CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> 2-sheaf in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> 2-sheaf </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/3651/#Item_14" title="Discuss this page in its dedicated thread on the nForum" 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="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>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-Topos theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-presheaf">(∞,2)-presheaf</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-site">(∞,2)-site</a>,</li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-sheaf">(∞,2)-sheaf</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a>, <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-category">tangent (∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/quasicoherent+%E2%88%9E-stack">quasicoherent ∞-stack</a></li> </ul> <h2 id="truncations">Truncations</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pretopos">2-pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-presheaf">2-presheaf</a>, <a class="existingWikiWord" href="/nlab/show/2-site">2-site</a>, <a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+2-topos">n-localic 2-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-Giraud+theorem">2-Giraud theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-site">(2,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/1-topos">1-topos</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a>, <a class="existingWikiWord" href="/nlab/show/site">site</a>, <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></li> </ul> </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> </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='#properties'>Properties</a></li> <ul> <li><a href='#characterization_of_over_sites'>Characterization of over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-sites</a></li> <ul> <li><a href='#over_a_1site'>Over a 1-site</a></li> <li><a href='#over_a_site__as_internal_categories'>Over 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>-site – As internal categories</a></li> </ul> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#codomain_fibrations__sheaves_of_modules'>Codomain fibrations / sheaves of modules</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#ReferencesInTermsOfInternalCategories'>In terms of categories internal to sheaf toposes</a></li> <li><a href='#InTermsOfFiberedCategories'>In terms of fibered categories</a></li> <li><a href='#2sites'>2-Sites</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>2-sheaf</em> is the generalization of the notion of <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> to the <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> of <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a>/<a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a>. A 2-category of 2-sheaves forms a <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a>.</p> <div class="num_remark"> <h6 id="remark_on_terminology">Remark on terminology</h6> <p>A <em>2-sheaf</em> is a higher sheaf of <a class="existingWikiWord" href="/nlab/show/categories">categories</a>. More restrictive than this is a higher sheaf with values in <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>, which would be a <em><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a></em>. Both these notions are often referred to as <strong><a class="existingWikiWord" href="/nlab/show/stack">stack</a></strong>, or sometimes “stack of groupoids” and “stack of categories” for definiteness. But moreover, traditionally a <a class="existingWikiWord" href="/nlab/show/stack">stack</a> (in either flavor) is considered only over a <a class="existingWikiWord" href="/nlab/show/1-site">1-site</a>, whereas it makes sense to consider <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaves">(2,1)-sheaves</a> more generally over <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sites">(2,1)-sites</a> and 2-sheaves over <a class="existingWikiWord" href="/nlab/show/2-sites">2-sites</a>.</p> <p>Therefore, saying “2-sheaf” serves to indicate the full generality of the notion of higher sheaves in <a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a>, as opposed to various special cases of this general notion which have traditionally been considered.