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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">Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="toc_internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <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+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <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/%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></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#GeneralDefinitionInComponents'>General definition in components</a></li> <li><a href='#GeneralDefinitionAbstractly'>General definition abstractly</a></li> <li><a href='#characterizations_over_special_sites'>Characterizations over special sites</a></li> <ul> <li><a href='#CharacterizationsOverSitesOfOpens'>Characterizations over sites of opens</a></li> <li><a href='#AsEtaleSpace'>As étale spaces</a></li> <li><a href='#CharacterizationOverCanonicalTopologies'>Characterization over canonical topologies</a></li> </ul> </ul> <li><a href='#sheaves_and_localization'>Sheaves and localization</a></li> <ul> <li><a href='#corollaries'>Corollaries</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on a <a class="existingWikiWord" href="/nlab/show/site">site</a> is a <em>sheaf</em> if its value on any object of the site is given by its compatible values on any <a class="existingWikiWord" href="/nlab/show/covering">covering</a> of that object.</p> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks">motivation for sheaves, cohomology and higher stacks</a>.</li> </ul> <p>A competing, though related, definition which one sometimes sees uses “sheaf” as a synonym for <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+space">étale space</a>; see <a href="#AsEtaleSpace">discussion below</a>.</p> <h2 id="definition">Definition</h2> <p>There are several equivalent ways to characterize sheaves. We start with the general but explicit componentwise definition and then discuss more <a class="existingWikiWord" href="/nlab/show/category+theory">general abstract</a> equivalent reformulations. Finally we give special discussion applicable in various common special cases.</p> <h3 id="GeneralDefinitionInComponents">General definition in components</h3> <p>The following is an explicit component-wise definition of sheaves that is fully general (for instance not assuming that the <a class="existingWikiWord" href="/nlab/show/site">site</a> has <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>).</p> <div class="num_defn" id="GeneralComponentwiseDefinition"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,J)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a> in the form of a <a class="existingWikiWord" href="/nlab/show/small+category">small category</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> equipped with a <a class="existingWikiWord" href="/nlab/show/coverage">coverage</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>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in PSh(C)</annotation></semantics></math> is a <strong>sheaf</strong> with respect to <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> if</p> <ul> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/covering">covering</a> family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>p</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> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{p_i : U_i \to U\}_{i \in I}</annotation></semantics></math> in <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></p> </li> <li> <p>and for every <em><a class="existingWikiWord" href="/nlab/show/matching+family">compatible family</a> of elements</em>, given by tuples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><mo>∈</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(s_i \in A(U_i))_{i \in 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>j</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">j,k \in I</annotation></semantics></math> and all <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub><mover><mo>←</mo><mi>f</mi></mover><mi>K</mi><mover><mo>→</mo><mi>g</mi></mover><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_j \stackrel{f}{\leftarrow} K \stackrel{g}{\to} U_k</annotation></semantics></math> in <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>j</mi></msub><mo>∘</mo><mi>f</mi><mo>=</mo><msub><mi>p</mi> <mi>k</mi></msub><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">p_j \circ f = p_k \circ g</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>A</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(f)(s_j) = A(g)(s_k) \in A(K)</annotation></semantics></math></p> </li> </ul> <p>then</p> <ul> <li>there is a <em>unique</em> element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s \in A(U)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>s</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">A(p_i)(s) = s_i</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \in I</annotation></semantics></math>.</li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>If in the above definition there is <em>at most</em> one such <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>, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a> with respect to <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>.</p> </div> <p>In this form the definition appears for instance in (<a href="#Johnstone">Johnstone, def. C2.1.2</a>).</p> <h3 id="GeneralDefinitionAbstractly">General definition abstractly</h3> <p>We now reformulate the <a href="#GeneralComponentwiseDefinition">above component-wise definition</a> in <a class="existingWikiWord" href="/nlab/show/category+theory">general abstract</a> terms.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><mi>C</mi><mo>↪</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> j : C \hookrightarrow PSh(C) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/covering">covering</a> 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><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{f_i : U_i \to U\}</annotation></semantics></math> in <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>, its <strong><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></strong> is the presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(\{U_i\})</annotation></semantics></math> defined as the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a></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>j</mi><mo>,</mo><mi>k</mi></mrow></munder><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mover><mo>→</mo><mrow></mrow></mover></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></munder><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \coprod_{j,k} j(U_j) \times_{j(U)} j(U_k) \stackrel{\overset{}{\to}}{\to} \coprod_i j(U_i) \to S(\{U_i\}) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(C)</annotation></semantics></math>.