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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="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> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#provisional_discussion'>Provisional discussion</a></li> <ul> <li><a href='#examples'>Examples</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The term <em>stack</em>, is a traditional synonym for <em><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a></em> or often, more restrictively, as a synonym for <em><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a></em> (see there for more details).</p> <p>This is part of a whole hierarchy of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categorical</a> generalizations of the notion of <em><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></em>. A <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a> / stack is equivalently 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>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>.</p> <p>Generally, then, <em><a class="existingWikiWord" href="/nlab/show/n-stack">n-stack</a></em> is a synonym for <em><a class="existingWikiWord" href="/nlab/show/n-sheaf">(n+1)-sheaf</a></em>, or more restrictively for <em><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-sheaf">(n+1,1)-sheaf</a></em>.</p> <p>More concretely this means that a 1-stack on a <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> (or more generally <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-site">(2,1)-site</a> or even <a class="existingWikiWord" href="/nlab/show/%282%2C2%29-site">(2,2)-site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>) is</p> <ul> <li> <p>a (<a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/functor">functor</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">S^{op}</annotation></semantics></math> to the 2-category <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of categories,</p> </li> <li> <p>that satisfies <a class="existingWikiWord" href="/nlab/show/descent">descent</a> for all covers.</p> </li> </ul> <p>If the pseudofunctor takes values in the 2-subcategory <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\subset Cat</annotation></semantics></math> of groupoids, the stack is sometimes referred to as a stack of groupoids. This is the more commonly occurring case so the term ‘stack’ has come to mean ‘stack of groupoids’ in much of the literature.</p> <p>In some circles the notion of a stack as a generalized <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> is almost more familiar than the notion of sheaf as a <a class="existingWikiWord" href="/nlab/show/space+and+quantity">generalized space</a>. For instance <a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a> have attracted much attention in the study of <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a> and <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>, while <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological spaces</a> are only beginning to be investigated more in <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>. Groupoid stacks are closely related to internal groupoids (see <a href="https://mathoverflow.net/questions/93948/link-between-internal-groupoids-and-stacks-on-a-topos">this MO post</a>).</p> <p>An <a class="existingWikiWord" href="/nlab/show/algebraic+stack">algebraic stack</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a> etc. is a stack over a site of schemes or differentiable manifolds with additional representability conditions.</p> <h2 id="provisional_discussion">Provisional discussion</h2> <p>The following is “provisional” material on stacks that <a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd Trimble</a> wrote in the course of a discussion with <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs</a>. Somebody should turn this here into a coherent entry on stacks.</p> <hr /> <p>(Todd speaking.) I don’t really speak “stacks”, but in an effort to build a bridge between sheaves and stacks, I’ll write down what I thought I understood, and ask someone such as Urs to come in and check. (Warning: I’m treating this edit box almost as a sandbox, in that what I say below is all a bit provisional until we get some discussion going.)</p> <div class="query"> <p>Hi Todd, thanks for this. I started making some remarks on the relation between descent <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>-categories and pseudofunctors from <a class="existingWikiWord" href="/nlab/show/covers">covers</a> regarded as <a class="existingWikiWord" href="/nlab/show/sieves">sieves</a> (hence as presheaves) at <a class="existingWikiWord" href="/nlab/show/descent+and+codescent">descent and codescent</a> in the section titled <em>Descent in terms of pseudo-functors</em>.</p> </div> <p>At the simplest level, 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 category. As we know, a presheaf 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 just a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">X: C^{op} \to Set</annotation></semantics></math>.