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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"> <p class="show_diff"> Showing changes from revision #155 to #156: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <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'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></strong>, <a class='existingWikiWord' href='/nlab/show/diff/model+%28%E2%88%9E%2C1%29-category'>model $\infty$-category</a></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/relative+category'>relative category</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibration'>fibration</a>, <a class='existingWikiWord' href='/nlab/show/diff/cofibration'>cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+factorization+system'>weak factorization system</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/resolution'>resolution</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+%28as+an+operation%29'>homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a><math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /></mrow><annotation encoding='application/x-tex'>\;</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Quillen+adjunction'>Quillen adjunction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Quillen+bifunctor'>Quillen bifunctor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/homotopy+Kan+extension'>homotopy Kan extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+weighted+colimit'>homotopy weighted (co)limit</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy (co)end</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bousfield-Kan+formula'>Bousfield-Kan map</a></p> </li> </ul> <p><strong>Refinements</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+model+category'>monoidal model category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/monoidal+Quillen+adjunction'>monoidal Quillen adjunction</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+model+category'>enriched model category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/enriched+Quillen+adjunction'>enriched Quillen adjunction</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+monoidal+model+category'>monoidal enriched model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+Quillen+adjunction'>simplicial Quillen adjunction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+monoidal+model+category'>simplicial monoidal model category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cofibrantly+generated+model+category'>cofibrantly generated model category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/combinatorial+model+category'>combinatorial model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cellular+model+category'>cellular model category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+model+category'>algebraic model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+model+category'>compactly generated model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+model+category'>proper model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+model+category'>cartesian closed model category</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+model+category'>locally cartesian closed model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+functors'>on functor categories (global)</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Reedy+model+structure'>Reedy model structure</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/slice+model+structure'>on slice categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>Bousfield localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/transferred+model+structure'>transferred model structure</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebraic+fibrant+objects'>on algebraic fibrant objects</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction+for+model+categories'>Grothendieck construction for model categories</a></p> </li> </ul> <p><strong>Presentation of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_4' xmlns='http://www.w3.org/1998/Math/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/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+localization'>simplicial localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>(∞,1)-categorical hom-space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentable (∞,1)-category</a></p> </li> </ul> <p><strong>Model structures</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Cisinski+model+structure'>Cisinski model structure</a></li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_5' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+for+infinity-groupoids'>for ∞-groupoids</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+topological+spaces'>on topological spaces</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+compactly+generated+topological+spaces'>on compactly generated spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+Delta-generated+topological+spaces'>on Delta-generated spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+diffeological+spaces'>on diffeological spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model structure</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Thomason+model+structure'>Thomason model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+simplicial+sets'>on simplicial sets</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+semi-simplicial+sets'>on semi-simplicial sets</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/constructive+model+structure+on+simplicial+sets'>constructive model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+left+fibrations'>for right/left fibrations</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+groupoids'>model structure on simplicial groupoids</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cubical+sets'>on cubical sets</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+strict+omega-groupoids'>on strict ∞-groupoids</a>, <a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure+on+groupoids'>on groupoids</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+chain+complexes'>on chain complexes</a>/<a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cosimplicial+abelian+groups'>model structure on cosimplicial abelian groups</a></p> <p>related by the <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cosimplicial+simplicial+sets'>model structure on cosimplicial simplicial sets</a></p> </li> </ul> <p><em>for equivariant <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_6' xmlns='http://www.w3.org/1998/Math/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/diff/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/diff/Borel+model+structure'>coarse model structure on topological G-spaces</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/Borel+model+structure'>Borel model structure</a>)</p> </li> </ul> <p><em>for rational <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_7' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+dg-algebras'>model structure on dgc-algebras</a></li> </ul> <p><em>for rational equivariant <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_8' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+equivariant+chain+complexes'>model structure on equivariant chain complexes</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+equivariant+dgc-algebras'>model structure on equivariant dgc-algebras</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_9' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+for+homotopy+n-types'>for n-groupoids</a>/<a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+homotopy+n-types'>for n-types</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure+on+groupoids'>for 1-groupoids</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_10' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+simplicial+groups'>model structure on simplicial groups</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+reduced+simplicial+sets'>model structure on reduced simplicial sets</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-algebras</em></p> <p><em>general <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_12' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+monoids+in+a+monoidal+model+category'>on monoids</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+algebras'>on simplicial T-algebras</a>, on <a class='existingWikiWord' href='/nlab/show/diff/homotopy+T-algebra'>homotopy T-algebra</a>s</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebras+over+a+monad'>on algebas over a monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebras+over+an+operad'>on algebras over an operad</a>, <a class='existingWikiWord' href='/nlab/show/diff/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 class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_13' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+dg-algebras'>model structure on differential-graded commutative algebras</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/model+structure+on+dg-algebras'>on dg-algebras</a> and on <a class='existingWikiWord' href='/nlab/show/diff/simplicial+ring'>on simplicial rings</a>/<a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cosimplicial+rings'>on cosimplicial rings</a></p> <p>related by the <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+L-infinity+algebras'>for L-∞ algebras</a>: <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-Lie+algebras'>on dg-Lie algebras</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-coalgebras'>on dg-coalgebras</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+Lie+algebras'>on simplicial Lie algebras</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+spectra'>model structure on spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+ring+spectra'>model structure on ring spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+parameterized+spectra'>model structure on parameterized spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+presheaves+of+spectra'>model structure on presheaves of spectra</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_14' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+relative+categories'>on categories with weak equivalences</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+quasi-categories'>Joyal model for quasi-categories</a> (and its <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+cubical+quasicategories'>cubical version</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+sSet-categories'>on sSet-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+complete+Segal+spaces'>for complete Segal spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+Cartesian+fibrations'>for Cartesian fibrations</a></p> </li> </ul> <p><em>for stable <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_15' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+dg-categories'>on dg-categories</a></li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_16' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+operads'>on operads</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+Segal+operads'>for Segal operads</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebras+over+an+operad'>on algebras over an operad</a>, <a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+dendroidal+sets'>on dendroidal sets</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+dendroidal+complete+Segal+spaces'>for dendroidal complete Segal spaces</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+dendroidal+Cartesian+fibrations'>for dendroidal Cartesian fibrations</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_17' xmlns='http://www.w3.org/1998/Math/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/diff/Theta-space'>for (n,r)-categories as ∞-spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+weak+complicial+sets'>for weak ∞-categories as weak complicial sets</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cellular+sets'>on cellular sets</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure'>on higher categories in general</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+strict+omega-categories'>on strict ∞-categories</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_18' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_19' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+homotopical+presheaves'>on homotopical presheaves</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>on simplicial presheaves</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/global+model+structure+on+simplicial+presheaves'>global model structure</a>/<a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+model+structure+on+simplicial+presheaves'>Cech model structure</a>/<a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+presheaves'>local model structure</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sheaves'>on simplicial sheaves</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+presheaves+of+simplicial+groupoids'>on presheaves of simplicial groupoids</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+sSet-enriched+presheaves'>on sSet-enriched presheaves</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+%282%2C1%29-sheaves'>model structure for (2,1)-sheaves</a>/for stacks</p> </li> </ul> </div> <h4 id='topos_theory'><math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_20' xmlns='http://www.w3.org/1998/Math/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'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%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/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-presheaf'>(∞,1)-presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%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/diff/elementary+%28infinity%2C1%29-topos'>elementary (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-site'>(∞,1)-site</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+an+%28infinity%2C1%29-category'>localization of an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+localization'>topological localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hypercompletion'>hypercompletion</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a>/<a class='existingWikiWord' href='/nlab/show/diff/derived+stack'>derived stack</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28n%2C1%29-topos'>(n,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/n-topos'>n-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/truncation'>n-truncated object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+object'>n-connected object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topos'>(1,1)-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-topos'>(2,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/2-topos'>2-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%282%2C1%29-presheaf'>(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-quasitopos'>(∞,1)-quasitopos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separated+%28infinity%2C1%29-presheaf'>separated (∞,1)-presheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quasitopos'>quasitopos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/separated+%282%2C1%29-presheaf'>separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C2%29-topos'>(∞,2)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-topos'>(∞,n)-topos</a></p> </li> </ul> <h2 id='sidebar_characterization'>Characterization</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pullback-stable+colimit'>universal colimits</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/effective+epimorphism'>effective epimorphism</a></li> </ul> </li> </ul> <h2 id='sidebar_morphisms'>Morphisms</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-geometric+morphism'>(∞,1)-geometric morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29Topos'>(∞,1)Topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/hypercomplete+%28infinity%2C1%29-topos'>hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hypercomplete+object'>hypercomplete object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-topos'>over-(∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-localic+%28infinity%2C1%29-topos'>n-localic (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+n-connected+%28n%2B1%2C1%29-topos'>locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/structured+%28infinity%2C1%29-topos'>structured (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/geometry+%28for+structured+%28infinity%2C1%29-toposes%29'>geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+n-connected+%28n%2B1%2C1%29-topos'>locally ∞-connected (∞,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+n-connected+%28n%2B1%2C1%29-topos'>∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-local+geometric+morphism'>local (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/concrete+%28infinity%2C1%29-sheaf'>concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%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/diff/presentations+of+%28infinity%2C1%29-sheaf+%28infinity%2C1%29-toposes'>models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+functors'>model structure on functors</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+site'>model site</a>/<a class='existingWikiWord' href='/nlab/show/diff/sSet-site'>sSet-site</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/cohesive+%28infinity%2C1%29-topos'>cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/shape+of+an+%28infinity%2C1%29-topos'>shape</a> / <a class='existingWikiWord' href='/nlab/show/diff/coshape+of+an+%28infinity%2C1%29-topos'>coshape</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+groups+in+an+%28infinity%2C1%29-topos'>homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical</a>/<a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric</a> homotopy