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(2,1)-sheaf in nLab
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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="locality_and_descent">Locality and descent</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/localization">localization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+object">local object</a>, <a class="existingWikiWord" href="/nlab/show/local+morphism">local morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+localization">reflective localization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/descent">descent</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cover">cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+object">descent object</a>, <a class="existingWikiWord" href="/nlab/show/descent+morphism">descent morphism</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/matching+family">matching family</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/stack">stack</a>, <a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>,<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+descent">cohomological descent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadic+descent">monadic descent</a>, <a class="existingWikiWord" href="/nlab/show/higher+monadic+descent">higher monadic descent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sweedler+coring">Sweedler coring</a>, <a class="existingWikiWord" href="/nlab/show/descent+in+noncommutative+algebraic+geometry">descent in noncommutative algebraic geometry</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/descent+and+locality+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="2category_theory">2-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a></p> </li> </ul> </div></div> <h4 id="topos_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#the_category_of_sheaves'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-sheaves</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-sheaf is a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> with values in <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>s. This is traditionally called a <a class="existingWikiWord" href="/nlab/show/stack">stack</a>.</p> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-site">(2,1)-site</a>. Write <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> for the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>s, <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s and <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>s.</p> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-sheaf</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is equivalently</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">C^{op} \to Grpd</annotation></semantics></math> that satisfies <a class="existingWikiWord" href="/nlab/show/descent">descent</a>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a> with values in <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>s;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/1-truncated">1-truncated</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </li> </ul> <h2 id="the_category_of_sheaves">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-sheaves</h2> <p>The <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-sheaves on a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-site">(2,1)-site</a> forms a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>.</p> <p>There are <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentations</a> of this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-topos. See <a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a> / <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-sheaf</strong> / <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-sheaf">(∞,n)-sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent">descent</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cover">cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+descent">cohomological descent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+morphism">descent morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadic+descent">monadic descent</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Sweedler+coring">Sweedler coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+monadic+descent">higher monadic descent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+in+noncommutative+algebraic+geometry">descent in noncommutative algebraic geometry</a></p> </li> </ul> </li> </ul> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/homotopy+level">homotopy level</a></th><th><a class="existingWikiWord" href="/nlab/show/n-truncated+object+in+an+%28infinity%2C1%29-category">n-truncation</a></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+theory">higher topos theory</a></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">h-level 0</td><td style="text-align: left;">(-2)-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28-2%29-groupoid">(-2)-groupoid</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a>/<a class="existingWikiWord" href="/nlab/show/contractible+type">contractible type</a></td></tr> <tr><td style="text-align: left;">h-level 1</td><td style="text-align: left;">(-1)-truncated</td><td style="text-align: left;">contractible-if-<a class="existingWikiWord" href="/nlab/show/inhabited+space">inhabited</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28-1%29-groupoid">(-1)-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-sheaf">(0,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/ideal">ideal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a>/<a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a></td></tr> <tr><td style="text-align: left;">h-level 2</td><td style="text-align: left;">0-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+0-type">homotopy 0-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-groupoid">0-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;">h-level 3</td><td style="text-align: left;">1-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/1-groupoid">1-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/stack">stack</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-groupoid">h-groupoid</a></td></tr> <tr><td style="text-align: left;">h-level 4</td><td style="text-align: left;">2-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+2-type">homotopy 2-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></td><td style="text-align: left;">(3,1)-sheaf/2-stack</td><td style="text-align: left;">h-2-groupoid</td></tr> <tr><td style="text-align: left;">h-level 5</td><td style="text-align: left;">3-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+3-type">homotopy 3-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></td><td style="text-align: left;">(4,1)-sheaf/3-stack</td><td style="text-align: left;">h-3-groupoid</td></tr> <tr><td style="text-align: left;">h-level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n+2</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28n%2B1%2C1%29-sheaf">(n+1,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/n-stack">n-stack</a></td><td style="text-align: left;">h-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoid</td></tr> <tr><td style="text-align: left;">h-level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td><td style="text-align: left;">untruncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a></td><td style="text-align: left;">h-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid</td></tr> </tbody></table> </div></body></html> </div> <div class="revisedby"> <p> Last revised on April 25, 2013 at 22:00:22. See the <a href="/nlab/history/%282%2C1%29-sheaf" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/%282%2C1%29-sheaf" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/%282%2C1%29-sheaf/5" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/%282%2C1%29-sheaf" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/%282%2C1%29-sheaf" accesskey="S" class="navlink" id="history" rel="nofollow">History (5 revisions)</a> <a href="/nlab/show/%282%2C1%29-sheaf/cite" style="color: black">Cite</a> <a href="/nlab/print/%282%2C1%29-sheaf" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/%282%2C1%29-sheaf" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>