#DuggerHollanderIsaksen02">Dugger-Hollander-Isaksen 02, prop. 7.2</a>).</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>If the <a class="existingWikiWord" href="/nlab/show/topos">topos</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><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C)</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a> a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : Y \to X</annotation></semantics></math> 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><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> is a hypercover if all its <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>s are acyclic <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>s.</p> </div> <p>In this form the notion of hypercover appears for instance in (<a href="#Brown73">Brown 73</a>).</p> <p>In some situations, we may be interested primarily in hypercovers that are built out of data entirely in the site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. We obtain such hypercovers by restricting <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> to be a discrete simplicial object which is representable, and each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Y_n</annotation></semantics></math> to be a coproduct of representables. This notion can equivalently be formulated in terms of diagrams <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">(\Delta/A) \to C</annotation></semantics></math>, where <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 some simplicial set and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Delta/A)</annotation></semantics></math> is its <a class="existingWikiWord" href="/nlab/show/category+of+simplices">category of simplices</a>.</p> <h2 id="examples">Examples</h2> <div class="num_example"> <h6 id="example">Example</h6> <p>Consider the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>const</mi><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X = const X_0</annotation></semantics></math> is simplicially constant. Then the conditions on a morphism <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> to be a hypercover is as follows.</p> <ul> <li> <p>in degree 0: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">Y_0 \to X_0</annotation></semantics></math> is a local epimorphism.</p> </li> <li> <p>in degree 1: The commuting diagram in question is</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>Y</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>diag</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>Y</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>0</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Y_1 &\to& X_0 \\ \downarrow && \downarrow^{\mathrlap{diag}} \\ Y_0 \times Y_0 &\to& X_0 \times X_0 } \,. </annotation></semantics></math></div> <p>Its <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Y</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><msub><mi>X</mi> <mn>0</mn></msub><mo>≃</mo><msub><mi>Y</mi> <mn>0</mn></msub><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><msub><mi>Y</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">(Y_0 \times Y_0)\times_{X_0 \times X_0} X_0 \simeq Y_0 \times_{X_0} Y_0</annotation></semantics></math>, hence the condition is that</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>Y</mi> <mn>0</mn></msub><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><msub><mi>Y</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">Y_1 \to Y_0 \times_{X_0} Y_0</annotation></semantics></math> is a local epimorphism.</p> </li> <li> <p>in degree 2: The commuting diagram in question is</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>Y</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>Id</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msub><mi>Y</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>Y</mi> <mn>0</mn></msub></mrow></msub><msub><mi>Y</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>Y</mi> <mn>0</mn></msub></mrow></msub><msub><mi>Y</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mo>×</mo> <mrow><msub><mi>Y</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>0</mn></msub></mrow></msub></mrow></msub><msub><mi>Y</mi> <mn>0</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Y_2 &\to& X_0 \\ \downarrow && \downarrow^{Id} \\ (Y_1 \times_{Y_0} Y_1 \times_{Y_0}Y_1)_{\times_{Y_0 \times Y_0}} Y_0 &\to& X_0 } \,. </annotation></semantics></math></div> <p>So the condition is that the vertical morphism is a local epi.</p> </li> <li> <p>Similarly, in any degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math> the condition is that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mi>n</mi></msub><mo>→</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> Y_n \to (\mathbf{cosk}_{n-1} Y)_n </annotation></semantics></math></div> <p>is a local epimorphism.</p> </li> </ul> </div> <h2 id="properties">Properties</h2> <h3 id="existence_and_refinements">Existence and refinements</h3> <h4 id="ech_nerves">Čech nerves</h4> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>=</mo><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 = \{U_i \to X\}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cover">cover</a>, the <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a> projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C(U) \to X</annotation></semantics></math> is a hypercover of height 0.</p> </div> <h4 id="OverGeneralSites">Over general sites</h4> <div class="num_prop" id="ExistenceOfSplitRefinements"> <h6 id="proposition_2">Proposition</h6> <p>Given any <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><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},J)</annotation></semantics></math> and given a diagram of simplicial presheaves</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>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>hcov</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && Y \\ && \downarrow^{\mathrlap{hcov}} \\ X' &\longrightarrow& X } </annotation></semantics></math></div> <p>where the vertical morphism is a hypercover, then there exists a completion to a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi><mo>′</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>hcov</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>hcov</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y' &\longrightarrow& Y \\ \downarrow^{\mathrlap{hcov}} && \downarrow^{\mathrlap{hcov}} \\ X' &\longrightarrow& X } </annotation></semantics></math></div> <p>where the left vertical morphism is a split hypercover, def. <a class="maruku-ref" href="#SplitHypercover"></a>.