groups</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Postnikov+tower+in+an+%28infinity%2C1%29-category'>Postnikov tower</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+tower+in+an+%28infinity%2C1%29-topos'>Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/function+algebras+on+infinity-stacks'>rational homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dimension'>dimension</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+dimension'>homotopy dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomological+dimension'>cohomological dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/covering+dimension'>covering dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#the_different_model_structures_and_their_interrelation'>The different model structures and their interrelation</a><ul><li><a href='#injectiveprojective__localglobal__presheavessheaves'>Injective/projective - local/global - presheaves/sheaves</a></li><li><a href='#reedy_and_intermediate_model_structures'>Reedy and intermediate model structures</a></li><li><a href='#DependencyOnSite'>Dependency on the underlying site</a></li></ul></li><li><a href='#PresentationOfInfiniToposes'>Presentation of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-toposes</a></li><li><a href='#FibAndCofibObjects'>Fibrant and cofibrant objects</a><ul><li><a href='#fibrant_objects'>Fibrant objects</a></li><li><a href='#CofibrantObjects'>Cofibrant objects</a></li><li><a href='#CofibrantReplacement'>Cofibrant replacement</a></li></ul></li><li><a href='#local_fibrations'>Local fibrations</a></li><li><a href='#Descent'>Localization and descent</a><ul><li><a href='#ČechLocalization'>Čech localization at Grothendieck (pre)topologies</a></li><li><a href='#for_values_in_strict_and_abelian_groupoids'>For values in strict and abelian <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids</a></li></ul></li><li><a href='#Properness'>Properness</a></li><li><a href='#MonoidalStructure'>Closed monoidal structure</a></li><li><a href='#HomotopyLimits'>Homotopy (co)limits</a></li><li><a href='#InclusionOfChainComplexes'>Inclusion of chain complexes of sheaves</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>Model structures</a> on <a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaves</a> <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>present</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-categories of (∞,1)-presheaves</a> and <a class='existingWikiWord' href='/nlab/show/diff/localization+of+an+%28infinity%2C1%29-category'>localizations of these</a>, such as notably the left exact localizations that are <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-categories of (∞,1)-sheaves</a>: these model structures are <a class='existingWikiWord' href='/nlab/show/diff/presentations+of+%28infinity%2C1%29-sheaf+%28infinity%2C1%29-toposes'>models for ∞-stack (∞,1)-toposes</a>.</p> <p>Recall that</p> <ul> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> is a way to <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>present</a> an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a>;</p> </li> <li> <p>in the context of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a> <a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaves</a> on an <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> are given by <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functors</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo></mrow><annotation encoding='application/x-tex'>C^{op} \to</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>SSet</a>.</p> </li> </ul> <p>This suggests that the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a> on some <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> can be <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presented</a> by a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> structure on the ordinary <a class='existingWikiWord' href='/nlab/show/diff/functor+category'>functor category</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>SSet</mi><mo stretchy='false'>]</mo><mo>≃</mo><mo stretchy='false'>[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> [C^{op},SSet] \simeq [\Delta^{op}, PSh(C)] </annotation></semantics></math></div> <p>– the category of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaves</a> .</p> <p>Various interrelated flavors of model structures on the category of simplicial presheaves on <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> have been introduced and studied since the 1970s, originally by K. Brown and <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>Andre Joyal</a> and then developed in detail by <a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>J. F. Jardine</a>.</p> <p>Notice that when regarded as a presentation of an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a>, i.e. of an <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a>, a simplicial presheaf – being an ordinary functor instead of a <a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctor</a> – corresponds to a <a class='existingWikiWord' href='/nlab/show/diff/rectified+infinity-stack'>rectified ∞-stack</a>. It might therefore seem that a model given by simplicial presheaves is too restrictive to capture the full expected flexibility of a notion of <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a>. But this is not so.</p> <p>In</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a></li> </ul> <p>a fully general definition of a <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of ∞-stacks</a> is given it is shown – proposition 6.5.2.1 – that, indeed, the Brown–Joyal–Jardine model is a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentation</a> of that.</p> <p>More precisely</p> <ul> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/global+model+structure+on+simplicial+presheaves'>global model structure on simplicial presheaves</a> on a category is a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentation</a> of the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+model+structure+on+simplicial+presheaves'>Čech model structure on simplicial presheaves</a> on a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> is a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentation</a> of the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+presheaves'>local model structure on simplicial presheaves</a> on a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> is a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentation</a> of the <em>hypercompletion</em> of the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a> (see the discussion at <a class='existingWikiWord' href='/nlab/show/diff/hypercover'>hypercover</a>).</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>Bousfield localization</a> of the global <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> structure to the local one presents the corresponding <a class='existingWikiWord' href='/nlab/show/diff/localization+of+an+%28infinity%2C1%29-category'>localization of an (∞,1)-category</a> from presheaves to sheaves, mimicking the corresponding statement for a <a class='existingWikiWord' href='/nlab/show/diff/category+of+sheaves'>category of sheaves</a>.</p> </li> </ul> <p>Originally K. Brown had considered in <a class='existingWikiWord' href='/nlab/show/diff/BrownAHT'>BrownAHT</a> not a model structure on simplicial presheaves but</p> <ul> <li>a <a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a> structure on locally Kan simplicial sheaves (see there for details)</li> </ul> <p>and Joyal had originally considered a</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+sheaves'>local model structure on simplicial sheaves</a>.</li> </ul> <p>Joyal’s <a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+sheaves'>local model structure on simplicial sheaves</a> is <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalent</a> to the injective <a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+presheaves'>local model structure on simplicial presheaves</a>.</p> <p>By repackaging <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>es as <a class='existingWikiWord' href='/nlab/show/diff/simplicial+groupoid'>simplicial groupoids</a> one obtains a <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+presheaves+of+simplicial+groupoids'>model structure on presheaves of simplicial groupoids</a> which is also Quillen equivalent to the above.</p> <p>If K. Brown’s <a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a> on locally Kan simplicial sheaves is restricted to globally Kan simplicial sheaves on a <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a> with <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+topos'>enough point</a> then it is the full subcategory on the fibrant objects in the projective <a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+sheaves'>local model structure on simplicial sheaves</a>.</p> <p>But since in all cases the weak equivalences are the same (where they apply, for Brown’s model if the topos has enough points), all these local <a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical categories</a> define equivalent <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy categories</a>.</p> <p>By Lurie’s result these are in each case in turn equivalent to the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of</a> the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a> of <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stacks</a>. So in particular they serve as a home for general <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a>.</p> <p>Various old results appear in a new light this way. For instance using the old result of <a class='existingWikiWord' href='/nlab/show/diff/BrownAHT'>BrownAHT</a> on the way ordinary <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a> is embedded in the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> of simplicial sheaves, one sees that the old right derived functor definition of <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a> really computes the <a class='existingWikiWord' href='/nlab/show/diff/infinity-stackification'>∞-stackification</a> of a sheaf of <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a>es regarded under the <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold–Kan correspondence</a> as a simplicial sheaf.</p> <h2 id='the_different_model_structures_and_their_interrelation'>The different model structures and their interrelation</h2> <p>It is the very point of <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> structures on a given <a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a> that there may be several of them: each <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presenting</a> the same <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> but also each suited for different computational purposes.</p> <p>So it is good that there are many model structures on simplicial (pre)sheaves, as there are.</p> <h3 id='injectiveprojective__localglobal__presheavessheaves'>Injective/projective - local/global - presheaves/sheaves</h3> <p>The following diagram is a map for part of the territory:</p> <div style='overflow: auto;'><div class='maruku-equation'> <math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_1' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <mrow> <mtable><mtr><mtd /> <mtd /> <mtd><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>Sh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd /> <mtd><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd /> <mtd><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mi>Sh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↑</mo> <mi>presentation</mi></msup></mtd> <mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↑</mo> <mi>presentation</mi></msup></mtd> <mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↑</mo> <mi>presentation</mi></msup></mtd></mtr> <mtr><mtd><mi>SSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msubsup><mo stretchy='false'>)</mo> <mi>inj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>→</mo><mi>embedding</mi></mover><mover><mo>←</mo><mi>sheafification</mi></mover></mover></mtd> <mtd><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msubsup><mo stretchy='false'>)</mo> <mi>inj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mo>←</mo><mrow /></mover><mo stretchy='false'>|</mo></mtd> <mtd><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mi>inj</mi></msub></mtd> <mtd><mover><mover><mo>→</mo><mi>Id</mi></mover><mover><mo>←</mo><mi>Id</mi></mover></mover></mtd> <mtd><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mtd> <mtd><mover><mo>↦</mo><mrow /></mover></mtd> <mtd><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msubsup><mo stretchy='false'>)</mo> <mi>proj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>←</mo><mi>embedding</mi></mover><mover><mo>→</mo><mi>sheafification</mi></mover></mover></mtd> <mtd><mi>SSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msubsup><mo stretchy='false'>)</mo> <mi>proj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd></mtr> <mtr><mtd><mi>Joyal</mi></mtd> <mtd><mover><mo>↔</mo><mrow><mi>Quillen</mi><mi>equivalence</mi></mrow></mover></mtd> <mtd><mi>Jardine</mi></mtd> <mtd><mover><mrow><mo>←</mo><mo stretchy='false'>|</mo></mrow><mrow><mi>left</mi><mi>Bousf</mi><mo>.</mo><mi>localization</mi></mrow></mover></mtd> <mtd><mi>Heller</mi></mtd> <mtd><mover><mo>↔</mo><mrow><mi>Quillen</mi><mi>equivalence</mi></mrow></mover></mtd> <mtd><mi>Bousfield</mi><mo>−</mo><mi>Kan</mi></mtd> <mtd><mover><mo>↦</mo><mrow><mi>left</mi><mi>Bousf</mi><mo>.</mo><mi>localization</mi></mrow></mover></mtd> <mtd><mi>Blander</mi></mtd> <mtd><mover><mo>↔</mo><mrow><mi>Quillen</mi><mi>equivalence</mi></mrow></mover></mtd> <mtd><mi>Brown</mi><mo>−</mo><mi>Gersten</mi></mtd></mtr> <mtr><mtd /></mtr> <mtr><mtd /> <mtd><mi>everything</mi><mi>cofibrant</mi><mo>;</mo></mtd></mtr> <mtr><mtd /> <mtd><mi>fibrant</mi><mo>=</mo><mi>global</mi><mi>injective</mi><mi>fib</mi><mo>.</mo><mo>.</mo><mo>.</mo></mtd></mtr> <mtr><mtd><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mtd> <mtd><mo>.</mo><mo>.</mo><mo>.</mo><mi>satisfying</mi><mi>descent</mi></mtd> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd><mi>cofibrant</mi><mo>=</mo><mi>global</mi><mi>projective</mi><mi>cofib</mi><mo>;</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd><mi>fibrant</mi><mo>=</mo><mi>Kan</mi><mi>valued</mi><mi>and</mi><mo>.</mo><mo>.</mo><mo>.</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>.</mo><mo>.</mo><mo>.</mo><mi>satisfying</mi><mi>descent</mi></mtd></mtr></mtable> </mrow> </mrow> <annotation encoding='application/x-tex'> \array{ &amp;&amp; (\infty,1)Sh(C) &amp;&amp;&amp; (\infty,1)PSh(C) &amp;&amp;&amp; (\infty,1)Sh(C) \\ &amp;&amp; \uparrow^{presentation} &amp;&amp;&amp; \uparrow^{presentation} &amp;&amp;&amp; \uparrow^{presentation} \\ SSh(C)^{l loc}_{inj} &amp; \stackrel{\stackrel{sheafification}{\leftarrow}} {\stackrel{embedding}{\to}}&amp; SPSh(C)^{l loc}_{inj} &amp;\stackrel{}{\leftarrow}|&amp; SPSh(C)_{inj} &amp;\stackrel{\stackrel{Id}{\leftarrow}} {\stackrel{Id}{\rightarrow}}&amp; SPSh(C)_{proj} &amp;\stackrel{}{\mapsto}&amp; SPSh(C)_{proj}^{l loc} &amp; \stackrel{\stackrel{sheafification}{\to}} {\stackrel{embedding}{\leftarrow}}&amp; SSh(C)_{proj}^{l loc} \\ Joyal &amp;\stackrel{Quillen equivalence}{\leftrightarrow}&amp; Jardine &amp;\stackrel{left Bousf. localization}{\leftarrow|}&amp; Heller &amp;\stackrel{Quillen equivalence}{\leftrightarrow}&amp; Bousfield-Kan &amp;\stackrel{left Bousf. localization}{\mapsto}&amp; Blander &amp;\stackrel{Quillen equivalence}{\leftrightarrow}&amp; Brown-Gersten \\ \\ &amp; everything cofibrant; \\ &amp; fibrant = global injective fib... \\ \;\;\; &amp; ...satisfying descent &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp; cofibrant = global projective cofib; \\ &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp; fibrant = Kan valued and... \\ &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp; \;\;\; ...satisfying descent } </annotation> </semantics> </math> </div></div> <p>Here</p> <ul> <li> <p>“inj” denotes the injective model structure: cofibrations are objectwise cofibrations</p> </li> <li> <p>“proj” denotes the projective model structure: fibrations are objectwise fibrations</p> </li> <li> <p>no “loc” subscript means global model structure: weak equivalences are the objectwise weak equivalences:</p> </li> <li> <p>“l loc” denotes <strong>left</strong> <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>Bousfield localization</a> at <a class='existingWikiWord' href='/nlab/show/diff/hypercover'>hypercovers</a> (at <a class='existingWikiWord' href='/nlab/show/diff/stalk'>stalk</a>wise acyclic fibrations if the <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+topos'>topos has enough points</a>)</p> </li> </ul> <p>The identity functor on the category <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>SPSh(C)</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaves</a> is a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a> for the projective and injective <a class='existingWikiWord' href='/nlab/show/diff/global+model+structure+on+simplicial+presheaves'>global model structure</a> and this is a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalence</a>.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+sheaves'>local model structures on simplicial sheaves</a> are just the restrictions of the <a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+presheaves'>those on simplicial presheaves</a>. (For the injective structure this is in <a href='#JardineLecture'>Jardine</a>, for the projective one in <a href='#Blander'>Blander, theorem 2.1, 2.2</a>).