</p> </div> <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/2-site">2-site</a> having finite <a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a>s (for convenience). For a covering family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">(f_i:U_i\to U)_i</annotation></semantics></math> we have the comma objects</p> <div style="text-align:center"><svg xmlns="http://www.w3.org/2000/svg" width="10em" height="10em" viewBox="-30 -20 180 150"> <desc>Comma Square</desc> <defs> <marker id="svg295arrowhead" viewBox="0 0 10 10" refX="0" refY="5" markerUnits="strokeWidth" markerWidth="8" markerHeight="5" orient="auto"> <path d="M 0 0 L 10 5 L 0 10 z"></path> </marker> </defs> <g font-size="16"> <foreignObject x="-10" y="0" width="45" height="25"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <msub><mi>f</mi> <mi>i</mi></msub> <mo stretchy="false">/</mo> <msub><mi>f</mi> <mi>j</mi></msub> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(f_i/f_j)</annotation> </semantics> </math></foreignObject> <foreignObject x="100" y="0" width="20" height="20"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>U</mi> <mi>i</mi></msub> </mrow> <annotation encoding="application/x-tex">U_i</annotation> </semantics> </math></foreignObject> <foreignObject x="0" y="100" width="20" height="20"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>U</mi> <mi>j</mi></msub> </mrow> <annotation encoding="application/x-tex">U_j</annotation> </semantics> </math></foreignObject> <foreignObject x="100" y="100" width="20" height="20"><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></foreignObject> <foreignObject x="110" y="50" width="20" height="25"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>f</mi> <mi>i</mi></msub> </mrow> <annotation encoding="application/x-tex">f_i</annotation> </semantics> </math></foreignObject> <foreignObject x="50" y="110" width="20" height="25"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>f</mi> <mi>j</mi></msub> </mrow> <annotation encoding="application/x-tex">f_j</annotation> </semantics> </math></foreignObject> <foreignObject x="-20" y="50" width="20" height="30"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub> </mrow> <annotation encoding="application/x-tex">q_{i j}</annotation> </semantics> </math></foreignObject> <foreignObject x="50" y="-20" width="20" height="30"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub> </mrow> <annotation encoding="application/x-tex">p_{i j}</annotation> </semantics> </math></foreignObject> <foreignObject x="60" y="60" width="20" height="25"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub> </mrow> <annotation encoding="application/x-tex">\mu_{i j}</annotation> </semantics> </math></foreignObject> </g> <g fill="none" stroke="#000" stroke-width="1.5" marker-end="url(#svg295arrowhead)"> <line x1="40" y1="10" x2="90" y2="10"></line> <line x1="30" y1="110" x2="90" y2="110"></line> <line y1="30" x1="10" y2="90" x2="10"></line> <line y1="30" x1="110" y2="90" x2="110"></line> <line x1="65" y1="55" x2="55" y2="65"></line> </g> </svg></div> <p>We also have the <a class="existingWikiWord" href="/nlab/show/double+comma+object">double comma objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_i/f_j/f_k) = (f_i/f_j)\times_{U_j} (f_j/f_k)</annotation></semantics></math> with projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>:</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_{i j k}:(f_i/f_j/f_k)\to (f_i/f_j)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>:</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s_{i j k}:(f_i/f_j/f_k)\to (f_j/f_k)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>:</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t_{i j k}:(f_i/f_j/f_k)\to (f_i/f_k)</annotation></semantics></math>.</p> <p>Now, 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>Cat</mi></mrow><annotation encoding="application/x-tex">X:C^{op} \to Cat</annotation></semantics></math> is called a <strong>2-presheaf</strong>. It is <strong>1-separated</strong> if</p> <ul> <li>For any covering family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">(f_i:U_i\to U)_i</annotation></semantics></math> and any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x,y\in X(U)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">a,b: x\to y</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(f_i)(a) = X(f_i)(b)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a=b</annotation></semantics></math>.</li> </ul> <p>It is <strong>2-separated</strong> if it is 1-separated and</p> <ul> <li>For any covering family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">(f_i:U_i\to U)_i</annotation></semantics></math> and any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x,y\in X(U)</annotation></semantics></math>, given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mi>i</mi></msub><mo>:</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b_i:X(f_i)(x) \to X(f_i)(y)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu_{i j}(y) \circ X(p_{i j})(b_i) = X(q_{i j})(b_i) \circ \mu_{i j}(x)</annotation></semantics></math>, there exists a (necessarily unique) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">b:x\to y</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mi>i</mi></msub><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b_i = X(f_i)(b)</annotation></semantics></math>.