</p> <p>Now let’s categorify just once: regard a category <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> as a bicategory whose local hom-categories are discrete. What I’ll call a “pre-stack” is then a homomorphism of bicategories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">X: C^{op} \to Cat</annotation></semantics></math>. Here I’m following Street’s terminology: a homomorphism of bicategories is the “pseudo” version of a weak map of bicategories, as opposed to the “lax” version. So, we have given coherent isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><mi>f</mi><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(f)X(g) \to X(f g)</annotation></semantics></math>, and so on.</p> <p>Now suppose that <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> also comes equipped with a topology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-covering sieve for <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>, so that in particular it’s a subfunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>F</mi><mo>↪</mo><mi>hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i: F \hookrightarrow \hom(-, c)</annotation></semantics></math>. We want to build a (truncated) simplicial object out of this, and to this end I’ll use some yoga which was basically developed in my Cafe post <a href="http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html">on the bar construction</a> [perhaps this may go partway to addressing your most recent query there, Urs].</p> <p>Namely, there is a canonical way of presenting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> as a colimit of representables. Officially, it’s given by a coend formula, but it’s probably more illuminating to think of it in terms of tensor products over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>C</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hom_C(-, -) \otimes_C F(-) \cong F(-)</annotation></semantics></math></div> <p>In the long-winded version, this says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is the coequalizer of a diagram having the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>c</mi><mo>,</mo><mi>d</mi></mrow></munder><msub><mi>hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>c</mi></munder><msub><mi>hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sum_{c, d} \hom_C(-, c) \times \hom_C(c, d) \times F(d) \stackrel{\to}{\to} \sum_c \hom_C(-, c) \times F(c) \to F(-)</annotation></semantics></math></div> <p>where the more visible one of the two parallel arrows involves the contravariant action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hom(c, d) \times F(d) \to F(c)</annotation></semantics></math></div> <p>and the less visible one uses <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> acting on itself:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hom(-, c) \times \hom(c, d) \times F(d) \to hom(-, d) \times F(d)</annotation></semantics></math></div> <p>The point now is that this coequalizer diagram represents the tail end of a simplicial object (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(-)</annotation></semantics></math> appearing in dimension -1), which in the notation of the bar construction one could call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(C, C, F)</annotation></semantics></math>. Let me explain this last bit.</p> <p>The point is that any category <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> can be regarded as a monad in the bicategory of spans. The underlying span is of course</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mover><mo>←</mo><mi>dom</mi></mover><msub><mi>C</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>cod</mi></mover><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">C_0 \stackrel{dom}{\leftarrow} C_1 \stackrel{cod}{\to} C_0</annotation></semantics></math></div> <p>and a presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> 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>, as a discrete op-fibration, has an underlying span</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo>←</mo><mi>F</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">C_0 \leftarrow F \to 1</annotation></semantics></math></div> <p>and is precisely an algebra over the monad <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>. Then, given the data of a monad and an algebra over that monad, one proceeds to build the bar construction as a simplicial object, and I think this is probably the simplicial thingy we want to base the category of descent data on (given a pre-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>).</p> <p>In fact, if memory serves the category of descent data can be efficiently expressed in bicategorical language as follows. The covering sieve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> becomes a homomorphism of bicategories by changing base from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mover><mo>→</mo><mi>F</mi></mover><mi>Set</mi><mover><mo>→</mo><mi>discrete</mi></mover><mi>Cat</mi></mrow><annotation encoding="application/x-tex">C^{op} \stackrel{F}{\to} Set \stackrel{discrete}{\to} Cat</annotation></semantics></math></div> <p>and, abbreviating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>discrete</mi></mrow><annotation encoding="application/x-tex">discrete</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>, it turns out that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Desc</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Nat</mi><mo stretchy="false">(</mo><mi>d</mi><mi>F</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Desc_F(X) \simeq Nat(d F, X)</annotation></semantics></math></div> <p>where the thing on the right side is the category of strong (i.e., pseudo) natural transformations between the indicated bicategory homomorphisms.