</p> <p>These are related by a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a> given by the usual <a class='existingWikiWord' href='/nlab/show/diff/geometric+embedding'>geometric embedding</a> of the <a class='existingWikiWord' href='/nlab/show/diff/category+of+sheaves'>category of sheaves</a> as a full <a class='existingWikiWord' href='/nlab/show/diff/subcategory'>subcategory</a> of that of presheaves – with <a class='existingWikiWord' href='/nlab/show/diff/sheafification'>sheafification</a> the <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> – and this is also <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalence</a>.</p> <p>The characteristic of the <em>left</em> Bousfield localizations is that for them the fibrant objects are those that satisfy <a class='existingWikiWord' href='/nlab/show/diff/descent'>descent</a>: see <a class='existingWikiWord' href='/nlab/show/diff/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a>.</p> <p>In either case</p> <ul> <li>the global model structures <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presents</a> the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a></li> </ul> <p>while</p> <ul> <li>the local model structures <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presents</a> the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a> (i.e. <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stacks</a>).</li> </ul> <p>The following diagram collection <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model categories</a> that are <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentations</a> for the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a>. All indicated morphism pairs are <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalences</a>.</p> <div style='overflow: auto;'><div class='maruku-equation'> <math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_2' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <mrow> <mtable><mtr><mtd><mi>PSh</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>SGrpd</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mover><mo>→</mo><mi>sheafification</mi></mover><mover><mo>←</mo><mi>embedding</mi></mover></mover></mtd> <mtd><mi>Sh</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>SGrpd</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>↔</mo><mrow /></mover></mtd> <mtd><mi>Sh</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>SSet</mi><msubsup><mo stretchy='false'>)</mo> <mi>inj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>→</mo><mi>embedding</mi></mover><mover><mo>←</mo><mi>sheafification</mi></mover></mover></mtd> <mtd><mi>PSh</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>SSet</mi><msubsup><mo stretchy='false'>)</mo> <mi>inj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>→</mo><mi>Id</mi></mover><mover><mo>←</mo><mi>Id</mi></mover></mover></mtd> <mtd><mi>PSh</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>SSet</mi><msubsup><mo stretchy='false'>)</mo> <mi>proj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd> <mtd><mover><mover><mo>→</mo><mi>sheafification</mi></mover><mover><mo>←</mo><mi>embedding</mi></mover></mover></mtd> <mtd><mi>Sh</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>SSet</mi><msubsup><mo stretchy='false'>)</mo> <mi>proj</mi> <mrow><mi>l</mi><mi>loc</mi></mrow></msubsup></mtd></mtr> <mtr><mtd /></mtr> <mtr><mtd><mi>Luo</mi><mo>−</mo><mi>Bubenik</mi><mo>−</mo><mi>Kim</mi></mtd> <mtd /> <mtd><mi>Joyal</mi><mo>−</mo><mi>Tierney</mi></mtd> <mtd /> <mtd><mi>Joyal</mi></mtd> <mtd /> <mtd><mi>Jardine</mi></mtd> <mtd /> <mtd><mi>Blander</mi></mtd> <mtd /> <mtd><mi>Brown</mi><mo>−</mo><mi>Gersten</mi></mtd></mtr></mtable> </mrow> </mrow> <annotation encoding='application/x-tex'> \array{ PSh(X, SGrpd) &amp;\stackrel{\stackrel{embedding}{\leftarrow}} {\stackrel{sheafification}{\to}}&amp; Sh(X,SGrpd) &amp;\stackrel{}{\leftrightarrow}&amp; Sh(X, SSet)^{l loc}_{inj} &amp;\stackrel{\stackrel{sheafification}{\leftarrow}} {\stackrel{embedding}{\to}}&amp; PSh(X, SSet)^{l loc}_{inj} &amp;\stackrel{\stackrel{Id}{\leftarrow}} {\stackrel{Id}{\to}}&amp; PSh(X, SSet)^{l loc}_{proj} &amp;\stackrel{\stackrel{embedding}{\leftarrow}} {\stackrel{sheafification}{\to}}&amp; Sh(X, SSet)^{l loc}_{proj} \\ \\ Luo-Bubenik-Kim &amp;&amp; Joyal-Tierney &amp;&amp; Joyal &amp;&amp; Jardine &amp;&amp; Blander &amp;&amp; Brown-Gersten } </annotation> </semantics> </math> </div></div> <p>On the right this lists the model structures on simplicial (pre)sheaves, here displayed as (pre)sheaves with values in <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a>, using <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>SSet</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>SPSh(C) \simeq PSh(C,SSet)</annotation></semantics></math>.</p> <p>On the left we have the Joyal–Tierney and Luo–Bubenik–Tim <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+presheaves+of+simplicial+groupoids'>model structures on presheaves of simplicial groupoids</a>.</p> <p>(…have to check here the relation <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sh</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>SGrpd</mi><mo stretchy='false'>)</mo><mo>↔</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>SGrpd</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sh(X,SGrpd)\leftrightarrow PSh(X, SGrpd)</annotation></semantics></math>)</p> <h3 id='reedy_and_intermediate_model_structures'>Reedy and intermediate model structures</h3> <p>To some extent the injective and projective model structures on simplicial presheaves are the two extremes of a larger family of model structures on simplicial presheaves that all have the same weak equivalences but different classes of cofibrations.</p> <p>Notably if the domain <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has the special property that it is a <a class='existingWikiWord' href='/nlab/show/diff/Reedy+category'>Reedy category</a> there is the <a class='existingWikiWord' href='/nlab/show/diff/Reedy+model+structure'>Reedy model structure</a> on <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[C, sSet]</annotation></semantics></math>. Its class of cofibrations is intermediate that of the projective and the injective <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+functors'>model structure on functors</a> and we have <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalence</a>s</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub><mover><munder><mo>→</mo><mi>Id</mi></munder><mover><mo>←</mo><mi>Id</mi></mover></mover><mo stretchy='false'>[</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>Reedy</mi></msub><mover><munder><mo>→</mo><mi>Id</mi></munder><mover><mo>←</mo><mi>Id</mi></mover></mover><mo stretchy='false'>[</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>inj</mi></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> [C,sSet]_{proj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [C,sSet]_{Reedy} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [C,sSet]_{inj} \,. </annotation></semantics></math></div> <p>For general <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, there is still a whole family of model structures on <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]</annotation></semantics></math> that interpolates between the injective and the projective model structure. See <a class='existingWikiWord' href='/nlab/show/diff/intermediate+model+structure'>intermediate model structure</a>.</p> <h3 id='DependencyOnSite'>Dependency on the underlying site</h3> <div class='num_prop' id='SiteDependence'> <h6 id='proposition'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>,</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>C,D</annotation></semantics></math> be <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a>s and let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>f : C \to D</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> that induces a morphism of <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a>s in that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f_* : PSh(D) \to PSh(C)</annotation></semantics></math> preserves <a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaves</a> and its <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f^* : PSh(C) \to PSh(D)</annotation></semantics></math> (given by left <a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a>) is left <a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>exact functor</a> in that it preserves <a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite limit</a>s.</p> <p>Then the induced <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>D</mi><msubsup><mo stretchy='false'>)</mo> <mi>inj</mi> <mi>loc</mi></msubsup><mover><mo>→</mo><mo>←</mo></mover><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msubsup><mo stretchy='false'>)</mo> <mi>inj</mi> <mi>loc</mi></msubsup><mo>:</mo><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'> f_* : SPSh(D)_{inj}^{loc} \stackrel{\leftarrow}{\to} SPSh(C)_{inj}^{loc} : f^* </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a> for the local injective model structure on presheaves on both sides.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>This is “little fact 5)” on page 10, 11 of (<a href='#JardineLecture'>JardineLectures</a>).</p> </div> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>D</mi><mo>↪</mo></mrow><annotation encoding='application/x-tex'>f : D \hookrightarrow</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full</a> <a class='existingWikiWord' href='/nlab/show/diff/dense+sub-site'>dense sub-site</a>. Then right <a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mo stretchy='false'>[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>f_* : [D^{op}, sSet] \to [C^{op}, sSet]</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> yields a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+Quillen+adjunction'>simplicial Quillen adjunction</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy='false'>)</mo><mo>:</mo><mo stretchy='false'>[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>inj</mi><mo>,</mo><mi>loc</mi></mrow></msub><mover><munder><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mover><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>inj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> (f^* \dashv f_*) : [D^{op}, sSet]_{inj,loc} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} [C^{op}, sSet]_{inj,loc} </annotation></semantics></math></div> <p>between the <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>left Bousfield localizations</a> of the projective model structures at the <a class='existingWikiWord' href='/nlab/show/diff/sieve'>sieve</a> inclusions <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>S(\{U_i\}) \to U</annotation></semantics></math> for each <a class='existingWikiWord' href='/nlab/show/diff/covering'>covering</a> family <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{U_i \to U\}</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>It is immediate that we have a simplicial Quillen adjunction on the global injective model structure: by definition of right <a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a> we have an <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>-<a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> and the <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> restriction functor <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> trivially preserves injective cofibrations and acyclic cofibrations.</p> <p>Since we have <a class='existingWikiWord' href='/nlab/show/diff/proper+model+category'>left proper model categories</a> it is sufficient (by the discussion at <a href='http://nlab.mathforge.org/nlab/show/simplicial%20Quillen%20adjunction#Recognition'>recognition of simplicial Quillen adjunctions</a>) for deducing that the Quillen adjunction descends to the local strucuture to check that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>f_*</annotation></semantics></math> preserves locally fibrant objects, which in turn by properties of <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>left Bousfield localization</a> is equivalent to checking that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> sends covering sieve inclusions to weak equivalences in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>[D^{op}, sSet]_{proj,loc}</annotation></semantics></math>.</p> <p>By the <a href='#GeneralizedCover'>result on generalized covers</a>, for this it is sufficient to check that for every covering sieve <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>S(\{U_i\}) \to X</annotation></semantics></math> and every representable <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>K \in D</annotation></semantics></math> and morphism <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>X</mi></mrow><annotation encoding='application/x-tex'>K \to f^* X</annotation></semantics></math>, there is a <a class='existingWikiWord' href='/nlab/show/diff/covering'>covering</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>K</mi> <mi>j</mi></msub><mo>→</mo><mi>K</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{K_j \to K\}</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> and local lifts</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>K</mi> <mi>j</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>K</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>f</mi> <mo>*</mo></msup><mi>X</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ K_j &\to& f^*(S(\{U_i\})) \\ \downarrow && \downarrow \\ K &\to& f^* X } \,. </annotation></semantics></math></div> <p>This follows directly from the single defining condition on a <a class='existingWikiWord' href='/nlab/show/diff/coverage'>coverage</a> on <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> </div> <h2 id='PresentationOfInfiniToposes'>Presentation of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-toposes</h2> <div class='num_def'> <h6 id='definition'>Definition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a>. Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{proj}</annotation></semantics></math> be the projective <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a> over <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo><mo stretchy='false'>{</mo><mi>C</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>W = \{C(\{U_i\}) \to U\}</annotation></semantics></math> be the set of <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a> projections in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[C, sSet]</annotation></semantics></math> for each <a class='existingWikiWord' href='/nlab/show/diff/covering'>covering</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{U_i \to U\}</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> <p>Write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Id</mi><mo>⊣</mo><mi>Id</mi><mo stretchy='false'>)</mo><mo>:</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><mover><munder><mo>→</mo><mi>Id</mi></munder><mover><mo>←</mo><mi>Id</mi></mover></mover><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'> (Id \dashv Id) : [C^{op}, sSet]_{proj,loc} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [C^{op}, sSet]_{proj} </annotation></semantics></math></div> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>left Bousfield localization</a> at <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math>.</p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>([C^{op}, sSet]_{proj})^\circ</annotation></semantics></math> for the full sub-<a class='existingWikiWord' href='/nlab/show/diff/simplicially+enriched+category'>simplicially enriched category</a> on the fibrant-cofibrant objects, similarly for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>([CartSp^{op}, sSet]_{proj,loc})^\circ</annotation></semantics></math>.</p> </div> <div class='num_prop' id='PresentationOfTheInfinTopos'> <h6 id='proposition_3'>Proposition</h6> <p>We have an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalence of (∞,1)-categories</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Sh</mi> <mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>↪</mo><mover><mo>←</mo><mi>L</mi></mover></mover></mtd> <mtd><msub><mi>PSh</mi> <mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mo>≃</mo></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mtd> <mtd><mover><munder><mo>→</mo><mrow><mi>ℝ</mi><mi>Id</mi></mrow></munder><mover><mo>←</mo><mrow><mi>𝕃</mi><mi>Id</mi></mrow></mover></mover></mtd> <mtd><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \array{ Sh_{(\infty,1)}(C) &\stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}& PSh_{(\infty,1)}(C) \\ \uparrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}} \\ ([C^{op}, sSet]_{proj,loc})^\circ & \stackrel { \overset{\mathbb{L} Id}{\leftarrow} } { \underset{\mathbb{R}Id}{\to} } & ([C^{op}, sSet]_{proj})^\circ } \,, </annotation></semantics></math></div> <p>where at the bottom we have the left and right <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functor</a>s of the identity functors, as discussed at <a class='existingWikiWord' href='/nlab/show/diff/simplicial+Quillen+adjunction'>simplicial Quillen adjunction</a>.</p> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>This follows using the arguments in the proof of <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, 6.5.2.14</a> and <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, prop. A.3.7.6</a>.