</p> </div> <p>Here the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> on the left is over the <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a></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>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mi>j</mi></msub></mrow></mover></mtd> <mtd><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mi>k</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>j</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>j</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ j(U_j) \times_{j(U)} j(U_k) &\stackrel{p_j}{\to}& j(U_j) \\ {}^{\mathllap{p_k}}\downarrow && \downarrow^{\mathrlap{j(f_j)}} \\ j(U_k) &\stackrel{j(f_k)}{\to}& j(U) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(C)</annotation></semantics></math>, and the two morphisms between the coproducts are those induced componentwise by the two projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>j</mi></msub><mo>,</mo><msub><mi>p</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">p_j, p_k</annotation></semantics></math> in this pullback diagram.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Using that <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> in a <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> are computed objectwise, we find that the <a class="existingWikiWord" href="/nlab/show/sieve">sieve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(\{U_i\})</annotation></semantics></math> defined this way is the presheaf that sends any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">K \in C</annotation></semantics></math> to the set of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">K \to U</annotation></semantics></math> in <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> that factor through one of the <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>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>For every <a class="existingWikiWord" href="/nlab/show/covering">covering</a> family there is a canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow></msub><mo>:</mo><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i_{\{U_i\}} : S(\{U_i\}) \to j(U) </annotation></semantics></math></div> <p>that is induced by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> from the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j(f_i) : j(U_i) \to j(U)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j(U_j) \times_{j(U)} j(U_k) \to j(U)</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>A <strong>sheaf</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,J)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in PSh(C)</annotation></semantics></math> that is a <a class="existingWikiWord" href="/nlab/show/local+object">local object</a> with respect to all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow></msub></mrow><annotation encoding="application/x-tex">i_{\{U_i\}}</annotation></semantics></math>: an object such that for all <a class="existingWikiWord" href="/nlab/show/covering">covering</a> families <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><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{f_i : U_i \to U\}</annotation></semantics></math> in <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> we have that the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_C(-,A)</annotation></semantics></math> sends the canonical morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow></msub><mo>:</mo><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i_{\{U_i\}} : S(\{U_i\}) \to j(U)</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mi>i</mi> <mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PSh_C(i_{\{U_i\}}, A) : PSh_C(j(U), A) \stackrel{\simeq}{\to} PSh_C(S(\{U_i\}), A) \,. </annotation></semantics></math></div></div> <p>Equivalently, using the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> and the fact that the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_C(-,A)</annotation></semantics></math> sends <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> to <a class="existingWikiWord" href="/nlab/show/limits">limits</a>, this says that the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow></munder><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A(U) \to \prod_i A(U_i) \stackrel{\to}{\to} \prod_{j,k} PSh_C(j(U_j) \times_{j(U)} j(U_k), A) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> diagram for each covering family.</p> <p>This is also called the <strong><a class="existingWikiWord" href="/nlab/show/descent">descent</a> condition</strong> for descent along the covering family.</p> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>For many examples of sites that appear in practice – but by far not for all – it happens that the pullback presheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j(U_j) \times_{j(U)} j(U_k)</annotation></semantics></math> are themselves again representable, hence that the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub><msub><mo>×</mo> <mi>U</mi></msub><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_j \times_U U_k</annotation></semantics></math> exists already in <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>, even before passing to the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>.</p> <p>In this special case we may apply the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> once more to deduce</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><msub><mo>×</mo> <mi>U</mi></msub><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PSh_C(j(U_j) \times_{j(U)} j(U_k), A) \simeq A(U_j \times_U U_k) \,. </annotation></semantics></math></div> <p>Then the sheaf condition is that all diagrams</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow></munder><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><msub><mo>×</mo> <mi>U</mi></msub><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A(U) \to \prod_i A(U_i) \stackrel{\to}{\to} \prod_{j,k} A(U_j \times_U U_k) </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>s.