</li> </ul> <p>It is a <strong>2-sheaf</strong> if it is 2-separated and</p> <ul> <li>For any covering family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><msub><mo stretchy="false">)</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">(f_i:U_i\to U)_i</annotation></semantics></math> and any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_i\in X(U_i)</annotation></semantics></math> together with morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>:</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\zeta_{i j}:X(p_{i j})(x_i) \to X(q_{i j})(x_j)</annotation></semantics></math> such that the following diagram commutes:<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>r</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>X</mi><mo stretchy="false">(</mo><msub><mi>r</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>r</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mo>≅</mo></mover></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo></mo><mo>≅</mo></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><mi>X</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>ζ</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>→</mo><mrow><mi>X</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>→</mo><mo>≅</mo></munder></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>q</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{X(r_{i j k})X(p_{i j})(x_i) & \overset{X(r_{i j k})(\zeta_{i j})}{\to} & X(r_{i j k})X(q_{i j})(x_j) & \overset{\cong}{\to} & X(s_{i j k})X(p_{j k})(x_j)\\ ^\cong \downarrow && && \downarrow ^{X(s_{i j k})(\zeta_{j k})}\\ X(t_{i j k}) X(p_{i k})(x_i) & \underset{X(t_{i j k})(\zeta_{i k})}{\to} & X(t_{i j k}) X(q_{i k})(x_k) & \underset{\cong}{\to} & X(s_{i j k}) X(q_{j k})(x_k)} </annotation></semantics></math></div> <p>there exists an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\in X(U)</annotation></semantics></math> and isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≅</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X(f_i)(x)\cong x_i</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i,j</annotation></semantics></math> the following square commutes:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mo>≅</mo></mover></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo></mo><mrow><mi>X</mi><mo stretchy="false">(</mo><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>→</mo><mo>≅</mo></munder></mtd> <mtd><mi>X</mi><mo stretchy="false">(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ X(p_{i j})X(f_i)(X) & \overset{\cong}{\to} & X(p_{i j})(x_i)\\ ^{X(\mu_{i j})}\downarrow && \downarrow^{\zeta_{i j}}\\ X(q_{i j})X(f_j)(x) & \underset{\cong}{\to} & X(q_{i j})(x_j).} </annotation></semantics></math></div></li> </ul> <p>A 2-sheaf, especially on a 1-site, is frequently called a <strong><a class="existingWikiWord" href="/nlab/show/stack">stack</a></strong>. However, this has the unfortunate consequence that a 3-sheaf is then called a 2-stack, and so on with the numbering all offset by one. Also, it can be helpful to use a new term because of the notable differences between 2-sheaves on 2-sites and 2-sheaves on 1-sites. The main novelty is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{i j}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\zeta_{i j}</annotation></semantics></math> <em>need not be invertible</em>.</p> <p>Note, though, they must be invertible as soon as <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 (2,1)-site: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{i j}</annotation></semantics></math> by definition and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\zeta_{i j}</annotation></semantics></math> since an inverse is provided by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ι</mi> <mrow><mi>i</mi><mi>j</mi></mrow> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota_{i j}^*(\zeta_{i j})</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota_{i j}\mapsto (f_i/f_j) \to (f_j/f_i)</annotation></semantics></math> is the symmetry equivalence.</p> <p>If <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> lacks finite limits, then in the definitions of “2-separated” and “2-sheaf” instead of the comma objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_i/f_j)</annotation></semantics></math>, we need to use arbitrary objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> equipped with maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>V</mi><mo>→</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p:V\to U_i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>V</mi><mo>→</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">q:V\to U_j</annotation></semantics></math>, and a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mi>p</mi><mo>→</mo><msub><mi>f</mi> <mi>j</mi></msub><mi>q</mi></mrow><annotation encoding="application/x-tex">f_i p \to f_j q</annotation></semantics></math>. We leave the precise definition to the reader.</p> <p>A 2-site is said to be <strong>subcanonical</strong> if for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">U\in C</annotation></semantics></math>, the representable functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(-,U)</annotation></semantics></math> is a 2-sheaf. When <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> has finite limits, it is easy to verify that this is true precisely when every covering family is a (necessarily pullback-stable) quotient of its kernel <span class="newWikiWord">2-polycongruence<a href="/nlab/new/2-polycongruence">?