</p> </div> <h2 id='FibAndCofibObjects'>Fibrant and cofibrant objects</h2> <h3 id='fibrant_objects'>Fibrant objects</h3> <p>The fibrant objects in the <a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+presheaves'>local model structure on simplicial presheaves</a> are those that</p> <ul> <li> <p>are fibrant with respect to the respective <a class='existingWikiWord' href='/nlab/show/diff/global+model+structure+on+simplicial+presheaves'>global model structure</a></p> </li> <li> <p>and satisfy <a class='existingWikiWord' href='/nlab/show/diff/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a>. See there for more details.</p> </li> </ul> <p>This descent condition is the analog in this model of the <a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a>-condition and the <a class='existingWikiWord' href='/nlab/show/diff/stack'>stack</a>-condition. In fact, it reduces to these for truncated simplicial presheaves.</p> <p>Since the fibrancy condition in the global projective model structure is simple – it just requires that the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaf</a> is in fact a presheaf of <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>es – the local projective model structure has slightly more immediate characterization of fibrant objects than the local injective model structures. (In fact, for suitable choices of <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a>s it may become very simple, as the above discussion of site dependence of the model structure shows).</p> <p>On the other hand the cofibrancy condition on objects is entirely <em>trivial</em> in the global and local injective model structure: since a cofibration there is just an objectwise cofibration, and since every <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> is cofibrant, every object is injective cofibrant.</p> <p>But the cofibrant objects in the projective structure are not too nasty either: every object that is degreewise a coproduct of representables is cofibrant. In particular the <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a>s of any <em><a class='existingWikiWord' href='/nlab/show/diff/good+open+cover'>good cover</a></em> (see below for more details) is a projectively cofibrant object.</p> <p>A <strong>cofibrant replacement</strong> functor in the local projective structure is discussed in <a href='#Dugger01'>Dugger 01</a>.</p> <p>Something related to a <strong>fibrant replacement</strong> functor (“<math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stackification”) is discussed in section 6.5.3 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a></li> </ul> <h3 id='CofibrantObjects'>Cofibrant objects</h3> <p>In the injective <a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+presheaves'>local model structure on simplicial presheaves</a> all objects are cofibrant. For the projective local structure</p> <p>necessary and sufficient conditions are given here:</p> <ul> <li id='Garner13'><a class='existingWikiWord' href='/nlab/show/diff/Richard+Garner'>Richard Garner</a>, <a href='https://mathoverflow.net/a/127187/381'>MO comment</a></li> </ul> <p>More specifically, there is the following useful statement (see also <em><a class='existingWikiWord' href='/nlab/show/diff/projectively+cofibrant+diagram'>projectively cofibrant diagram</a></em>) (see also <a href='#Low'>Low, remark 8.2.3</a>).</p> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>A simplicial presheaf <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \in sPSh(C)</annotation></semantics></math> is said to have <strong>free degeneracies</strong> or the <strong>degenerate cells split off</strong> if in each degree there is a <a class='existingWikiWord' href='/nlab/show/diff/subobject'>sub</a>-presheaf <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>k</mi></msub><mo>↪</mo><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>N_k \hookrightarrow X_k</annotation></semantics></math> such that the canonical mophism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <munder><mrow><mi>σ</mi><mo>:</mo><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><mrow><mi>surj</mi><mo>.</mo></mrow></munder></munder><msub><mi>N</mi> <mi>n</mi></msub><mover><mo>→</mo><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <munder><mrow><mi>σ</mi><mo>:</mo><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><mrow><mi>surj</mi><mo>.</mo></mrow></munder></munder><msup><mi>σ</mi> <mo>*</mo></msup></mrow></mover><msub><mi>F</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'> \coprod_{\underset{surj.}{\sigma : [k] \to [n]}} N_n \stackrel{\coprod_{\underset{surj.}{\sigma : [k] \to [n]}} \sigma^*}{\to} F_k </annotation></semantics></math></div> <p>is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>.</p> </div> <p>So if degenerate cells split off we have in particular that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>k</mi></msub><mo>=</mo><msubsup><mi>X</mi> <mi>k</mi> <mi>nd</mi></msubsup><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><msubsup><mi>X</mi> <mi>k</mi> <mi>dg</mi></msubsup><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> X_k = X_k^{nd} \coprod X_k^{dg} \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>X</mi> <mi>k</mi> <mi>nd</mi></msubsup></mrow><annotation encoding='application/x-tex'>X_k^{nd}</annotation></semantics></math> is the presheaf of non-degenerate <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-cells and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>X</mi> <mi>k</mi> <mi>dg</mi></msubsup></mrow><annotation encoding='application/x-tex'>X_k^{dg}</annotation></semantics></math> is a separate presheaf containing all the degenerate cells (and itself a coproduct over separate presheaves for each degree and order of degeneracy).</p> <div class='num_prop'> <h6 id='proposition_4'>Proposition</h6> <p>In the <em>projective</em> <a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+presheaves'>local model structure</a> all objects that are</p> <ol> <li> <p>degreewise <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproducts</a> of <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representables</a></p> </li> <li> <p>and whose degenerate cells split off</p> </li> </ol> <p>are cofibrant.</p> </div> <p>This is in <a href='#Dugger01'>Dugger 01, Cor. 9.4</a>.</p> <div class='num_example'> <h6 id='example'>Example</h6> <p><strong>(split hypercovers)</strong></p> <p>If <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>Y \to X</annotation></semantics></math> is an acyclic fibration in the local projective model structure with <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a representable and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> cofibration in the above way, it is called a <strong><a class='existingWikiWord' href='/nlab/show/diff/split+hypercover'>split hypercover</a></strong> .</p> <p>All <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a>s <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(\{U_i\})</annotation></semantics></math> coming from an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> have split degeneracies. The condition that the Čech nerve be degreewise a coproduct of representables is a condition akin to that of <a class='existingWikiWord' href='/nlab/show/diff/good+open+cover'>good open cover</a>s (which is precisely the special case for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C = </annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/CartSp'>CartSp</a>). This is then a split hypercover of <em>height</em> 0.</p> </div> <div class='num_defn'> <h6 id='definition_3'>Definition</h6> <p><strong>(good cover)</strong></p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> with a weak equivalence <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \stackrel{\simeq}{\to} X</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msup><mo stretchy='false'>)</mo> <mi>loc</mi></msup></mrow><annotation encoding='application/x-tex'>SPSh(C)^{loc}</annotation></semantics></math> is a <strong><a class='existingWikiWord' href='/nlab/show/diff/good+open+cover'>good cover</a></strong> if it is degreewise a coproduct of <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable</a>s.</p> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>This reduces to the ordinary notion of <a class='existingWikiWord' href='/nlab/show/diff/good+open+cover'>good cover</a> as an open cover by contractible spaces such that all finite intersections of these are again contractibe when using a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> like <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C = </annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/CartSp'>CartSp</a>.</p> </div> <h3 id='CofibrantReplacement'>Cofibrant replacement</h3> <p>In</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Dugger'>Daniel Dugger</a>, <em><a class='existingWikiWord' href='/nlab/files/DuggerUniv.pdf' title='Universal homotopy theories'>Universal homotopy theories</a></em></li> </ul> <p>a useful cofibrant replacement functor for the projective local model structure is discussed.</p> <div class='num_defn'> <h6 id='definition_4'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>↪</mo><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>A \in PSh(C) \hookrightarrow SPSh(C)</annotation></semantics></math> an ordinary presheaf (simplicially discrete simplicial presheaf) let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>Q</mi><mo stretchy='false'>˜</mo></mover><mi>A</mi></mrow><annotation encoding='application/x-tex'>\tilde Q A</annotation></semantics></math> be the simplicial presheaf which in degree <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mover><mi>Q</mi><mo stretchy='false'>˜</mo></mover><mi>A</mi><msub><mo stretchy='false'>)</mo> <mi>k</mi></msub><mo>:</mo><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>U</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo>→</mo><msub><mi>U</mi> <mn>0</mn></msub><mo>→</mo><mi>A</mi></mrow></munder><msub><mi>U</mi> <mi>k</mi></msub><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> (\tilde Q A)_k := \coprod_{U_k \to U_{k-1} \to \cdots \to U_0 \to A} U_k \,, </annotation></semantics></math></div> <p>where the <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>U_k</annotation></semantics></math> range over the representables, i.e. the objects in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>↪</mo><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C \hookrightarrow SPSh(C)</annotation></semantics></math>. The face and degeneracy maps are the obvious ones coming from composing maps and inserting identity maps in the labels over which the coproduct ranges.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>A \in SPSh(C)</annotation></semantics></math> an arbitrary simplicial presheaf let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>A</mi></mrow><annotation encoding='application/x-tex'>Q A</annotation></semantics></math> be the diagonal of the bisimplicial presheaf obtained by applying <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>Q</mi><mo stretchy='false'>˜</mo></mover></mrow><annotation encoding='application/x-tex'>\tilde Q</annotation></semantics></math> degreewise</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>A</mi><mo>=</mo><mrow><mo>(</mo><mi>⋯</mi><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>U</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow></munder><msub><mi>U</mi> <mn>1</mn></msub><mover><mo>→</mo><mo>→</mo></mover><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>0</mn></msub></mrow></munder><msub><mi>U</mi> <mn>0</mn></msub><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Q A = \left( \cdots \coprod_{U_1 \to U_0 \to A_1} U_1 \stackrel{\to}{\to}\coprod_{U_0 \to A_0} U_0 \right) \,. </annotation></semantics></math></div></div> <div class='num_prop'> <h6 id='proposition_5'>Proposition</h6> <p>For all <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>A \in SPSh(C)</annotation></semantics></math> the object <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>A</mi></mrow><annotation encoding='application/x-tex'>Q A</annotation></semantics></math> is cofibrant and is weakly equivalent to <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msubsup><mo stretchy='false'>)</mo> <mi>proj</mi> <mi>loc</mi></msubsup></mrow><annotation encoding='application/x-tex'>SPSh(C)_{proj}^{loc}</annotation></semantics></math>.</p> </div> <p>This is in prop 2.8 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Dugger'>Daniel Dugger</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Universal+Homotopy+Theories'>Universal Homotopy Theories</a></em></li> </ul> <h2 id='local_fibrations'>Local fibrations</h2> <p>A <em><a class='existingWikiWord' href='/nlab/show/diff/local+fibration'>local fibration</a></em> or <em>local weak equivalence</em> of simplicial (pre)sheaves is defined to be one whose lifting property is satisfied after refining to some cover.</p> <p><strong>Warning</strong>. Notice that this is a priori unrelated to equivalences and fibrations with respect to any local model structure.</p> <p>If the <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+topos'>enough points</a>, then local fibrations of simplicial presheaves are equivalently those that are <a class='existingWikiWord' href='/nlab/show/diff/stalk'>stalk</a>wise fibrations of simplicial sets.</p> <p>This is discussed in (<a href='#Jardine96'>Jardine 96</a>).</p> <h2 id='Descent'>Localization and descent</h2> <h3 id='ČechLocalization'>Čech localization at Grothendieck (pre)topologies</h3> <p>We discuss some aspects of the <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>left Bousfield localization</a> of the projective global model structure on simplicial presheaves at <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topology'>Grothendieck topologies</a> and <a class='existingWikiWord' href='/nlab/show/diff/covering'>covering</a> families. By the discussion at <a class='existingWikiWord' href='/nlab/show/diff/topological+localization'>topological localization</a> these are models for <a class='existingWikiWord' href='/nlab/show/diff/topological+localization'>topological localization</a>s leading to <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-categories of (∞,1)-sheaves</a>.</p> <p>The central reference is (<a href='#DuggerHollanderIsaksen'>DuggerHollanderIsaksen</a>) with the central theorem being this one:</p> <div class='num_theorem'> <h6 id='theorem'>Theorem</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> given by a <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topology'>Grothendieck topology</a>. The left <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>Bousfield localization</a> of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>sPSh(C)_{proj}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mi>inj</mi></msub></mrow><annotation encoding='application/x-tex'>sPSh(C)_{inj}</annotation></semantics></math>, respectively, at the following classes of morphisms exist and coincide:</p> <ol> <li> <p>the set of all <a class='existingWikiWord' href='/nlab/show/diff/covering'>covering</a> <a class='existingWikiWord' href='/nlab/show/diff/sieve'>sieve</a> <a class='existingWikiWord' href='/nlab/show/diff/subfunctor'>subfunctor</a>s <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>↪</mo><mi>j</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>R \hookrightarrow j(X)</annotation></semantics></math>;</p> </li> <li> <p>the set of all morphisms <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>hocolim</mi> <mi>R</mi></msub><mo>→</mo><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>hocolim_R \to U \to X</annotation></semantics></math> for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> a covering sieve of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>;</p> </li> <li> <p>the set of all <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a> projections <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>C(\{U_i\}) \to X</annotation></semantics></math> for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{U_i \to X\}</annotation></semantics></math> a covering sieve;</p> </li> <li> <p>the class of all bounded <a class='existingWikiWord' href='/nlab/show/diff/hypercover'>hypercover</a>s <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \to X</annotation></semantics></math>;</p> </li> <li> <p>the class of morphisms <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>→</mo><mover><mi>F</mi><mo stretchy='false'>¯</mo></mover></mrow><annotation encoding='application/x-tex'>F \to \bar F</annotation></semantics></math> from a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaf</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> to the simplicial <a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a> obtained by degreewise <a class='existingWikiWord' href='/nlab/show/diff/sheafification'>sheafification</a>.</p> </li> <li> <p>if the topology is generated from a <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>basis</a>, then: the set of covering sieve subfunctors <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mi>U</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>R_U \to X</annotation></semantics></math> for each covering family <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{U_i \to X\}</annotation></semantics></math> in the basis.</p> </li> </ol> </div> <p>This is theorem A5 in <a href='http://front.math.ucdavis.edu/0205.5027'>DugHolIsak</a>.