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_C(S(\{U_i\}), A)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> is equivalent to the condition that the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(U)</annotation></semantics></math> is isomorphic to the set of <a class="existingWikiWord" href="/nlab/show/matching+families">matching families</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><mo>∈</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s_i \in A(U_i))</annotation></semantics></math> as it appears in the <a href="#GeneralComponentwiseDefinition">above component-wise definition</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We may express the set of <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_C(j(U_j) \times_{j(U)} j(U_k), A)</annotation></semantics></math> (as described there) by the <a class="existingWikiWord" href="/nlab/show/end">end</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>j</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>j</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>K</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>Set</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PSh_C(j(U_j) \times_{j(U)} j(U_k), A) \simeq \int_{K \in C} Set( C(K,U_j) \times_{C(K,U)} C(K,U_k) , A(K)) \,. </annotation></semantics></math></div> <p>Using this in the expression of the <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a></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> <mi>i</mi></munder><mi>A</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><msub><mo>∫</mo> <mrow><mi>K</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>Set</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>→</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow></munder><msub><mo>∫</mo> <mrow><mi>K</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>Set</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><msub><mi>U</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \prod_i A(U_i) \simeq \prod_i \int_{K \in C} Set( C(K,U_i), A(K)) \stackrel{\to}{\to} \prod_{j,k} \int_{K \in C} Set( C(K,U_j) \times_{C(K,U)} C(K,U_k) , A(K)) </annotation></semantics></math></div> <p>as a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of the product set on the left manifestly yields the componenwise definition above.</p> </div> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>A <strong>morphism of sheaves</strong> is just a morphism of the underlying presheaves. So the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_J(C)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> on the sheaves:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sh_J(C) \hookrightarrow PSh(C) </annotation></semantics></math></div></div> <h3 id="characterizations_over_special_sites">Characterizations over special sites</h3> <p>We discuss equivalent characterizations of sheaves that are applicable if the underlying <a class="existingWikiWord" href="/nlab/show/site">site</a> enjoys certain special properties.</p> <h4 id="CharacterizationsOverSitesOfOpens">Characterizations over sites of opens</h4> <p>An important special case of sheaves is those over a <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a> such as a <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Op(X)</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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>. We consider some equivalent ways of characterizing sheaves among presheaves in such a situation.</p> <p>(The following was mentioned in Peter LeFanu Lumsdaine’s comment <a href="http://mathoverflow.net/questions/23268/geometric-intuition-for-limits/23276#23276">here</a>).</p> <div class="num_prop" id="AsContinuousFunctorOverOpenSubsets"> <h6 id="proposition_2">Proposition</h6> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi><mo>=</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Op = Op(X)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> of some <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> regarded as a <a class="existingWikiWord" href="/nlab/show/site">site</a> with the canonical <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \hookrightarrow U\}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/covering">covering</a> if the <a class="existingWikiWord" href="/nlab/show/union">union</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∪</mo> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo>≃</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">\cup_i U_i \simeq U</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi></mrow><annotation encoding="application/x-tex">Op</annotation></semantics></math>.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <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">\mathcal{F}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi></mrow><annotation encoding="application/x-tex">Op</annotation></semantics></math> is a <strong>sheaf</strong> precisely if for every <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a> <a class="existingWikiWord" href="/nlab/show/subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo>↪</mo><mi>Op</mi></mrow><annotation encoding="application/x-tex">\mathcal{U} \hookrightarrow Op</annotation></semantics></math>, <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">\mathcal{F}</annotation></semantics></math> takes the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi></mrow><annotation encoding="application/x-tex">Op</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo>↪</mo><mi>Op</mi></mrow><annotation encoding="application/x-tex">\mathcal{U} \hookrightarrow Op</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/limit">limit</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo stretchy="false">(</mo><munder><mi>lim</mi><mo>→</mo></munder><mi>𝒰</mi><mo stretchy="false">)</mo><mo>≃</mo><munder><mi>lim</mi><mo>←</mo></munder><mi>ℱ</mi><mo stretchy="false">(</mo><mi>𝒰</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{F}(\underset{\to}{lim} \mathcal{U}) \simeq \underset{\leftarrow}{lim} \mathcal{F}(\mathcal{U}) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>A complete full subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo>↪</mo><mi>Op</mi></mrow><annotation encoding="application/x-tex">\mathcal{U} \hookrightarrow Op</annotation></semantics></math> is a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \hookrightarrow X\}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> that