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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 #22 to #23: <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='homotopy_theory'>Homotopy theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a></strong></p> <p>flavors: <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/diff/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/diff/cohesive+homotopy+theory'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/diff/directed+homotopy+theory'>directed</a>…</p> <p>models: <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/diff/localic+homotopy+theory'>localic</a>, …</p> <p>see also <strong><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>higher homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/spherical+object'>spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</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/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/diff/cofibration+category'>cofibration category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Waldhausen+category'>Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+universal+bundle'>universal bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+at+geometric+homotopies'>homotopy localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+interval+object'>infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+a+topos'>fundamental group of a topos</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a></li> </ul> </li> <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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+category'>fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Galois+theory'>Galois theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p> </li> </ul> </div> <h4 id='topos_theory'><math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_1' 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='#definition'>Definition</a></li><li><a href='#interpretation_as_stacks'>Interpretation as <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stacks</a></li><li><a href='#examples'>Examples</a></li><li><a href='#remarks'>Remarks</a></li><li><a href='#properties'>Properties</a></li><li><a href='#related_entries'>Related entries</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p><em>Simplicial presheaves</em> over some <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> are</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/presheaf'>Presheaves</a> with values in the category <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>SimpSet</a> of simplicial sets, i.e., functors <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>op</mi></msup><mo>→</mo><mo lspace='0em' rspace='thinmathspace'>Simp</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo></mrow><annotation encoding='application/x-tex'>S^{op} \to \Simp\Set</annotation></semantics></math>, i.e., functors <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>S^{op} \to [\Delta^{op}, \Set]</annotation></semantics></math>;</li> </ul> <p>or equivalently, using the Hom-<a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjunction</a> and symmetry of the <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal structure</a> on <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> <ul> <li>simplicial objects in the category of presheaves, i.e. functors <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>,</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta^{op} \to [S^{op},\Set]</annotation></semantics></math>.</li> </ul> <h2 id='interpretation_as_stacks'>Interpretation as <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-stacks</h2> <p>Regarding <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>Simp</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo></mrow><annotation encoding='application/x-tex'>\Simp\Set</annotation></semantics></math> as a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> using the standard <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sets'>model structure on simplicial sets</a> and inducing from that a model structure on <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>,</mo><mo lspace='0em' rspace='thinmathspace'>Simp</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[S^{op}, \Simp\Set]</annotation></semantics></math> makes simplicial presheaves a model for <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>stacks</a>, as described at <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack+homotopically'>infinity-stack homotopically</a>.</p> <p>In more illustrative language this means that a simplicial presheaf on <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> can be regarded as an <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>groupoid</a> (in particular a <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>) whose space of <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-morphisms is modeled on the objects of <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> in the sense described at <a class='existingWikiWord' href='/nlab/show/diff/space+and+quantity'>space and quantity</a>.</p> <h2 id='examples'>Examples</h2> <ul> <li> <p>Notice that most definitions of <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/infinity-category'>category</a> the <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-category is itself defined to be a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> with extra structure (in a <a class='existingWikiWord' href='/nlab/show/diff/geometric+definition+of+higher+categories'>geometric definition of higher category</a>) or gives rise to a simplicial set under taking its <a class='existingWikiWord' href='/nlab/show/diff/nerve'>nerve</a> (in an <a class='existingWikiWord' href='/nlab/show/diff/algebraic+definition+of+higher+categories'>algebraic definition of higher category</a>). So most notions of presheaves of higher categories will naturally induce presheaves of simplicial sets.