</p> <p>This localization <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>sPSh(C)_{proj,cov}</annotation></semantics></math> is the <strong>Čech localization</strong> of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>sPSh(C)</annotation></semantics></math> with respect to the given <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topology'>Grothendieck topology</a>. It is a presentation of <a class='existingWikiWord' href='/nlab/show/diff/topological+localization'>topological localization</a> of an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a> to an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Sh</mi> <mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>↪</mo><mover><mo>→</mo><mi>L</mi></mover></mover></mtd> <mtd><msub><mi>Psh</mi> <mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mo>≃</mo></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mtd> <mtd><mover><munder><mo>→</mo><mrow /></munder><mover><mo>←</mo><mrow><mi>left</mi><mo>.</mo><mi>Bousf</mi><mo>.</mo></mrow></mover></mover></mtd> <mtd><mo stretchy='false'>(</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ Sh_{(\infty,1)}(C) &\stackrel{\overset{L}{\to}}{\hookrightarrow}& Psh_{(\infty,1)}(C) \\ \uparrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}} \\ (sPSh(C)_{proj,cov})^\circ &\stackrel{\overset{left. Bousf.}{\leftarrow}}{\underset{}{\to}}& (sPSh(C)_{proj})^\circ } \,. </annotation></semantics></math></div> <p>The following definition and proposition provides information on what the general morphisms are which become weak equivalences after localization at</p> <div class='num_def' id='GeneralizedCover'> <h6 id='defintion'>Defintion</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/local+epimorphism'>generalized cover</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a>. A <strong><a class='existingWikiWord' href='/nlab/show/diff/local+epimorphism'>local epimorphism</a></strong> (or <strong><a class='existingWikiWord' href='/nlab/show/diff/local+epimorphism'>generalized cover</a></strong>) in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>sPSh(C)</annotation></semantics></math> is a morphism <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>f : E \to B</annotation></semantics></math> of simplicial presheaves with the property that for every <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> and every morphism <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>j(U) \to B</annotation></semantics></math> there exists a <a class='existingWikiWord' href='/nlab/show/diff/covering'>covering</a> <a class='existingWikiWord' href='/nlab/show/diff/sieve'>sieve</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{U_i \to U\}</annotation></semantics></math> such that for every <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>U_i \to U</annotation></semantics></math> the composite <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>U_i \to U \to B</annotation></semantics></math> has a lift <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> through <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>j</mi><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>∃</mo><mi>σ</mi></mrow></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>j</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mo>∀</mo></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ j(U_i) &\stackrel{\exists \sigma}{\to}& E \\ \downarrow && \downarrow \\ j(U) &\stackrel{\forall}{\to} & B } \,. </annotation></semantics></math></div></div> <p>(<a href='#DuggerHollanderIsaksen'>Dugger-Hollander-Isaksen, corollary A.3</a>)</p> <div class='num_prop'> <h6 id='proposition_6'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>f : E \to B</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/local+epimorphism'>local epimorphism</a> in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>sPSh(C)</annotation></semantics></math> in the above sense, its <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a> projection</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>E</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'> C(E) \to B </annotation></semantics></math></div> <p>is a weak equivalence in the projective local model structure <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>sPSh(C)_{proj, loc}</annotation></semantics></math>.</p> </div> <p>This is <a href='#DuggerHollanderIsaksen'>Dugger-Hollander-Isaksen, corollary A.3</a>.</p> <p><math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></p> <h3 id='for_values_in_strict_and_abelian_groupoids'>For values in strict and abelian <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids</h3> <p>Many simplicial presheaves appearing in practice are (equivalent) to objects in <a class='existingWikiWord' href='/nlab/show/diff/sub-%28infinity%2C1%29-category'>sub-(∞,1)-categories</a> of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sh_{(\infty,1)}(C)</annotation></semantics></math> of abelian or at least <a class='existingWikiWord' href='/nlab/show/diff/strict+omega-groupoid'>strict ∞-groupoid</a>s. These subcategories typically offer convenient and desireable contexts for formulating and proving statements about special cases of general simplicial presheaves.</p> <p>One well-known such notion is given by the <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a>. This identifies <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a>es of <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>s with strict and strictly <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids.</p> <p>Dropping the condition on symmetric monoidalness we obtain a more general such inclusion, a kind of non-abelian Dold-Kan correspondence:</p> <p>the identification of <a class='existingWikiWord' href='/nlab/show/diff/crossed+complex'>crossed complex</a>es of groupoids as precisely the <a class='existingWikiWord' href='/nlab/show/diff/strict+omega-groupoid'>strict ∞-groupoid</a>s. This has been studied in particular in <a class='existingWikiWord' href='/nlab/show/diff/Nonabelian+Algebraic+Topology'>nonabelian algebraic topology</a>.</p> <p>So we have a sequence of inclusions</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>ChainCplx</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>CrsCpl</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>KanCplx</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>simeq</mi></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>simeq</mi></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>simeq</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>StrAb</mi><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mn>∞</mn><mi>Grpd</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ ChainCplx &\hookrightarrow& CrsCpl &\hookrightarrow& KanCplx \\ \downarrow^{\mathrlap{simeq}} && \downarrow^{\mathrlap{simeq}} && \downarrow^{\mathrlap{simeq}} \\ StrAb Str\infty Grpd &\hookrightarrow& Str \infty Grpd &\hookrightarrow& \infty Grpd } </annotation></semantics></math></div> <p>of strict <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids into all <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids. See also the <a class='existingWikiWord' href='/nlab/show/diff/cosmic+cube'>cosmic cube</a> of <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a>.</p> <p>Among the special tools for handling <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stacks on <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> that factor at some point through the above inclusion are the following:</p> <ul> <li> <p><strong>restriction to abelian sheaf cohomology</strong> – Using the fact that the objects of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sh_{(\infty,1)}(C)</annotation></semantics></math> are modeled by <a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaves</a> symmetric monoidal <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Lie groupoids are identified under the <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a> with <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{N}</annotation></semantics></math>-graded <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complexes</a> of sheaves. To these the rich set of tools for <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a> apply.</p> </li> <li> <p><strong>descent for strict <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoid valued sheaves</strong> – There is a good theory of <a class='existingWikiWord' href='/nlab/show/diff/descent'>descent</a> for (presheaves) with values in strict <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids (more restrictive than the fully general theory but more general than <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a>). This goes back to <a class='existingWikiWord' href='/nlab/show/diff/Ross+Street'>Ross Street</a> and its relation to the full theory has been clarified by <a class='existingWikiWord' href='/nlab/show/diff/Dominic+Verity'>Dominic Verity</a> in <a href='#Verity'>Verity09</a>.</p> </li> </ul> <p>We state a useful theorem for the computation of <a class='existingWikiWord' href='/nlab/show/diff/descent'>descent</a> for presheaves with values in <a class='existingWikiWord' href='/nlab/show/diff/strict+omega-groupoid'>strict ∞-groupoid</a>s. Recall the standard terminology for <a class='existingWikiWord' href='/nlab/show/diff/descent'>descent</a>, i.e. for the <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-categorical <a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a>-condition:</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \in C</annotation></semantics></math> a representable, <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>Y,A \in [C^{op}, sSet]</annotation></semantics></math> simplicial presheaves and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>p : Y \to U</annotation></semantics></math> a morphism, we say that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> <em>satisfies <a class='existingWikiWord' href='/nlab/show/diff/descent'>descent</a></em> along <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> or equivalently that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is a <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/local+object'>local object</a> if the canonical morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>=</mo></mover><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> A(U) \stackrel{=}{\to} [C^{op}, sSet](U,A) \to [C^{op}, sSet](Y,A) </annotation></semantics></math></div> <p>is a weak equivalence. Here the first equality is the enriched <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a>. By the <a class='existingWikiWord' href='/nlab/show/diff/co-Yoneda+lemma'>co-Yoneda lemma</a> we may decompose <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> into its cells as</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>⋅</mo><msub><mi>Y</mi> <mi>n</mi></msub><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> Y = \int^{[n] \in \Delta} \Delta[n] \cdot Y_n \,, </annotation></semantics></math></div> <p>where in the integrand we have the <a class='existingWikiWord' href='/nlab/show/diff/copower'>tensoring</a> of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]</annotation></semantics></math> over <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>. Using that the enriched <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom-functor</a> sends coends to ends, the enriched <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom-functor</a> on the right we may equivalently write out as an <a class='existingWikiWord' href='/nlab/show/diff/end'>end</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>⋅</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>⋅</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mi>sSet</mi><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>,</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><msub><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mi>sSet</mi><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>,</mo><mi>A</mi><mo stretchy='false'>(</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mo>:</mo><mi>Desc</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} [C^{op}, sSet](Y,A) & = [C^{op}, sSet](\int^{[n] \in \Delta} \Delta[n] \cdot Y_n ,A) \\ & = \int_{[n] \in \Delta}[C^{op}, sSet](\Delta[n] \cdot Y_n ,A) \\ & = \int_{[n] \in \Delta} sSet(\Delta[n], [C^{op}, sSet](Y_n, A)) \\ & = \int_{[n] \in \Delta} sSet(\Delta[n], A(Y_n)) \\ & =:Desc(Y,A) \end{aligned} </annotation></semantics></math></div> <p>(equality signs denote <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>s), where in the second but last line we again used the <a class='existingWikiWord' href='/nlab/show/diff/copower'>tensoring</a> of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaves</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]</annotation></semantics></math> over <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>.</p> <p>In the last line we have the <em><a class='existingWikiWord' href='/nlab/show/diff/totalization'>totalization</a></em> of the cosimplicial <a class='existingWikiWord' href='/nlab/show/diff/simplicial+object'>simplicial object</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo stretchy='false'>(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> A(Y_\bullet) : \Delta \to sSet \,, </annotation></semantics></math></div> <p>sometimes called the <em>descent object</em> of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> relative to <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, even though in this case it is really nothing but the hom-object of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> into <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. If <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is fibrant and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> cofibrant, then <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Desc</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Desc(Y,A)</annotation></semantics></math> is a Kan complex: the <em>descent <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoid</em> .</p> <p>Now suppose that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A} : C^{op} \to Str \infty Grpd</annotation></semantics></math> is a presheaf with values in <a class='existingWikiWord' href='/nlab/show/diff/strict+omega-groupoid'>strict ∞-groupoid</a>s. In the context of strict <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids the standard <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/simplex'>simplex</a> is given by the <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th <a class='existingWikiWord' href='/nlab/show/diff/oriental'>oriental</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>O</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>O(n)</annotation></semantics></math>. This allows to perform a construction that looks like a descent object in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>Str\infty Grpd</annotation></semantics></math>:</p> <div class='num_defn'> <h6 id='definition_5'>Definition</h6> <p><strong>(Ross Street)</strong></p> <p>The descent object for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi><mo>∈</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\mathcal{A} \in [C^{op}, Str \infty Grpd]</annotation></semantics></math> relative to <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>∈</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>Y \in [C^{op}, sSet]</annotation></semantics></math> is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Desc</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mi>Str</mi><mn>∞</mn><mi>Cat</mi><mo stretchy='false'>(</mo><mi>O</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>𝒜</mi><mo stretchy='false'>(</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>∈</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> Desc(Y,\mathcal{A}) := \int_{[n] \in \Delta} Str\infty Cat(O(n), \mathcal{A}(Y_n)) \;\in Str \infty Grpd \,, </annotation></semantics></math></div> <p>where the <a class='existingWikiWord' href='/nlab/show/diff/end'>end</a> is taken in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>Str \infty Grpd</annotation></semantics></math>.</p> </div> <p>This objects had been suggested by <a class='existingWikiWord' href='/nlab/show/diff/Ross+Street'>Ross Street</a> to be the right descent object for strict <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-category-valued presheaves in <a href='#Street03'>Street03</a></p> <p>Under the <a class='existingWikiWord' href='/nlab/show/diff/omega-nerve'>∞-nerve</a> functor <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mo>:</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>N_O : Str\infty Grpd \to sSet</annotation></semantics></math> this yields a <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mn>0</mn></msub><mi>Desc</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N_0 Desc(Y,\mathcal{A})</annotation></semantics></math>. On the other hand, applying the <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-nerve directly to <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> yields a simplicial presheaf <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>N_O\mathcal{A}</annotation></semantics></math> to which the above simplicial descent applies.</p> <p>The following theorem asserts that under certain conditions both notions coincide.</p> <div class='num_theorem'> <h6 id='theorem_2'>Theorem</h6> <p><strong>(Dominic Verity)</strong></p> <p>If <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A} : C^{op}, Str \infty Grpd</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>Y : C^{op} \to sSet</annotation></semantics></math> are such that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy='false'>(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>N_O \mathcal{A}(Y_\bullet) : \Delta \to sSet</annotation></semantics></math> is fibrant in the <a class='existingWikiWord' href='/nlab/show/diff/Reedy+model+structure'>Reedy model structure</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>Δ</mi><mo>,</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy='false'>]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding='application/x-tex'>[\Delta, sSet_{Quillen}]_{Reedy}</annotation></semantics></math>, then</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>Desc</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Desc</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A}) </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a> of <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>es.