</a></span>. In particular, the regular coverage on a regular 2-category is subcanonical, as is the coherent coverage on a coherent 2-category.</p> <p>The 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2Sh(C)</annotation></semantics></math> of 2-sheaves on a small 2-site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is, by definition, a <a class="existingWikiWord" href="/nlab/show/Grothendieck+2-topos">Grothendieck 2-topos</a>.</p> <h2 id="properties">Properties</h2> <h3 id="characterization_of_over_sites">Characterization of over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-sites</h3> <p>If the underlying <a class="existingWikiWord" href="/nlab/show/2-site">2-site</a> happens to be an <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-site">(n,r)-site</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> lower than 2, there may be other equivalent ways to think of 2-sheaves.</p> <p>A <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a> with a <a class="existingWikiWord" href="/nlab/show/2-site">2-site</a> of definition that happens to be just a 1-site or <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-site">(2,1)-site</a> is <em>1-localic</em> or <em>(2,1)-localic</em>.</p> <h4 id="over_a_1site">Over a 1-site</h4> <p>Over a 1-site, the <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> says that <a class="existingWikiWord" href="/nlab/show/2-functors">2-functors</a> on the site are equivalent to <a class="existingWikiWord" href="/nlab/show/fibered+categories">fibered categories</a> over the site. Hence in this case the theory of 2-sheaves can be entirely formulated in terms of fibered categories. See <em><a href="#InTermsOfFiberedCategories">References – In terms of fibered categories</a></em>.</p> <p>Also, over a 1-site a 2-sheaf is essentially a <em><a class="existingWikiWord" href="/nlab/show/indexed+category">indexed category</a></em>. Therefore stacks over 1-sites can also be discussed in this language, see notably the work (<a href="#BungePare">Bunge-Pare</a>).</p> <p>In particular, if the 1-site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, then every topos <em>over</em> <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 its <a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a> (a <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>-topos) induces an <a class="existingWikiWord" href="/nlab/show/indexed+category">indexed category</a>.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>If <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 a topos and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a <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>-topos, then (the <a class="existingWikiWord" href="/nlab/show/indexed+category">indexed category</a> corresponding to) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a 2-sheaf 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> with respect to the <a class="existingWikiWord" href="/nlab/show/canonical+topology">canonical topology</a>.</p> </div> <p>This appears as (<a href="#BungePare">Bunge-Pare, corollary 2.6</a>).</p> <p>Moreover, over a <a class="existingWikiWord" href="/nlab/show/1-site">1-site</a> the <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a> of 2-sheaves ought to be equivalent to the (suitably defined) <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of <a class="existingWikiWord" href="/nlab/show/internal+categories">internal categories</a> in the underlying <a class="existingWikiWord" href="/nlab/show/1-topos">1-topos</a>. See <em><a href="#ReferencesInTermsOfInternalCategories">References – In terms of internal categories</a></em>.</p> <h4 id="over_a_site__as_internal_categories">Over 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>-site – As internal categories</h4> <p>Over a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-site">(2,1)-site</a> the <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a> of 2-sheaves ought to be equivalent to the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of <a class="existingWikiWord" href="/nlab/show/internal+%28infinity%2C1%29-categories">internal (infinity,1)-categories</a> in the corresponding <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>.</p> <p>This is discussed at <em><a href="2-topos#InTermsOfInternalCategories">2-Topos – In terms of internal categories</a></em>.</p> <h2 id="examples">Examples</h2> <h3 id="codomain_fibrations__sheaves_of_modules">Codomain fibrations / sheaves of modules</h3> <p>A classical class of examples for 2-sheaves are <a class="existingWikiWord" href="/nlab/show/codomain+fibrations">codomain fibrations</a> over suitable sites, or rather their <a class="existingWikiWord" href="/nlab/show/tangent+categories">tangent categories</a>. As discussed there, this includes the case of sheaves of categories of <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over sites of <a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebras</a>.