</p> <p>In that case, the stack condition on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> becomes the statement that the canonical functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mover><mo>≅</mo><mi>Yoneda</mi></mover><mi>Nat</mi><mo stretchy="false">(</mo><mi>d</mi><mi>hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Nat</mi><mo stretchy="false">(</mo><mi>d</mi><mi>F</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(c) \stackrel{Yoneda}{\cong} Nat(d \hom(-, c), X) \to Nat(d F, X)</annotation></semantics></math></div> <p>(where the first equivalence comes from the bicategorical Yoneda lemma, and the second functor is induced from the subfunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>F</mi><mo>→</mo><mi>hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i: F \to \hom(-, c)</annotation></semantics></math>) is an equivalence for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-covering sieves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>. This formulation connects up nicely, that is, is a straight categorification of what was put down in the entry <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>.</p> <h3 id="examples">Examples</h3> <ul> <li>The stack of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BG</mi></mrow><annotation encoding="application/x-tex">BG</annotation></semantics></math> is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Mfd</mi> <mi>op</mi></msup><mo>→</mo><mi>Gpd</mi></mrow><annotation encoding="application/x-tex">Mfd^{op} \to Gpd</annotation></semantics></math>, sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>Mfd</mi></mrow><annotation encoding="application/x-tex">U\in Mfd</annotation></semantics></math> to the groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant morphisms; sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mover><mo>→</mo><mi>f</mi></mover><mi>V</mi></mrow><annotation encoding="application/x-tex">U\xrightarrow{f} V</annotation></semantics></math> to the functor induced by pullbacks of principal bundles via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BG</mi></mrow><annotation encoding="application/x-tex">BG</annotation></semantics></math> comes from the prestack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>BG</mi> <mi>p</mi></msup><mo>:</mo><msup><mi>Mfd</mi> <mi>op</mi></msup><mo>→</mo><mi>Gpd</mi></mrow><annotation encoding="application/x-tex">BG^p: Mfd^{op} \to Gpd</annotation></semantics></math> sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>Mfd</mi></mrow><annotation encoding="application/x-tex">U\in Mfd</annotation></semantics></math> to the groupoid of trivial principal bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant morphisms (then it is just a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mover><mo>→</mo><mi>g</mi></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">U\xrightarrow{g} G</annotation></semantics></math>; sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mover><mo>→</mo><mi>f</mi></mover><mi>V</mi></mrow><annotation encoding="application/x-tex">U\xrightarrow{f}V</annotation></semantics></math> to the functor induced by pullbacks of principal bundles via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. Then sheafification or stackification will give us <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BG</mi></mrow><annotation encoding="application/x-tex">BG</annotation></semantics></math> back.</li> </ul> <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/presheaf+of+groupoids">presheaf of groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+stack">group stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a> / <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a> / <strong>stack</strong> / <a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+morphism+of+stacks">representable morphism of stacks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a>, <a class="existingWikiWord" href="/nlab/show/quotient+stack">quotient stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+stack">geometric stack</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>, <a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stack">Deligne-Mumford stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigidification+of+a+stack">rigidification of a stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Picard+stack">Picard stack</a></p> </li> </ul> </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/axiom+of+stack+completion">axiom of stack completion</a></p> </li> </ul> <p>Special kinds of stacks include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+stacks">geometric stacks</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gerbes">gerbes</a>.