</p> </div> <p>This is proven in <a href='#Verity'>Verity09</a>.</p> <div class='num_corollary'> <h6 id='corollary'>Corollary</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>∈</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>Y \in [C^{op}, sSet]</annotation></semantics></math> is such that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Y</mi> <mo>•</mo></msub><mo>:</mo><mi>Δ</mi><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mo>↪</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>Y_\bullet : \Delta \to [C^{op}, Set] \hookrightarrow [C^{op}, sSet]</annotation></semantics></math> is cofibrant in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>Δ</mi><mo>,</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub><msub><mo stretchy='false'>]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding='application/x-tex'>[\Delta, [C^{op}, sSet]_{proj}]_{Reedy}</annotation></semantics></math> then for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A} : C^{op} \to Str \infty Grpd</annotation></semantics></math> we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>Desc</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Desc</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A}) \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_4'>Proof</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Y</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>Y_\bullet</annotation></semantics></math> is Reedy cofibrant, then by definition the canonical morphisms</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>→</mo><mo lspace='verythinmathspace' rspace='0em'>+</mo></mover><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>↦</mo><msub><mi>Y</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>Y</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'> \lim_{\to}( ([n] \stackrel{+}{\to} [k]) \mapsto Y_k ) \to Y_n </annotation></semantics></math></div> <p>are cofibrations in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{proj}</annotation></semantics></math>. Since the latter is an <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding='application/x-tex'>sSet_{Quillen}</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/enriched+model+category'>enriched model category</a> and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>N_O \mathcal{A}</annotation></semantics></math> is fibrant, it follows that the <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom-functor</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet](-, N_O \mathcal{A})</annotation></semantics></math> sends cofibrations to fibrations, so that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy='false'>(</mo><msub><mi>Y</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo>→</mo><munder><mi>lim</mi> <mo>←</mo></munder><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>→</mo><mo lspace='verythinmathspace' rspace='0em'>+</mo></mover><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo>↦</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy='false'>(</mo><msub><mi>Y</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> N_O\mathcal{A}(Y_n) \to \lim_{\leftarrow}( [n]\stackrel{+}{\to} [k] \mapsto N_O\mathcal{A}(Y_k)) </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/Kan+fibration'>Kan fibration</a>. But this says that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy='false'>(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N_O \mathcal{A}(Y_\bullet)</annotation></semantics></math> is Reedy fibrant, so that the assumption of Verity’s theorem is met.</p> </div> <div class='num_corollary'> <h6 id='corollary_2'>Corollary</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a> of a <a class='existingWikiWord' href='/nlab/show/diff/good+open+cover'>good open cover</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{U_i \to X\}</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifold</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and any <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi><mo>:</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>→</mo><mi>Str</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A} : CartSp^{op} \to Str \infty Grpd</annotation></semantics></math> we have that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>𝒜</mi><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>N</mi> <mi>O</mi></msub><mi>Desc</mi><mo stretchy='false'>(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo>,</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> [C^{op}, sSet](Y,N_O \mathcal{A}) \simeq N_O Desc(Y_\bullet, \mathcal{A}) \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_5'>Proof</h6> <p>By the above is sufices to note that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Y</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>Y_\bullet</annotation></semantics></math> is cofibrant in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub><msub><mo stretchy='false'>]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding='application/x-tex'>[\Delta^{op}, [C^{op}, sSet]_{proj}]_{Reedy}</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a> of a good open cover. By the assumption of good open cover we have that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is degreewise a coproduct of representables and that the inclusion of all degenerate <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-cells into all <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-cells is a full inclusion into a coproduct, i.e. an inlusion of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mi>j</mi></munder><msub><mi>U</mi> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \coprod_{i \in I} U_i \to \coprod_j U_{j \in J} </annotation></semantics></math></div> <p>induced from an inclusion of subsets <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>↪</mo><mi>J</mi></mrow><annotation encoding='application/x-tex'>I \hookrightarrow J</annotation></semantics></math>. Since all representables are cofibrant in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{proj}</annotation></semantics></math> such an inclusion is a cofibration.</p> </div> <p>In conclusion we find that for determining the <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stack condition for <em>strict</em> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Lie groupoids we may equivalently use Street’s formula for strict <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupid valued presheaves. This is sometimes useful for computations in low categorical degree.</p> <h2 id='Properness'>Properness</h2> <p>The global model structures on simplicial presheaves are all <a class='existingWikiWord' href='/nlab/show/diff/proper+model+category'>left</a> and <a class='existingWikiWord' href='/nlab/show/diff/proper+model+category'>right proper model categories</a>. Since left <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>Bousfield localization of model categories</a> preserves left properness (as discussed <a href='Bousfield+localization+of+model+categories#ExistenceForLeftProperCombinatorialSimplicialModelCategories'>there</a>), all local model structures on simplicial presheaves are also left proper.</p> <p>But the local model structures are not in general right proper anymore. A sufficient condition for them to be right proper is given in the following Prop. <a class='maruku-ref' href='#StalkwiseWeakEquivalencesImpliesRightProperness'>8</a>.</p> <p>The injective local model structures which are right proper are <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+model+category'>locally cartesian closed model categories</a> and <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>present</a> <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+%28infinity%2C1%29-category'>locally cartesian closed (infinity,1)-categories</a> (by the discussion <a href='locally+cartesian+closed+infinity,1-category#PresentationTheorem'>there</a>).</p> <div class='num_prop' id='OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks'> <h6 id='proposition_7'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> with <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+topos'>enough points</a>. Then the weak equivalences in the local model structures on <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>sPSh(C)</annotation></semantics></math> are the <a class='existingWikiWord' href='/nlab/show/diff/stalk'>stalk</a>-wise <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalences</a> of simplicial sets.</p> </div> <p>(p. 12 <a href='http://www.math.uiuc.edu/K-theory/0175/'>here</a>)</p> <div class='num_prop' id='StalkwiseWeakEquivalencesImpliesRightProperness'> <h6 id='proposition_8'>Proposition</h6> <p>A sufficient condition for an injective or projective local model structure of simplicial presheaves over a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> to be right proper is that the weak equivalences are precisely the <a class='existingWikiWord' href='/nlab/show/diff/stalk'>stalk</a>-wise <a class='existingWikiWord' href='/nlab/show/diff/simplicial+weak+equivalence'>weak equivalences of simplicial sets</a>.</p> </div> <p>This is mentioned for instance in (<a href='#Olsson'>Olsson, remark 4.3</a>).</p> <p>By Prop. <a class='maruku-ref' href='#OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks'>7</a> this sufficient condition holds, for instance, for the injective Jardine model structure when <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+topos'>enough points</a>.</p> <div class='proof'> <h6 id='proof_6'>Proof</h6> <p>The key is that forming <a class='existingWikiWord' href='/nlab/show/diff/stalk'>stalks</a> is, being the <a class='existingWikiWord' href='/nlab/show/diff/inverse+image'>inverse image</a> of a <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+topos'>geometric morphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>x</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>x</mi> <mo>*</mo></msub><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>Set</mi><mover><munder><mo>→</mo><mrow><msub><mi>x</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>x</mi> <mo>*</mo></msup></mrow></mover></mover><mi>Sh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (x^* \dashv x_*) := Set \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} Sh(C) </annotation></semantics></math></div> <p>an operation that preserves <a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite limits</a>.</p> <p>Let therefore <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>f : X \to A</annotation></semantics></math> be a stalkwise weak equivalence of simplicial presheaves and let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_231' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>g : A \to B</annotation></semantics></math> be a fibration. Notice that in all the model structures (injective, projective, global, local) the fibrations are always <em>in particular</em> objectwise fibrations.</p> <p>Then the pullback <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>g</mi> <mo>*</mo></msup><mi>f</mi></mrow><annotation encoding='application/x-tex'>g^* f</annotation></semantics></math> in</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>g</mi> <mo>*</mo></msup><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mrow><msup><mi>g</mi> <mo>*</mo></msup><mi>f</mi></mrow></mpadded></msup></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ g^* X &\to& X \\ \big\downarrow {}^{\mathrlap{g^* f}} && \big\downarrow {}^{\mathrlap{f}} \\ A &\stackrel{g}{\to}& B } </annotation></semantics></math></div> <p>is a weak equivalence if for all topos points <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> the stalk <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><msup><mi>g</mi> <mo>*</mo></msup><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x^* (g^* f)</annotation></semantics></math> is a weak equivalence of simplicial sets. But since stalks preserve <a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite limits</a>, we have a pullback diagram of simplicial sets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><msup><mi>g</mi> <mo>*</mo></msup><mi>X</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><msup><mi>g</mi> <mo>*</mo></msup><mi>f</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ x^*(g^* X) &\to& x^*( X) \\ \downarrow^{\mathrlap{x^*(g^* f)}} && \downarrow^{\mathrlap{x^*(f)}} \\ x^*(A) &\stackrel{x^*(g)}{\to}& x^*(B) } \,. </annotation></semantics></math></div> <p>It is now sufficient to observe that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup><mi>g</mi></mrow><annotation encoding='application/x-tex'>x^* g</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/Kan+fibration'>Kan fibration</a>, which implies the result by the fact that the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical model structure on simplicial sets</a> is right proper.</p> <p>To see this, notice that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x^*(g)</annotation></semantics></math> is a Kan fibration precisely if for all <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>1 \leq k </annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_240' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>0 \leq i \leq k</annotation></semantics></math> the morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_241' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>x</mi> <mo>*</mo></msup><mi>A</mi><msup><mo stretchy='false'>)</mo> <mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo></mrow></msup><mo>→</mo><mo stretchy='false'>(</mo><msup><mi>x</mi> <mo>*</mo></msup><mi>A</mi><msup><mo stretchy='false'>)</mo> <mrow><mi>Λ</mi><mo stretchy='false'>[</mo><mi>k</mi><msup><mo stretchy='false'>]</mo> <mi>i</mi></msup></mrow></msup><msub><mo>×</mo> <mrow><mo stretchy='false'>(</mo><msup><mi>x</mi> <mo>*</mo></msup><mi>B</mi><msup><mo stretchy='false'>)</mo> <mrow><mi>Λ</mi><mo stretchy='false'>[</mo><mi>k</mi><msup><mo stretchy='false'>]</mo> <mi>i</mi></msup></mrow></msup></mrow></msub><mo stretchy='false'>(</mo><msup><mi>x</mi> <mo>*</mo></msup><mi>B</mi><msup><mo stretchy='false'>)</mo> <mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo></mrow></msup></mrow><annotation encoding='application/x-tex'> (x^* A)^{\Delta[k]} \to (x^* A)^{\Lambda[k]^i} \times_{(x^* B)^{\Lambda[k]^i} } (x^* B)^{\Delta[k]} </annotation></semantics></math></div> <p>is an <a class='existingWikiWord' href='/nlab/show/diff/epimorphism'>epimorphism</a> of sets. Since stalks commute with finite limits, this is equivalent to</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_242' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup><mrow><mo>(</mo><msup><mi>A</mi> <mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo></mrow></msup><mo>→</mo><msup><mi>A</mi> <mrow><mi>Λ</mi><mo stretchy='false'>[</mo><mi>k</mi><msup><mo stretchy='false'>]</mo> <mi>i</mi></msup></mrow></msup><msub><mo>×</mo> <mrow><msup><mi>B</mi> <mrow><mi>Λ</mi><mo stretchy='false'>[</mo><mi>k</mi><msup><mo stretchy='false'>]</mo> <mi>i</mi></msup></mrow></msup></mrow></msub><msup><mi>B</mi> <mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo></mrow></msup><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> x^* \left( A^{\Delta[k]} \to A^{\Lambda[k]^i} \times_{ B^{\Lambda[k]^i} } B^{\Delta[k]} \right) </annotation></semantics></math></div> <p>being an epimorphism. Now the morphism in parenthesis is an epimorphism since the fibration <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_243' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is in particular an objectwise Kan fibration, and <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> functors such as <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_244' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>x^*</annotation></semantics></math> preserve epimorphisms.</p> </div> <h2 id='MonoidalStructure'>Closed monoidal structure</h2> <p>If the underlying site has <a class='existingWikiWord' href='/nlab/show/diff/finite+product'>finite products</a>, then both the injective and the projective, the global and the local model structure on simplicial presheaves becomes a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+model+category'>monoidal model category</a> with respect to the standard <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+structure+on+presheaves'>closed monoidal structure on presheaves</a>.</p> <p>See for instance <a href='http://www.math.univ-toulouse.fr/~toen/crm-2008.pdf#page=24'>here</a>.</p> <div class='num_lemma'> <h6 id='lemma'>Lemma</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_245' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a category with <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>products</a>. Then the <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+structure+on+presheaves'>closed monoidal structure on presheaves</a> makes <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_246' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{proj}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+model+category'>monoidal model category</a>.