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>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> an <a class="existingWikiWord" href="/nlab/show/exact+category">exact category</a> with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>, the <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cod</mi><mo>:</mo><msup><mi>C</mi> <mi>I</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Cod : C^I \to C</annotation></semantics></math> or equivalently (under the <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a>), the self-<a class="existingWikiWord" href="/nlab/show/indexed+category">indexing</a> 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> is a 2-sheaf with respect to the <a class="existingWikiWord" href="/nlab/show/canonical+topology">canonical topology</a>.</p> </div> <p>This is for instance (<a href="#BungePare">Bunge-Pare, corollary 2.4</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> / <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> / <a class="existingWikiWord" href="/nlab/show/cosheaf">cosheaf</a></p> </li> <li> <p><strong>2-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%2C2%29-sheaf">(∞,2)-sheaf</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> </li> </ul> <h2 id="references">References</h2> <p>Historically, the original definition of <em><a class="existingWikiWord" href="/nlab/show/stack">stack</a></em> included the case of category-valued functors, hence of 2-sheaves, in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean+Giraud">Jean Giraud</a>, <em>Cohomologie non abélienne</em> 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> <li> <p><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 <strong>274</strong>, Springer (1972) [<a href="https://doi.org/10.1007/BFb0073964">doi:10.1007/BFb0073964</a>]</p> </li> </ul> <h3 id="ReferencesInTermsOfInternalCategories">In terms of categories internal to sheaf toposes</h3> <p>Category-valued stacks as <a class="existingWikiWord" href="/nlab/show/internal+categories">internal categories</a> in the underlying <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>:</p> <ul> <li id="BungePare"> <p><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Par%C3%A9">Robert Paré</a>, <em>Stacks and equivalence of indexed categories</em>, <a class="existingWikiWord" href="/nlab/show/Cahiers">Cahiers de Top. et Géom. Diff. Catég</a> <strong>20</strong> 4 (1979) 373-399 [<a href="http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0">numdam:CTGDC_1979__20_4_373_0</a>]</p> </li> <li id="Bunge"> <p><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <em>Stack completions and Morita equivalence for categories in a topos</em>, <a class="existingWikiWord" href="/nlab/show/Cahiers">Cahiers de Top. et Géom. Diff. Catég</a> <strong>20</strong> 4, (1979) 401-436 [<a href="http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0">numdam</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=558106">MR558106</a>]</p> </li> <li id="JoyalTierney"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, section 3 of: <em>Strong stacks and classifying spaces</em>, in: <em>Category Theory</em> (<a class="existingWikiWord" href="/nlab/show/Como">Como</a>, 1990), Lecture Notes in Mathematics <strong>1488</strong>, Springer (1991) 213-236 [<a href="https://doi.org/10.1007/BFb0084222">doi:10.1007/BFb0084222</a>]</p> <blockquote> <p>(establishing the <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+Cat">canonical model structure on Cat</a> in the internal generality)</p> </blockquote> </li> </ul> <h3 id="InTermsOfFiberedCategories">In terms of fibered categories</h3> <p>A discussion of stacks over <a class="existingWikiWord" href="/nlab/show/1-sites">1-sites</a> in terms of their <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">associated</a> <a class="existingWikiWord" href="/nlab/show/fibered+categories">fibered categories</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Angelo+Vistoli">Angelo Vistoli</a>, <em>Notes on Grothendieck topologies, fibered categories and descent theory</em> (<a href="http://homepage.sns.it/vistoli/descent.pdf">pdf</a>)</li> </ul> <h3 id="2sites">2-Sites</h3> <p>The above text involves content transferred from</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, <em><a class="existingWikiWord" href="/michaelshulman/show/2-site">2-site</a></em></li> </ul> <p>2-sites were earlier considered in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em><a class="existingWikiWord" href="/nlab/show/StreetCBS">StreetCBS</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 8, 2023 at 07:04:43. See the <a href="/nlab/history/2-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/2-sheaf" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3651/#Item_14">Discuss</a><span class="backintime"><a href="/nlab/revision/2-sheaf/20" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/2-sheaf" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/2-sheaf" accesskey="S" class="navlink" id="history" rel="nofollow">History (20 revisions)</a> <a href="/nlab/show/2-sheaf/cite" style="color: black">Cite</a> <a href="/nlab/print/2-sheaf" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/2-sheaf" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>