</p> <p>Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒟</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{D}, K)\to (\mathcal{C},J)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/dense+subsite">dense subsite</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">Y'</annotation></semantics></math> as above exists such that it is simplicial-degree wise a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of (<a class="existingWikiWord" href="/nlab/show/representable+functor">representables</a> by) objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="#Low">Low 14-05-26, lemma 8.2.20</a>)</p> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>In particular taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X'\to X</annotation></semantics></math> in prop. <a class="maruku-ref" href="#ExistenceOfSplitRefinements"></a> to be an identity, the proposition says that every hypercover may be refined by a split hypercover.</p> </div> <p>(see also <a href="#Low">Low 14-05-26, lemma 8.2.23</a>)</p> <h4 id="OverVerdierSites">Over Verdier sites</h4> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/Verdier+site">Verdier site</a></strong> is a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> with finite <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> equipped with a <a class="existingWikiWord" href="/nlab/show/basis+for+a+Grothendieck+topology">basis for a Grothendieck topology</a> such that the generating <a class="existingWikiWord" href="/nlab/show/covering">covering</a> maps <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>X</mi></mrow><annotation encoding="application/x-tex">U_i \to X</annotation></semantics></math> all have the property that their <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> U_i \to U_i \times_X U_i </annotation></semantics></math></div> <p>is also a generating covering. We say that <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>X</mi></mrow><annotation encoding="application/x-tex">U_i \to X</annotation></semantics></math> is <strong>basal</strong>.</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>It is sufficient that all the <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>X</mi></mrow><annotation encoding="application/x-tex">U_i \to X</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>s.</p> <p>Examples include the standard <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a>-topology on <a class="existingWikiWord" href="/nlab/show/Top">Top</a>.</p> </div> <div class="num_defn" id="BasalHypercover"> <h6 id="definition_5">Definition</h6> <p>A <strong>basal hypercover</strong> over a <a class="existingWikiWord" href="/nlab/show/Verdier+site">Verdier site</a> is a hypercover <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> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the components of the maps into the matching object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>n</mi></msub><mo>→</mo><mi>M</mi><msub><mi>U</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">U_n \to M U_n</annotation></semantics></math> are basal maps, as above.</p> </div> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Over a Verdier site, every hypercover may be refined by a split (def. <a class="maruku-ref" href="#SplitHypercover"></a>) and basal hypercover (def. <a class="maruku-ref" href="#BasalHypercover"></a>).</p> </div> <p>This is (<a href="#DuggerHollanderIsaksen02">Dugger-Hollander-Isaksen 02, theorem 8.6</a>).</p> <h3 id="HypercoverHomology">Hypercover homology</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : Y \to X</annotation></semantics></math> be a hypercover. We may regard this as an object in the <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</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><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Sh(C)/X</annotation></semantics></math>. By the discussion <a href="http://ncatlab.org/nlab/show/category+of+presheaves#RelWithOvercategories">here</a> this is equivalently <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><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C/X)</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ab(Sh(C/X))</annotation></semantics></math> for the category of abelian <a class="existingWikiWord" href="/nlab/show/group+object">group object</a>s in the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</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><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C/X)</annotation></semantics></math>. This is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>.</p> <p>Forming in the sheaf topos the <a class="existingWikiWord" href="/nlab/show/free+functor">free</a> abelian group on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_n</annotation></semantics></math> for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, we obtain a simplicial abelian group object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo>∈</mo><mi>Ab</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">\bar f \in Ab(Sh(C/X))^{\Delta}</annotation></semantics></math>. As such this has a <a class="existingWikiWord" href="/nlab/show/Moore+complex">normalized chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_\bullet(\bar f)</annotation></semantics></math>.</p> <div class="num_prop"> <h6 id="proposition_3">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>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : Y \to X</annotation></semantics></math> a hypercover, the <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\bar f)</annotation></semantics></math> vanishes in positive degree and is the group of <a class="existingWikiWord" href="/nlab/show/integer">integer</a>s in degree 0, as an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ab(Sh(C)(X)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mi>for</mi><mspace width="thickmathspace"></mspace><mi>p</mi><mo>≥</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mi>ℤ</mi></mtd> <mtd><mi>for</mi><mspace width="thickmathspace"></mspace><mi>p</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_p(N(f)) \simeq \left\{ \array{ 0 & for \; p \geq 1 \\ \mathbb{Z} & for \; p = 0 } \right. \,. </annotation></semantics></math></div></div> <h3 id="DescentAndCohomology">Descent and cohomology</h3> <p>The following theorem characterizes the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>-condition for the <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentation</a> of an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> by a <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a> in terms of descent along hypercovers.