</p> </div> <div class='proof'> <h6 id='proof_7'>Proof</h6> <p>It is sufficient to check that the Cartesian product of presheaves</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_247' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊗</mo><mo>:</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub><mo>×</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub><mo>→</mo><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'> \otimes : sPSh(C)_{proj} \times sPSh(C)_{proj} \to sPSh(C)_{proj} </annotation></semantics></math></div> <p>is a left <a class='existingWikiWord' href='/nlab/show/diff/Quillen+bifunctor'>Quillen bifunctor</a>. As discussed there, since <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_248' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sPSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>sPSh(C)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/cofibrantly+generated+model+category'>cofibrantly generated model category</a> for that it is sufficient to check that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_249' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊗</mo></mrow><annotation encoding='application/x-tex'>\otimes</annotation></semantics></math> satisfies the <span class='newWikiWord'>pushout-prodct axiom<a href='/nlab/new/pushout-prodct+axiom'>?</a></span> on <a class='existingWikiWord' href='/nlab/show/diff/cofibrantly+generated+model+category'>generating (acyclic) cofibrations</a>.</p> <p>As discussed at <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+functors'>model structure on functors</a>, these are those morphisms of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_250' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Id</mi><mo>×</mo><mi>i</mi><mo>:</mo><mi>U</mi><mo>⋅</mo><mi>S</mi><mo>→</mo><mi>U</mi><mo>⋅</mo><mi>T</mi></mrow><annotation encoding='application/x-tex'> Id \times i : U \cdot S \to U \cdot T </annotation></semantics></math></div> <p>for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_251' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \in C</annotation></semantics></math> representable and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_252' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding='application/x-tex'>i : S \to T</annotation></semantics></math> an (acylic) cofibration in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_253' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding='application/x-tex'>sSet_{Quillen}</annotation></semantics></math>. For these morphisms checking the pushout-product axiom amounts to checking it in <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_254' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>, where it is evident.</p> </div> <div class='num_lemma'> <h6 id='lemma_2'>Lemma</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_255' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> with <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a>s and let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_256' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{proj,cov}</annotation></semantics></math> be the left <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>Bousfield localization</a> at the <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a> projections.</p> <p>Then for <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_257' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> any cofibrant object, the <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+structure+on+presheaves'>closed monoidal structure on presheaves</a>-adjunction</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_258' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊣</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>:</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> (X \times (-) \dashv [X,-]) : [C^{op}, sSet]_{proj,cov} \to [C^{op}, sSet]_{proj,cov} </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a>.</p> </div> <div class='proof'> <h6 id='proof_8'>Proof</h6> <p>The above lemma implies that the <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_259' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \times (-)</annotation></semantics></math> preserves cofibrations. As discussed in the <a href='http://ncatlab.org/nlab/show/Quillen+adjunction#sSet'>section on sSet-enriched adjunctions</a> at <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a> since the adjunction is <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_260' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>-enriched and since <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_261' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{proj,cov}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/proper+model+category'>left proper</a> <a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a> it suffices to check that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_262' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,-]</annotation></semantics></math> preserves fibrant objects.</p> <p>For that let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_263' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{U_i \to U\}</annotation></semantics></math> be a covering family and <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_264' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(\{U_i\})</annotation></semantics></math> the corresponding <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Čech nerve</a>. We need to check that if <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_265' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>A \in [C^{op}, sSet]_{proj,cov}</annotation></semantics></math> is fibrant, then</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_266' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>,</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> [C^{op}, sSet](U, [X,A]) \to [C^{op},sSet](C(\{U_i\}), [X,A]) </annotation></semantics></math></div> <p>is an equivalence of <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>es.</p> <p>Writing <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_267' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow></msup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>⋅</mo><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>C(\{U_i\}) = \int^{[n]} \Delta[n] \cdot \coprod U_{i_0, \cdots, i_n}</annotation></semantics></math> and using that the <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom-functor</a> preserves <a class='existingWikiWord' href='/nlab/show/diff/end'>end</a>s, this is eqivalent to</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_268' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>C</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> [C^{op},sSet]( X \times C(\{U_i\}) \to X \times U , A) </annotation></semantics></math></div> <p>being an equivalence. Now we observe that <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_269' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>C</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>X \times C(\{U_i\}) \to X\times U</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/local+epimorphism'>local epimorphism</a> in the above sense, namely a morphism such that for every morphism <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_270' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>V \to X \times U</annotation></semantics></math> out of a representable, there is a lift <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_271' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_272' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>X</mi><mo>×</mo><mi>C</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>U</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ && X \times C(\{U_i\}) \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow \\ V &\to& X \times U } \,. </annotation></semantics></math></div> <p>By the above discussion of the Čech-localization of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_273' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{proj}</annotation></semantics></math>, this is a local morphism, hence does produce an equivalence when hommed into the fibrant object <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_274' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> </div> <h2 id='HomotopyLimits'>Homotopy (co)limits</h2> <p>Properties of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a>s and <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a>s of simplicial presheaves are discussed at</p> <ul> <li><a href='http://ncatlab.org/nlab/show/homotopy+limit#SimpSheaves'>Homotopy (co)limits of simplicial (pre)sheaves</a></li> </ul> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_275' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a>.</p> <div class='num_prop'> <h6 id='proposition_9'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_276' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>F : D \to [C^{op}, sSet]</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite</a> diagram.</p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_277' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℝ</mi> <mi>glob</mi></msub><msub><mi>lim</mi> <mo>←</mo></msub><mi>F</mi><mo>∈</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\mathbb{R}_{glob}\lim_{\leftarrow} F \in [C^{op}, sSet]</annotation></semantics></math> for any representative of the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a> over <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_278' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> computed in the global model structure <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_279' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{proj}</annotation></semantics></math>, well defined up to <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> in the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a>.</p> <p>Then <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_280' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℝ</mi> <mi>glob</mi></msub><msub><mi>lim</mi> <mo>←</mo></msub><mi>F</mi><mo>∈</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\mathbb{R}_{glob}\lim_{\leftarrow} F \in [C^{op},sSet]</annotation></semantics></math> presents also the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a> of <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_281' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> computed in the local model structure <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_282' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{proj,loc}</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_9'>Proof</h6> <p>By the discussion at <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a> the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_283' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi><msub><mi>lim</mi> <mo>←</mo></msub></mrow><annotation encoding='application/x-tex'>\mathbb{R}\lim_{\leftarrow}</annotation></semantics></math> computes the corresponding <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a> and <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheafification'>(∞,1)-sheafification</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_284' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L </annotation></semantics></math> is a left <a class='existingWikiWord' href='/nlab/show/diff/flat+%28infinity%2C1%29-functor'>exact (∞,1)-functor</a> and preserves these finite <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a>s:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_285' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>D</mi><mo>,</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msub><mo stretchy='false'>]</mo> <mi>inj</mi></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>L</mi> <mo>*</mo></msub></mrow></mover></mtd> <mtd><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>D</mi><mo>,</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub><msub><mo stretchy='false'>]</mo> <mi>inj</mi></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><mi>ℝ</mi><munder><mi>lim</mi> <mo>←</mo></munder></mrow></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><mi>ℝ</mi><munder><mi>lim</mi> <mo>←</mo></munder></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mtd> <mtd><mover><mo>←</mo><mrow><mi>L</mi><mo>≃</mo><mi>𝕃</mi><mi>Id</mi></mrow></mover></mtd> <mtd><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ ([D, [C^{op}, sSet]_{proj, loc}]_{inj})^\circ &\stackrel{L_*}{\leftarrow}& ([D, [C^{op}, sSet]_{proj}]_{inj})^\circ \\ \downarrow^{\mathrlap{\mathbb{R} \lim_\leftarrow}} && \downarrow^{\mathrlap{\mathbb{R} \lim_\leftarrow}} \\ ([C^{op}, sSet]_{proj,loc})^\circ &\stackrel{L \simeq \mathbb{L} Id}{\leftarrow}& ([C^{op}, sSet]_{proj})^\circ } \,. </annotation></semantics></math></div> <p>Here <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_286' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>≃</mo><mi>𝕃</mi><mi>Id</mi></mrow><annotation encoding='application/x-tex'>L \simeq \mathbb{L} Id</annotation></semantics></math> is the left <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functor</a> of the identity for the <a href='#PresentationOfTheInfinTopos'>above</a> left Bousfield localization. Since left Bousfield localization does not change the cofibrations and includes the global weak equivalences into the local weak equivalences, the postcomposition of the diagram <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_287' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_288' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕃</mi><mi>Id</mi></mrow><annotation encoding='application/x-tex'>\mathbb{L} Id</annotation></semantics></math> is given by cofibrant replacement in the local structure, too. But the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a> of the diagram is invariant, up to equivalence, under cofibrant replacement, and hence a finite homotopy limit diagram in the global structure is also one in the local structure.</p> </div> <h2 id='InclusionOfChainComplexes'>Inclusion of chain complexes of sheaves</h2> <p>We discuss how <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a>es of presheaves of <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>s embed into the model structure on simplicial presheaves. Under passing to the intrinsic <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a> of the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a> <a href='#PresentationOfInfiniToposes'>presented by</a> by <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_289' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>loc</mi></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, sSet]_{loc}</annotation></semantics></math>, this realizes traditional <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a> over <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_290' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and generalizes it to general base objects.</p> <p>Observe from the discussion at <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+abelian+groups'>model structure on simplicial abelian groups</a> that the degreewise <a class='existingWikiWord' href='/nlab/show/diff/free+functor'>free functor</a>-<a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_291' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>Ab</mi><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><mi>Set</mi></mrow><annotation encoding='application/x-tex'>(F \dashv U) : Ab \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Set</annotation></semantics></math> (see <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+Lawvere+theory'>algebra over a Lawvere theory</a> for details) induces a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_292' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>:</mo><msub><mi>sAb</mi> <mi>Quillen</mi></msub><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding='application/x-tex'> (F \dashv U) : sAb_{Quillen } \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet_{Quillen} </annotation></semantics></math></div> <p>between the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+abelian+groups'>model structure on simplicial abelian groups</a> and the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical model structure on simplicial sets</a>, which exhibits <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_293' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>sAb</mi> <mi>Quillen</mi></msub></mrow><annotation encoding='application/x-tex'>sAb_{Quillen}</annotation></semantics></math> as the corresponding <a class='existingWikiWord' href='/nlab/show/diff/transferred+model+structure'>transferred model structure</a>.</p> <p>Moreover, the <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a> constitutes in particular a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalence</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_294' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>N</mi> <mo>•</mo></msub><mo>⊣</mo><mi>Γ</mi><mo stretchy='false'>)</mo><mo>:</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><msub><mo /><mi>proj</mi></msub><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mrow><msub><mi>N</mi> <mo>•</mo></msub></mrow></mover></mover><msub><mi>sAb</mi> <mi>Quillen</mi></msub></mrow><annotation encoding='application/x-tex'> (N_\bullet \dashv \Gamma) : Ch_\bullet^+_{proj} \stackrel{\overset{N_\bullet}{\leftarrow}}{\underset{\Gamma}{\to}} sAb_{Quillen} </annotation></semantics></math></div> <p>between the projective <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+chain+complexes'>model structure on chain complexes</a> of <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>s in non-negative degree and simplicial abelian groups.