is closed under forming <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a> of subsets. The <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>→</mo></munder><mo stretchy="false">(</mo><mi>𝒰</mi><mo>↪</mo><mi>Op</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mo>∪</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> \underset{\to}{\lim} (\mathcal{U} \hookrightarrow Op) \simeq \cup_{i \in I} U_i </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/union">union</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>≔</mo><msub><mo>∪</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U \coloneqq \cup_{i \in I} U_i</annotation></semantics></math> of all these open subsets. Notice that by construction the component maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \hookrightarrow U\}</annotation></semantics></math> of the colimit are a <a class="existingWikiWord" href="/nlab/show/covering">covering</a> family of <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>.</p> <p>Inspection then shows that the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>←</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mi>ℱ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim}_{i \in I} \mathcal{F}(U_i)</annotation></semantics></math> is the corresponding set of <a class="existingWikiWord" href="/nlab/show/matching+families">matching families</a> (use the description of <a href="http://ncatlab.org/nlab/show/limit#ConstructionFromProductsAndEqualizers">limits in terms of products and equalizers</a> ). Hence the statement follows with def. <a class="maruku-ref" href="#GeneralComponentwiseDefinition"></a>.</p> </div> <h4 id="AsEtaleSpace">As étale spaces</h4> <p>Further in the case where the <a class="existingWikiWord" href="/nlab/show/site">site</a> is the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> <p>Some authors (e.g., <a class="existingWikiWord" href="/nlab/show/Robert+Goldblatt">Goldblatt</a> in <em>Topoi: The Categorial Analysis of Logic</em>, §4.5, p. 96) use <em>sheaf</em> to mean what we call an <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+space">étale space</a>: a topological <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> where the projection map is a <a class="existingWikiWord" href="/nlab/show/local+homeomorphism">local homeomorphism</a>.</p> <p>As discussed at <em><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+space#RelationToSheaves">étale space#RelationToSheaves</a></em>, there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> between the “sheaves” in this sense over a given base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> (i.e., the étale spaces over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>), and the sheaves as defined above over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> <h4 id="CharacterizationOverCanonicalTopologies">Characterization over canonical topologies</h4> <p>The above prop. <a class="maruku-ref" href="#AsContinuousFunctorOverOpenSubsets"></a> shows that often sheaves are characterized as contravariant functors that take some <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> to <a class="existingWikiWord" href="/nlab/show/limits">limits</a>. This is true in full generality for the following case</p> <div class="num_prop" id="AsContinuousFunctorsOnCanonicalTopology"> <h6 id="proposition_3">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> be be a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/large+site">large site</a> when equipped with the <a class="existingWikiWord" href="/nlab/show/canonical+topology">canonical topology</a>. Then a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> (with values in <a class="existingWikiWord" href="/nlab/show/small+sets">small sets</a>) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> is a sheaf precisely if it sends all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> to <a class="existingWikiWord" href="/nlab/show/limits">limits</a>.</p> </div> <h2 id="sheaves_and_localization">Sheaves and localization</h2> <p>We now describe the derivation and the detailed description of various aspects of sheaves, the <a class="existingWikiWord" href="/nlab/show/descent">descent</a> condition for sheaves and <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a>, relating it to all the related notions</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization">localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">Lawvere-Tierney topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+isomorphism">local isomorphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cover">cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercover">hypercover</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dense+monomorphism">dense monomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+epimorphism">local epimorphism</a></p> </li> </ul> </li> </ul> <p>We start by assuming that a <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> into a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> category is given and derive the consequences.</p> <p>So let <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> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>PSh</mi> <mi>S</mi></msub><mo>=</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">PSh(S) = PSh_S = [S^{op}, Set]</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/topos">topos</a> of <a class="existingWikiWord" href="/nlab/show/presheaf">presheaves</a>.