</p> </li> <li> <p>In particular, regarding a <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> as a one object category <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding='application/x-tex'>\mathbf{B}G</annotation></semantics></math> and then taking the nerve <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><mo stretchy='false'>)</mo><mo>∈</mo><mo lspace='0em' rspace='thinmathspace'>Simp</mo><mo lspace='0em' rspace='thinmathspace'>Set</mo></mrow><annotation encoding='application/x-tex'>N(\mathbf{B}G) \in \Simp\Set</annotation></semantics></math> of these (the “classifying simplicial set of the group whose <a class='existingWikiWord' href='/nlab/show/diff/geometric+realization'>geometric realization</a> is the <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}G</annotation></semantics></math>), which is clearly a functorial operation, turns any presheaf with values in groups into a simplicial presheaf.</p> </li> </ul> <h2 id='remarks'>Remarks</h2> <ul> <li>There are various useful <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> structures on the category of simplicial presheaves. See <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a>.</li> </ul> <h2 id='properties'>Properties</h2> <p>Here are some basic but useful facts about simplicial presheaves.</p> <div class='un_prop'> <h6 id='proposition'>Proposition</h6> <p>Every simplicial presheaf <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a> over a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> of <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>-valued sheaves regarded as discrete simplicial sheaves.</p> <p>More precisely, for <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>→</mo><mi>SSet</mi></mrow><annotation encoding='application/x-tex'>X : S^{op} \to SSet</annotation></semantics></math> a simplicial presheaf, let <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo><mo>↪</mo><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>,</mo><mi>SSet</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>D_X : \Delta^{op} \to [S^{op},Set] \hookrightarrow [S^{op},SSet]</annotation></semantics></math> be given by <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub><mo>:</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>↦</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>D_X : [n] \mapsto X_n</annotation></semantics></math>. Then there is a weak equivalence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>hocolim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>D</mi> <mi>X</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> hocolim_{[n] \in \Delta} D_X([n]) \stackrel{\simeq}{\to} X \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>See for instance <a href='http://www.math.uiuc.edu/K-theory/0563/spre.pdf#page=6'>remark 2.1, p. 6</a></p> <ul> <li><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>, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 136 Issue 1, 2004 (<a href='https://arxiv.org/abs/math/0205027'>arXiv:math/0205027</a>, <a href='http://www.math.uiuc.edu/K-theory/0563'>K-theory:0563</a>, <a href='https://doi.org/10.1017/S0305004103007175'>doi:10.1017/S0305004103007175</a>)</li> </ul> <p>(which is otherwise about <a class='existingWikiWord' href='/nlab/show/diff/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a>).</p> </div> <div class='un_cor'> <h6 id='corollary'>Corollary</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo>:</mo><mo stretchy='false'>(</mo><msup><mi>SSet</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mo>×</mo><msup><mi>SSet</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>SSet</mi></mrow><annotation encoding='application/x-tex'>[-,-] : (SSet^{S^{op}})^{op} \times SSet^{S^{op}} \to SSet</annotation></semantics></math> be the canonical <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SSet</mi></mrow><annotation encoding='application/x-tex'>SSet</annotation></semantics></math>-enrichment of the category of simplicial presheaves (i.e. the assignment of <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>SSet</a>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+functor+category'>enriched functor categories</a>).</p> <p>It follows in particular from the above that every such <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-object</a> <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,A]</annotation></semantics></math> of simplical presheaves can be written as a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a> (in <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>SSet</a> for instance realized as a <a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a>, as described there) over evaluations of <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_29' 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> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>First the above yields</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_30' 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><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd> <mtd><mo>≃</mo><mo stretchy='false'>[</mo><msub><mi>hocolim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd></mtr> <mtr><mtd /> <mtd><msub><mi>holim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mo stretchy='false'>[</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned} [X, A ] & \simeq [ hocolim_{[n] \in \Delta} X_n , A ] \\ & holim_{[n] \in \Delta} [X_n, A] \end{aligned} \,. </annotation></semantics></math></div> <p>Next from the <a class='existingWikiWord' href='/nlab/show/diff/co-Yoneda+lemma'>co-Yoneda lemma</a> we know that the <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>-valued presheaves <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>X_n</annotation></semantics></math> are in turn colimits over representables in <math class='maruku-mathml' display='inline' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, so that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_33' 