</p> <p>We write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_295' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>N</mi> <mo>•</mo></msub><mi>F</mi><mo>⊣</mo><mi>Ξ</mi><mo stretchy='false'>)</mo><mo>:</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><msub><mo /><mi>proj</mi></msub><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mrow><msub><mi>N</mi> <mo>•</mo></msub></mrow></mover></mover><msub><mi>sAb</mi> <mi>Quillen</mi></msub><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding='application/x-tex'> (N_\bullet F \dashv \Xi) : Ch_\bullet^+_{proj} \stackrel{\overset{N_\bullet}{\leftarrow}}{\underset{\Gamma}{\to}} sAb_{Quillen} \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet_{Quillen} </annotation></semantics></math></div> <p>for the composite <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a>. For <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_296' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> any <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>, postcomposition with <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_297' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ξ</mi></mrow><annotation encoding='application/x-tex'>\Xi</annotation></semantics></math> induces a Quillen adjunction</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_298' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>N</mi> <mo>•</mo></msub><mi>F</mi><mo>⊣</mo><mi>Ξ</mi><mo stretchy='false'>)</mo><mo>:</mo><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><msub><mo /><mi>proj</mi></msub><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub><mover><munder><mo>→</mo><mi>Ξ</mi></munder><mover><mo>←</mo><mrow><msub><mi>N</mi> <mo>•</mo></msub><mi>F</mi></mrow></mover></mover><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'> (N_\bullet F \dashv \Xi) : [C^{op}, Ch_\bullet^+_{proj}]_{proj} \stackrel{\overset{N_\bullet F}{\leftarrow}}{\underset{\Xi}{\to}} [C^{op}, sSet]_{proj} </annotation></semantics></math></div> <p>between the projective <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+functors'>model structure on functors</a> <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_299' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><msub><mo /><mi>proj</mi></msub><msub><mo stretchy='false'>]</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>[C^{op}, Ch_\bullet^+_{proj}]_{proj}</annotation></semantics></math> and the global projective model structure on simplicial presheaves, which by convenient abuse of notation we denote by the same symbols.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/local+fibration'>local fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brown-Gersten+property'>Brown-Gersten property</a></p> </li> </ul> <p><a class='existingWikiWord' href='/nlab/show/diff/model+topos'>model topos</a></p> <ul> <li> <p><strong>model structure on simplicial presheaves</strong></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sheaves'>model structure on simplicial sheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+sSet-enriched+presheaves'>model structure on sSet-enriched presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cubical+presheaves'>model structure on cubical presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+presheaves+of+spectra'>model structure on presheaves of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+%282%2C1%29-sheaves'>model structure for (2,1)-sheaves</a></p> </li> </ul> <p><strong>Locally presentable categories:</strong> <a class='existingWikiWord' href='/nlab/show/diff/cocomplete+category'>Cocomplete</a> possibly-<a class='existingWikiWord' href='/nlab/show/diff/large+category'>large categories</a> generated under <a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimits</a> by <a class='existingWikiWord' href='/nlab/show/diff/small+object'>small</a> <a class='existingWikiWord' href='/nlab/show/diff/generator'>generators</a> under <a class='existingWikiWord' href='/nlab/show/diff/small+limit'>small</a> <a class='existingWikiWord' href='/nlab/show/diff/relation'>relations</a>. Equivalently, <a class='existingWikiWord' href='/nlab/show/diff/accessible+functor'>accessible</a> <a class='existingWikiWord' href='/nlab/show/diff/reflective+localization'>reflective localizations</a> of <a class='existingWikiWord' href='/nlab/show/diff/free+cocompletion'>free cocompletions</a>. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a <a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>left exact</a> localization.</p> <table><thead><tr><th><math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_300' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/%28n%2Cr%29-category'>(n,r)-categories</a><math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_301' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th><th><math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_302' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/topos'>toposes</a><math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_303' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th><th>locally presentable</th><th>loc finitely pres</th><th>localization theorem</th><th><a class='existingWikiWord' href='/nlab/show/diff/free+cocompletion'>free cocompletion</a></th><th>accessible</th></tr></thead><tbody><tr><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-category+theory'>(0,1)-category theory</a></strong></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/suplattice'>suplattice</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/algebraic+lattice'>algebraic lattices</a></td><td style='text-align: left;'><a href='algebraic+lattice#RelationToLocallyFinitelyPresentableCategories'>Porst’s theorem</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/power+set'>powerset</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a></td></tr> <tr><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></strong></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>toposes</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+category'>locally presentable categories</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/locally+finitely+presentable+category'>locally finitely presentable categories</a></td><td style='text-align: left;'><a href='locally+presentable+category#AsLocalizationsOfPresheafCategories'>Gabriel–Ulmer’s theorem</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>presheaf category</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/accessible+category'>accessible categories</a></td></tr> <tr><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category theory</a></strong></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/model+topos'>model toposes</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/combinatorial+model+category'>combinatorial model categories</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Dugger%27s+theorem'>Dugger's theorem</a></td><td style='text-align: left;'>global <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structures on simplicial presheaves</a></td><td style='text-align: left;'>n/a</td></tr> <tr><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a></strong></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-toposes</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>locally presentable (∞,1)-categories</a></td><td style='text-align: left;' /><td style='text-align: left;'><a href='locally+presentable+infinity-category#Definition'>Simpson’s theorem</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-presheaf (∞,1)-categories</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/accessible+%28infinity%2C1%29-category'>accessible (∞,1)-categories</a></td></tr> </tbody></table> <h2 id='references'>References</h2> <p><span><del class='diffmod'> A</del><ins class='diffmod'> Introductions:</ins><del class='diffdel'> nice</del><del class='diffdel'> introduction</del><del class='diffdel'> and</del><del class='diffdel'> survey</del><del class='diffdel'> is</del><del class='diffdel'> provided</del><del class='diffdel'> in</del><del class='diffdel'> the</del><del class='diffdel'> notes</del></span></p> <ul> <li id='Dugger98'><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Dugger'>Dan Dugger</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Dugger'>Dan Dugger</a>, <em>Sheaves and homotopy theory</em>, (1998) [[web](http://www.uoregon.edu/~ddugger/cech.html), <a href='http://www.uoregon.edu/~ddugger/cech.dvi'>dvi</a>, <a href='http://ncatlab.org/nlab/files/cech.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/DuggerSheavesAndHomotopyTheory.pdf' title='pdf'>pdf</a>]</p></ins><del class='diffmod'><em>Sheaves and homotopy theory</em></del><ins class='diffmod'> </ins><del class='diffdel'>, (1998) [[web](http://www.uoregon.edu/~ddugger/cech.html), </del><del class='diffdel'><a href='http://www.uoregon.edu/~ddugger/cech.dvi'>dvi</a></del><del class='diffdel'>, </del><del class='diffdel'><a href='http://ncatlab.org/nlab/files/cech.pdf'>pdf</a></del><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/files/DuggerSheavesAndHomotopyTheory.pdf' title='pdf'>pdf</a></del><del class='diffdel'>]</del></li><ins class='diffins'> </ins><ins class='diffins'><li id='Jardine07'> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>John F. Jardine</a>, <em>Simplicial Presheaves</em>, lecture notes, Fields Institute (2007) [[pdf](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf), <a href='https://www.uwo.ca/math/faculty/jardine/courses/fields/fields_lectures.html'>webpage</a>]</p> </li></ins> </ul> <p>Detailed discussion of the injective model structures on simplicial presheaves:</p> <ul> <li id='JardineLecture'> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>John F. Jardine</a>, <em>Simplicial presheaves</em><span> , Journal of Pure and Applied Algebra<del class='diffdel'> 47</del><del class='diffdel'> (1987),</del><del class='diffdel'> 35-87</del><del class='diffdel'> (</del></span><del class='diffmod'><a href='http://math.uchicago.edu/~amathew/simplicialpresheaves.pdf'>pdf</a></del><ins class='diffmod'><strong>47</strong></ins><span><del class='diffmod'> )</del><ins class='diffmod'> </ins><ins class='diffins'> (1987)</ins><ins class='diffins'> 35-87</ins><ins class='diffins'> []</ins></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>John F. Jardine</a>, <em>Stacks and the homotopy theory of simplicial sheaves</em>, Homology, homotopy and applications, vol. 3 (2), 2001, pp.361–384 (<a href='https://projecteuclid.org/euclid.hha/1139840259'>euclid:hha/1139840259</a>)</p> </li> <li id='Jardine96'> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>John F. Jardine</a>, <em>Boolean localization, in practice</em><span> , Documenta<del class='diffmod'> Mathematica,</del><ins class='diffmod'> Mathematica</ins><del class='diffdel'> Vol.</del><del class='diffdel'> 1</del><del class='diffdel'> (1996),</del><del class='diffdel'> 245-275</del><del class='diffdel'> (</del></span><ins class='diffins'><strong>1</strong></ins><ins class='diffins'> (1996), 245-275 (</ins><a href='https://www.math.uni-bielefeld.de/documenta/vol-01/13.html'>documenta:vol-01/13</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>John F. Jardine</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Local+homotopy+theory'>Local homotopy theory</a></em>, Springer Monographs in Mathematics (2015) [[doi:10.1007/978-1-4939-2300-7](https://doi.org/10.1007/978-1-4939-2300-7)]</p> </li> </ul> <p>The projective model structure is discussed in</p> <ul> <li id='Dugger01'><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Dugger'>Daniel Dugger</a>, <em>Universal homotopy theories</em>, Advances in Mathematics Volume 164, Issue 1, (2001) Pages 144-176 (<a href='http://hopf.math.purdue.edu/Dugger/dduniv.pdf'>pdf</a>, <a href='https://arxiv.org/abs/math/0007070'>arXiv:math/0007070</a>, <a href='https://doi.org/10.1006/aima.2001.2014'>doi:10.1006/aima.2001.2014</a>)</li> </ul> <p>See also</p> <ul> <li>Benjamin Blander, <em>Local projective model structures on simplicial presheaves</em>, K-Theory, Volume 24, Number 3, (2001), 283–301, <a href='http://dx.doi.org/10.1023/a:1013302313123'>doi</a>.</li> </ul> <p>On <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>derived hom-spaces</a> (<a class='existingWikiWord' href='/nlab/show/diff/function+complex'>function complexes</a>) in the projective model structure on simplicial presheaves:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/William+Dwyer'>William Dwyer</a>, <a class='existingWikiWord' href='/nlab/show/diff/Daniel+Kan'>Daniel Kan</a>, <em>Function complexes for diagrams of simplicial sets</em>, Indagationes Mathematicae (Proceedings) <strong>86</strong> 2 (1983) 139-147 [, pdf]</li> </ul> <p>A brief review in the context of <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+Hodge+theory'>nonabelian Hodge theory</a> is in section 4 of</p> <ul> <li id='Olsson'>Martin Olsson, <em>Towards non-abelian <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_304' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>-adic Hodge theory in the good reduction case</em> (<a href='http://math.berkeley.edu/~molsson/PHT3-24-08.pdf'>pdf</a>)</li> </ul> <p>A detailed study of <a class='existingWikiWord' href='/nlab/show/diff/descent'>descent</a> for simplicial presheaves is given in</p> <ul> <li id='DuggerHollanderIsaksen'> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Dugger'>Daniel Dugger</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sharon+Hollander'>Sharon Hollander</a>, <a class='existingWikiWord' href='/nlab/show/diff/Daniel+Isaksen'>Daniel Isaksen</a>, <em>Hypercovers and simplicial presheaves</em>, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9–51 (<a href='http://www.math.uiuc.edu/K-theory/0563/'>web</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Dugger'>Daniel Dugger</a>, <a class='existingWikiWord' href='/nlab/show/diff/Daniel+Isaksen'>Daniel Isaksen</a>, <em>Weak equivalences of simplicial presheaves</em> (<a href='http://arxiv.org/abs/math/0205025'>arXiv</a>)</p> </li> </ul> <p>A survey of many of the model structures together with a treatment of the left local projective one is in</p> <ul> <li id='Blander'><a class='existingWikiWord' href='/nlab/show/diff/Benjamin+Blander'>Benjamin Blander</a>, <em>Local projective model structure on simplicial presheaves</em> (<a href='http://www.math.uiuc.edu/K-theory/0462/combination2.pdf'>pdf</a>)</li> </ul> <p>See also</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Isaksen'>Daniel Isaksen</a>, <em>Flasque model structure for simplicial presheaves</em> (<a href='http://www.math.uiuc.edu/K-theory/0679/'>web</a>, <a href='http://www.math.uiuc.edu/K-theory/0679/flasque.pdf'>pdf</a>)</li> </ul> <p>The characterization of the model category of simplicial presheaves as the canonical <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentation</a> of the (hypercompletion of) the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a> on a site is in</p> <ul> <li> <p><a href='http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=528'>proposition 6.5.2.1</a> of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a></em></li> </ul> </li> </ul> <p>A set of lecture notes on simplicial presheaves with an eye towards algebraic sites and <a class='existingWikiWord' href='/nlab/show/diff/derived+algebraic+geometry'>derived algebraic geometry</a> is</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Bertrand+To%C3%ABn'>Bertrand Toen</a>, <em>Simplicial presheaves and derived algebraic geometry</em> , lecture at <a href='http://www.crm.es/HigherCategories/'>Simplicial methofs in higher categories</a> (<a href='http://www.crm.cat/HigherCategories/hc1.pdf'>pdf</a>)</li> </ul> <p>Last not least, it is noteworthy that the idea of localizing simplicial sheaves at stalkwise weak equivalences is already described and applied in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kenneth+Brown'>Kenneth Brown</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/BrownAHT'>Abstract Homotopy Theory and Generalized Sheaf cohomology</a></em> ,</li> </ul> <p>using instead of a full <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> structure the more lightweight one of a Brown <a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a>.</p> <p>A comparison between Brown-Gersten and Joyal-Jardine approach:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Vladimir+Voevodsky'>Vladimir Voevodsky</a>, <em>Homotopy theory of simplicial presheaves in completely decomposable topologies</em>, <a href='http://arxiv.org/abs/0805.4578'>arxiv/0805.4578</a></li> </ul> <p>The proposal for descent objects for strict <math class='maruku-mathml' display='inline' id='mathml_a0fbe322239e0702170053ecbededfd07a8fc020_305' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoid-valued presheaves discussed in <a href='#DescentForStrictInf'>Descent for strict infinity-groupoids</a> appeared in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ross+Street'>Ross Street</a>, <em>Categorical and combinatorial aspects of descent theory</em> (<a href='http://arxiv.org/abs/math/0303175'>arXiv</a>)</li> </ul> <p>The relation to the general descent conditionF is discussed in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Dominic+Verity'>Dominic Verity</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves'>Relating descent notions</a></em></li> </ul> <p>A useful collection of facts is in</p> <ul> <li id='Low'><a class='existingWikiWord' href='/nlab/show/diff/Zhen+Lin+Low'>Zhen Lin Low</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Notes+on+homotopical+algebra'>Notes on homotopical algebra</a></em></li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on October 15, 2023 at 12:03:45. 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