</p> <p>Assume then that another topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Sh</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">Sh(S) = Sh_S</annotation></semantics></math> is given together with a <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f : Sh(S) \to PSh(S) </annotation></semantics></math></div> <p>i.e. with a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_* : Sh(S) \to PSh(S) </annotation></semantics></math></div> <p>and a left <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f^* : PSh(S) \to Sh(S) </annotation></semantics></math></div> <p>Such that both form a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex"> f^* \dashv f_* </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_*</annotation></semantics></math>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> for the category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Core</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>W</mi><mo>↪</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Core(PSh(S)) \hookrightarrow W \hookrightarrow PSh(S) </annotation></semantics></math></div> <p>consisting of all those morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S)</annotation></semantics></math> that are sent to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>s under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></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>W</mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Core</mi><mo stretchy="false">(</mo><msub><mi>Sh</mi> <mi>S</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> W = (f^*)^{-1}(Core(Sh_S)) \,. </annotation></semantics></math></div> <p>From the discussion at <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> we know that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(S)</annotation></semantics></math> is equivalent to the full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S)</annotation></semantics></math> on all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+object">local object</a>s.</p> <p>Recall that an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in PSh(S)</annotation></semantics></math> is called a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+object">local object</a> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p : Y \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mo>:</mo><msub><mi>PSh</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>PSh</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> p^* : PSh_S(X,A) \to PSh_S(Y,A) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. This we call the <a class="existingWikiWord" href="/nlab/show/descent">descent</a> condition on presheaves (saying that a presheaf “descends” along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> “down to” <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>). Our task is therefore to identify the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>, show how it determines and is determed by a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a> on <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> – equipping <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> with the structure of a <a class="existingWikiWord" href="/nlab/show/site">site</a> – and characterize the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+object">local object</a>s. These are (up to equivalence of categories) the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi></mrow><annotation encoding="application/x-tex">Sh</annotation></semantics></math>, i.e. the sheaves with respect to the given <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>.</p> <div class="un_lemma"> <h6 id="lemma">Lemma</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> if and only if for every <a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a> <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> and every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U\to X</annotation></semantics></math> the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>U</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">Y \times_X U \to U</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></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>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><mo>∈</mo><mi>W</mi></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><mo>⇔</mo><mo>∈</mo><mi>W</mi></mrow></msup></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Y \times_X U &\to& Y \\ \downarrow^{\in W} && \downarrow^{\Leftrightarrow \in W} \\ U &\to& X } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is stable under <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> (as described at <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a>: simply because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> preserves finite limits) it is clear that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>U</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">Y \times_X U \to U</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> is.</p> <p>To get the other direction, use the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a> to write <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> as a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <a class="existingWikiWord" href="/nlab/show/representable+functor">representables</a> over the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">/</mo><msub><mi>const</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y/const_X)</annotation></semantics></math> (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>):</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>≃</mo><msub><mi>colim</mi> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \simeq colim_{U_i \to X} U_i \,. </annotation></semantics></math></div> <p>Then pull back <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><msub><mi>colim</mi> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow></msub><mi>U</mi></mrow><annotation encoding="application/x-tex">Y \to colim_{U_i \to X} U</annotation></semantics></math> over the entire colimiting cone, so that over each component we have</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>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mi>i</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mi>i</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Y \times_X U_i &\to& Y \\ \downarrow && \downarrow \\ U_i &\to& X } \,. </annotation></semantics></math></div> <p>Using that in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">colimits are stable under base change</a> we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>colim</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>colim</mi> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> colim_i (Y \times_X U_i) \simeq (colim_i U_i) \times_X Y \,. </annotation></semantics></math></div> <p>But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><msub><mi>colim</mi> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X \simeq colim_i U_i</annotation></semantics></math> the right hand is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times_X Y</annotation></semantics></math>, which is just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>. So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><msub><mi>colim</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y = colim_i (Y \times_X U_i)</annotation></semantics></math> and