</p> <div class="num_theorem" id="DescentFromDescentAlongHypercovers"> <h6 id="theorem_2">Theorem</h6> <p>In the <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">PSh(C)^{\Delta^{op}}</annotation></semantics></math> an object is fibrant precisely if it is fibrant in the global <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> and in addition satisfies <a class="existingWikiWord" href="/nlab/show/descent">descent</a> along all hypercovers over representables that are degreewise <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>s of representables.</p> </div> <p>This is the central theorem in (<a href="#DuggerHollanderIsaksen02">Dugger-Hollander-Isaksen 02</a>).</p> <p>The following theorem is a corollary of this theorem, using the discussion at <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>. But historically it predates the above- theorem.</p> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p><strong>(Verdier’s hypercovering theorem)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and <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> a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, we have that the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with coefficients in <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 given</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow></msub><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><msub><mi>Hom</mi> <mi>Sh</mi></msub><mo stretchy="false">(</mo><msup><mi>Y</mi> <mo>•</mo></msup><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^q(X, F) \simeq {\lim_{\to}}_{Y \to X} H^q(Hom_{Sh}(Y^\bullet,F)) </annotation></semantics></math></div> <p>by computing for each hypercover <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> the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a> of the cosimplicial abelian group obtained by evaluating <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> degreewise on the hypercover, and then taking the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of the result over the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> of all hypercovers over <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> </div> <p>A proof of this result in terms of the structure of a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> on the category of simplicial presheaves appears in (<a href="#Brown73">Brown 73, section 3</a>).</p> <h2 id="related_entries">Related entries</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/hypersheaf">hypersheaf</a></li> </ul> <h2 id="reference">Reference</h2> <p>The concept of hypercovers was introduced for <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Verdier">Jean-Louis Verdier</a>, Exposé V, sect. 7 of <a class="existingWikiWord" href="/nlab/show/SGA4">SGA4</a>,</li> </ul> <p>An early standard reference founding <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+homotopy+theory">étale homotopy theory</a> is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Artin">Michael Artin</a>, <a class="existingWikiWord" href="/nlab/show/Barry+Mazur">Barry Mazur</a>, <em>Étale Homotopy</em> , Lecture Notes in Mathematics 100, Springer- Verlag, Berline-Heidelberg-New York (1972).</li> </ul> <p>The modern reformulation of their notion of hypercover in terms of simplicial presheaves is mentioned for instance at the end of section 2, on <a href="http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf#page=6">p. 6</a> of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Frederick+Jardine">John Frederick Jardine</a>, <em>Fields Lectures: Simplicial presheaves</em> (<a href="http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf">pdf</a>)</li> </ul> <p>A discussion of hypercovers of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> and relation to <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+homotopy+type">étale homotopy type</a> of <a class="existingWikiWord" href="/nlab/show/smooth+schemes">smooth schemes</a> and <a class="existingWikiWord" href="/nlab/show/A1-homotopy+theory">A1-homotopy theory</a> is in</p> <ul> <li id="Isaksen01"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Isaksen">Daniel Isaksen</a>, <em>Étale realization of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1</annotation></semantics></math>-homotopy theory of schemes</em>, 2001 (<a href="http://www.math.uiuc.edu/K-theory/0495/">K-theory archive</a>)</p> </li> <li id="DuggerIsaksen05"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Isaksen">Daniel Isaksen</a>, <em>Hypercovers in topology</em>, 2005 (<a href="http://www.math.uiuc.edu/K-theory/0528/hypercover.pdf">pdf</a>, <a href="http://www.math.uiuc.edu/K-theory/0528/">K-Theory archive</a>)</p> </li> </ul> <p>A discussion in a topos with enough points in in</p> <ul> <li id="Brown73"><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>, 1973</li> </ul> <p>A thorough discussion of hypercovers over representables and their role in <a class="existingWikiWord" href="/nlab/show/descent">descent</a> for simplicial presheaves is in</p> <ul> <li id="DuggerHollanderIsaksen02"><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>, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 1., 2004. (<a href="http://arxiv.org/abs/math/0205027">arXiv:0205027</a>, <a href="http://www.math.uiuc.edu/K-theory/0563/">K-theory archive</a>)</li> </ul> <p>On the Verdier hypercovering theorem see</p> <ul> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Frederick+Jardine">John Frederick Jardine</a>, <em>The Verdier hypercovering theorem</em> (<a href="https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/verdier-hypercovering-theorem/13FA7ACF61C486C2FD5C06FFF6483562">doi</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zhen+Lin+Low">Zhen Lin Low</a>, <em>Cocycles in categories of fibrant objects</em>, (<a href="http://arxiv.org/abs/1502.03925v3">pdf</a>)</p> </li> </ul> <p>Split hypercover refinement over general sites is discussed 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 December 11, 2023 at 12:07:45. 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