</p> </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> <h2 id="references">References</h2> <p>The concept originates (under the French term <em>champ</em> and for the purpose of defining <a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian cohomology</a>) in:</p> <ul> <li id="Giraud66"> <p><a class="existingWikiWord" href="/nlab/show/Jean+Giraud">Jean Giraud</a>, <em>Cohomologie non abélienne</em>, Columbia University (1966) [<a href="https://books.google.de/books/about/Cohomologie_non_ab%C3%A9lienne.html?id=EQ1CAAAAIAAJ&redir_esc=y">GoogleBooks</a>]</p> <p>published as: Grundlehren <strong>179</strong>, Springer (1971) [<a href="https://www.springer.com/gp/book/9783540053071">doi:10.1007/978-3-662-62103-5</a>]</p> </li> </ul> <p>and under the English term <em>stack</em> (for specialization to <a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stacks">Deligne-Mumford stacks</a>/<a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>) in:</p> <ul> <li id="DeligneMumford69"><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <a class="existingWikiWord" href="/nlab/show/David+Mumford">David Mumford</a>, <em>The irreducibility of the space of curves of given genus</em>, Publications Mathématiques de l’IHÉS (Paris) <strong>36</strong> (1969) 75-109 [<a href="https://doi.org/10.1007/BF02684599">doi:10.1007/BF02684599</a>, <a href="http://www.numdam.org/item?id=PMIHES_1969__36__75_0">numdam:PMIHES_1969__36__75_0</a>]</li> </ul> <p>and in the context of <a class="existingWikiWord" href="/nlab/show/classifying+toposes">classifying toposes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean+Giraud">Jean Giraud</a>, <em>Classifying topos</em>, in: <a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a> (ed.) <em>Toposes, Algebraic Geometry and Logic</em>, Lecture Notes in Mathematics, vol 274. Springer (1972) (<a href="https://doi.org/10.1007/BFb0073964">doi:10.1007/BFb0073964</a>)</li> </ul> <p>Further early discussion:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Pare">Robert Pare</a>, <em>Stacks and equivalence of indexed categories</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no.4 (1979) [<a href="http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0">numdam</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <em>Stack completions and Morita equivalence for categories in a topos</em>, Cahiers de topologie et géométrie différentielle xx-4 (1979)</p> <p>401-436 [<a href="http://www.ams.org/mathscinet-getitem?mr=558106">MR558106</a>, <a href="http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0">numdam</a>]</p> </li> </ul> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a></em> and for considerably more literature see at <em><a class="existingWikiWord" href="/nlab/show/algebraic+stack">algebraic stack</a></em>.</p> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Introduction to the language of stacks and gerbes</em> [<a href="http://arxiv.org/abs/math/0212266">arXiv:math/0212266</a>]</p> <blockquote> <p>(including discussion of <a class="existingWikiWord" href="/nlab/show/gerbes">gerbes</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jochen+Heinloth">Jochen Heinloth</a>, <em>Notes on differentiable stacks</em> (2004) [<a href="https://math.nyu.edu/~tschinke/WS04/pdf/heinloth.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Heinloth-DifferentiableStacks.pdf" title="pdf">pdf</a>]</p> <blockquote> <p>(focused on <a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a>)</p> </blockquote> </li> </ul> <p>Discussion of stacks in their incarnation (under the <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a>) as <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibrations">Grothendieck fibrations</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Angelo+Vistoli">Angelo Vistoli</a>, <em>Grothendieck topologies, fibered categories and descent theory</em> [<a href="http://arxiv.org/abs/math/0412512">math.AG/0412512</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=2223406">MR2223406</a>] in: Fantechi et al. (eds.), <em>Fundamental algebraic geometry. Grothendieck’s <a class="existingWikiWord" href="/nlab/show/FGA+explained">FGA explained</a></em>, Mathematical Surveys and Monographs <strong>123</strong>, Amer. Math. Soc. (2005) 1-104 [<a href="https://bookstore.ams.org/surv-123-s">ISBN:978-0-8218-4245-4</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=2007f:14001">MR2007f:14001</a>]</li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <a class="existingWikiWord" href="/nlab/show/presheaves+of+groupoids">presheaves of groupoids</a>, presenting stacks, by a <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">localization</a> of a <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> (modeling <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> before localization):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+F.+Jardine">John F. Jardine</a>, <em>Stacks and the homotopy theory of simplicial sheaves</em>, Homology, Homotopy and Applications, <strong>3</strong> 2 (2001) 361-384 [<a href="https://projecteuclid.org/euclid.hha/1139840259">euclid:hha/1139840259</a>]</p> </li> <li id="Hollander01"> <p><a class="existingWikiWord" href="/nlab/show/Sharon+Hollander">Sharon Hollander</a>, <em>A homotopy theory for stacks</em>, Israel Journal of Mathematics <strong>163</strong> 1 (2008) 93-124 [<a href="http://arxiv.org/abs/math.AT/0110247">arXiv:math.AT/0110247</a>, <a href="https://doi.org/10.1007/s11856-008-0006-5">doi:10.1007/s11856-008-0006-5</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 14, 2023 at 04:48:11. 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