we find that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> is a morphism of colimits. But under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> the two respective diagrams become isomorphic, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Y \times_X U_i \to U_i</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>. That means that the corresponding morphism of colimits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^*(Y \to X)</annotation></semantics></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> preserves colimits) is an isomorphism, which finally means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</p> </div> <div class="un_lemma"> <h6 id="lemma_2">Lemma</h6> <p>A presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in PSh(S)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/local+object">local object</a> with respect to all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> already if it is local with respect to those morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> whose codomain is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a></p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Rewriting the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> in terms of colimits as in the above proof</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><msub><mi>colim</mi> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>colim</mi> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow></msub><mi>U</mi></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ colim_{U \to X} U_i \times_X Y &\stackrel{\simeq}{\to}& Y \\ \downarrow && \downarrow \\ colim_{U \to X} U &\stackrel{\simeq}{\to}& X } </annotation></semantics></math></div> <p>we find that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(X) \to A(Y)</annotation></semantics></math> equals</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> lim_{U \to X} (A(U) \to A(U \times_X Y)) \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is local with respect to morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> with representable codomain, then by the above if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> all the morphisms in the limit here are isomorphisms, hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><msub><mi>Id</mi> <mrow><mi>A</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots = Id_{A(X)} \,. </annotation></semantics></math></div></div> <div class="un_lemma"> <h6 id="lemma_3">Lemma</h6> <p>Every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>⊂</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \subset PSh(S)</annotation></semantics></math> factors as an epimorphism followed by a monomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S)</annotation></semantics></math> with both being morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>Use factorization through <a class="existingWikiWord" href="/nlab/show/image">image</a> and <a class="existingWikiWord" href="/nlab/show/coimage">coimage</a>, use exactness of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> to deduce that the factorization exists not only in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S)</annotation></semantics></math> but even in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</p> <p>More in detail, given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> we get the diagram</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>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mi>Y</mi><msub><mo>⊔</mo> <mrow><mi>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi></mrow></msub><mi>Y</mi></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Y \times_X Y &&\to&& Y \\ &&& \swarrow \\ \downarrow &&Y \sqcup_{Y \times_X Y} Y && \downarrow \\ & \nearrow && \searrow \\ Y && \to && X } \,. </annotation></semantics></math></div> <p>Because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> is exact, the pullbacks and pushouts in this diagram remain such under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math>. But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^*(Y \to X)</annotation></semantics></math> is an isomorphism by assumption, the all these are pullbacks and pushouts along isomorphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(S)</annotation></semantics></math>, so all morphisms in the above diagram map to isomorphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(S)</annotation></semantics></math>, hence the entire diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S)</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</p> <p>Since the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><msub><mo>⊔</mo> <mrow><mi>Y</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>Y</mi></mrow></msub><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \sqcup_{Y \times_X Y} Y \to X</annotation></semantics></math> out of the <a class="existingWikiWord" href="/nlab/show/coimage">coimage</a> is at the same time the <a class="existingWikiWord" href="/nlab/show/equalizer">equalizing</a> morphism into the <a class="existingWikiWord" href="/nlab/show/image">image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mo stretchy="false">(</mo><mi>X</mi><mover><mo>→</mo><mo>→</mo></mover><mi>X</mi><msub><mo>⊔</mo> <mi>Y</mi></msub><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">lim(X \stackrel{\to}{\to} X \sqcup_Y X)</annotation></semantics></math>, it is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>.</p> </div> <div class="un_definition"> <h6 id="definition_6">Definition</h6> <p>The monomorphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S)</annotation></semantics></math> which are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> are called <a class="existingWikiWord" href="/nlab/show/dense+monomorphism">dense monomorphism</a>s.</p> </div> <div class="un_lemma"> <h6 id="lemma_4">Lemma</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> with <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> <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>colim</mi><mo stretchy="false">(</mo><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>U</mi><mo>→</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Y = colim ( U \times_X U \to U ) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>=</mo><msub><mo>⊔</mo> <mi>α</mi></msub><msub><mi>U</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">U = \sqcup_{\alpha} U_\alpha</annotation></semantics></math> a disjoint union of representables</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>This is a direct consequence of the standard fact that subfunctors are in bijection with <a class="existingWikiWord" href="/nlab/show/sieve">sieve</a>s.