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><mi>⋯</mi></mtd> <mtd><mo>≃</mo><msub><mi>holim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mo stretchy='false'>[</mo><msub><mi>colim</mi> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>holim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>lim</mi> <mi>i</mi></msub><mo stretchy='false'>[</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned} \cdots & \simeq holim_{[n] \in \Delta} [ colim_i U_{i}, A] \\ & \simeq holim_{[n] \in \Delta} lim_i [ U_{i}, A] \end{aligned} \,. </annotation></semantics></math></div> <p>And finally the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a> reduces this to</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0763f6d99838e3bf8649ccba2abf9a613d17651a_34' 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><mi>⋯</mi></mtd> <mtd><msub><mi>holim</mi> <mrow><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>lim</mi> <mi>i</mi></msub><mi>A</mi><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned} \cdots & holim_{[n] \in \Delta} lim_i A(U_i) \end{aligned} \,. </annotation></semantics></math></div></div> <p>Notice that these kinds of computations are in particular often used when checking/computing <a class='existingWikiWord' href='/nlab/show/diff/descent'>descent and codescent</a> along a <a class='existingWikiWord' href='/nlab/show/diff/cover'>cover</a> or <a class='existingWikiWord' href='/nlab/show/diff/hypercover'>hypercover</a>. For more on that in the context of simplicial presheaves see <a class='existingWikiWord' href='/nlab/show/diff/descent+for+simplicial+presheaves'>descent for simplicial presheaves</a>.</p> <h2 id='related_entries'>Related entries</h2> <ul> <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/sheaf+of+spectra'>presheaf of spectra</a></p> </li> </ul> <p>Applications appear for instance at</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/geometric+infinity-function+theory'>geometric infinity-function theory</a></li> </ul> <h2 id='references'>References</h2> <p>The original articles are</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kenneth+Brown'>Kenneth S. Brown</a>, <em>Abstract homotopy theory and generalized sheaf cohomology</em>. Transactions of the American Mathematical Society 186 (1973), 419-419. <a href='http://dx.doi.org/10.1090/s0002-9947-1973-0341469-9'>doi</a>.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kenneth+Brown'>Kenneth S. Brown</a>, <a class='existingWikiWord' href='/nlab/show/diff/Stephen+M.+Gersten'>Stephen M. Gersten</a>, <em>Algebraic K-theory as generalized sheaf cohomology</em>. In: Higher K-Theories. Lecture Notes in Mathematics (1973), 266–292. <a href='http://dx.doi.org/10.1007/bfb0067062'>doi</a>.</p> </li> </ul> <p>The Joyal model structure on <a class='existingWikiWord' href='/nlab/show/diff/simplicial+sheaf'>simplicial sheaves</a> was constructed in 1984 by <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>Joyal</a> in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, <em>Lettre d’André Joyal à Alexandre Grothendieck</em>, April 11, 1984, <a href='https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf'>PDF</a>.</li> </ul> <p>Simplicial sheaves were endowed with a structure of a <a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a> in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>J. F. Jardine</a>, <em>Simplicial objects in a Grothendieck topos</em>. In: Applications of algebraic K-theory to algebraic geometry and number theory. Contemporary Mathematics (1986), 193-239. <a href='http://dx.doi.org/10.1090/conm/055.1/862637'>doi</a></li> </ul> <p>The model structure on simplicial presheaves is due to <a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>Jardine</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'><span><del class='diffmod'> J.</del><ins class='diffmod'> John</ins> F. Jardine</span></a>, <em>Simplical presheaves</em><span><del class='diffmod'> .</del><ins class='diffmod'> ,</ins> Journal of Pure and Applied Algebra<del class='diffdel'> 47:1</del><del class='diffdel'> (1987),</del><del class='diffdel'> 35-87.</del></span><del class='diffmod'><a href='http://dx.doi.org/10.1016/0022-4049(87)90100-9'>doi</a></del><ins class='diffmod'><strong>47</strong></ins><ins class='diffins'> 1 (1987) 35-87 []</ins></li> </ul> <p>A modern expository account is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>John F. Jardine</a>, <em>Local Homotopy Theory</em>, Springer, 2015. <a href='http://dx.doi.org/10.1007/978-1-4939-2300-7'>doi</a>.</li> </ul> <p>The local <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+functors'>projective model structure</a> (as opposed to the injective model structures used above) on <a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaves</a> and <a class='existingWikiWord' href='/nlab/show/diff/simplicial+sheaf'>simplicial sheaves</a> is due to</p> <ul> <li><span class='newWikiWord'>Benjamin A. Blander<a href='/nlab/new/Benjamin+A.+Blander'>?</a></span>, <em>Local Projective Model Structures on Simplicial Presheaves</em>, K-Theory 24:3, 283-301. <a href='http://dx.doi.org/10.1023/a:1013302313123'>doi</a>.</li> </ul> <p>Further articles:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>J. F. Jardine</a>, <em>Stacks and the homotopy theory of simplicial sheaves</em>. Homology, Homotopy and Applications 3:2 (2001), 361-384. <a href='http://dx.doi.org/10.4310/hha.2001.v3.n2.a5'>doi</a>.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>J. F. Jardine</a>, <em>Fields Lectures: Simplicial presheaves</em>. <a href='https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf'>PDF</a>.</p> </li> </ul> <p>For their interpretation in the more general context of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(infinity,1)-sheaves</a> see Section 6.5.2 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> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on November 14, 2023 at 13:44:14. 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