</p> </div> <div class="un_corollary"> <h6 id="corollary">Corollary</h6> <p>If a presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/local+object">local</a> with respect to all <a class="existingWikiWord" href="/nlab/show/dense+monomorphism">dense monomorphism</a>s, then it is already local with respect to all morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> of the form</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>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>=</mo><mi>colim</mi><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>W</mi></mtd> <mtd><mover><mo>→</mo><mo>→</mo></mover></mtd> <mtd><mi>U</mi></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↓</mo> <mrow><mi>dense</mi><mi>mono</mi></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>Id</mi></msup></mtd></mtr> <mtr><mtd><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>U</mi></mtd> <mtd><mover><mo>→</mo><mo>→</mo></mover></mtd> <mtd><mi>U</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y \\ \downarrow \\ X } = colim \left( \array{ W &\stackrel{\to}{\to}& U \\ \;\;\downarrow^{dense mono} && \downarrow^{Id} \\ U \times_X U & \stackrel{\to}{\to}& U } \right) </annotation></semantics></math></div> <p>with the left vertical morphism a <a class="existingWikiWord" href="/nlab/show/dense+monomorphism">dense monomorphism</a></p> <p>(and with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>=</mo><msub><mo>⊔</mo> <mi>α</mi></msub><msub><mi>U</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">U = \sqcup_\alpha U_\alpha</annotation></semantics></math> the disjoint union (of representable presheaves) over a <a class="existingWikiWord" href="/nlab/show/cover">cover</a>ing family of objects.)</p> </div> <div class="un_definition"> <h6 id="definition_7">Definition</h6> <p>The morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> with representable codomain</p> <ul> <li> <p>of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>colim</mi><mo stretchy="false">(</mo><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>U</mi><mover><mo>→</mo><mo>→</mo></mover><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">colim (U \times_X U \stackrel{\to}{\to} U) \to X</annotation></semantics></math> as above are <a class="existingWikiWord" href="/nlab/show/cover">cover</a>s:</p> </li> <li> <p>of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>colim</mi><mo stretchy="false">(</mo><mi>W</mi><mover><mo>→</mo><mo>→</mo></mover><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">colim (W \stackrel{\to}{\to} U) \to X</annotation></semantics></math> (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> a cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>U</mi></mrow><annotation encoding="application/x-tex">U \times_X U</annotation></semantics></math>) as above are <a class="existingWikiWord" href="/nlab/show/hypercover">hypercover</a>s</p> </li> </ul> <p>of the representable <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> </div> <div class="un_proposition"> <h6 id="proposition_4">Proposition</h6> <p>A presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-local, i.e. a sheaf, already if it is local (satisfies <a class="existingWikiWord" href="/nlab/show/descent">descent</a>) with respect to all <a class="existingWikiWord" href="/nlab/show/cover">cover</a>s, i.e. all <a class="existingWikiWord" href="/nlab/show/dense+monomorphism">dense monomorphism</a>s with codomain a <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a>.</p> </div> <blockquote> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs</a>: the above shows this almost. I am not sure yet how to see the remaining bit directly, without making recourse to the full machinery leading up to section VII, 4, corollary 7 in <a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a>.</p> </blockquote> <p>So we finally conclude:</p> <div class="un_corollary"> <h6 id="corollaries">Corollaries</h6> <p>We have:</p> <ul> <li> <p>Systems <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> of weak equivalences defined by choice of <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : Sh(S) \to PSh(S)</annotation></semantics></math> are in canonical bijection with choice of <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>.</p> </li> <li> <p>A presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-local, i.e. local with respect to all <a class="existingWikiWord" href="/nlab/show/local+isomorphism">local isomorphism</a>s, if and only if it is local already with respect to all <a class="existingWikiWord" href="/nlab/show/dense+monomorphism">dense monomorphism</a>, i.e. if and only if it satisfies sheaf condition for all covering <a class="existingWikiWord" href="/nlab/show/sieve">sieve</a>s.</p> </li> </ul> </div> <p>From the <em>assumption</em> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : Sh(S) \to PSh(S)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> follows at once the following explicit description of the <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^* : PSh(S) \to Sh(S)</annotation></semantics></math>.</p> <div class="un_lemma"> <h6 id="lemma_sheafification">Lemma (Sheafification)</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in PSh(S)</annotation></semantics></math> a presheaf, its <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo>:</mo><mo>=</mo><msub><mi>f</mi> <mo>*</mo></msub><msup><mi>f</mi> <mo>*</mo></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\bar A := f_* f^* A</annotation></semantics></math> is the presheaf given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo>:</mo><mi>U</mi><mo>↦</mo><msub><mi>colim</mi> <mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>→</mo><mi>U</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>W</mi></mrow></msub><mi>A</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \bar A : U \mapsto colim_{(Y \to U) \in W} A(Y) </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>By the discussion at <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(S)</annotation></semantics></math> is equivalent to the <a class="existingWikiWord" href="/nlab/show/localization">localization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">PSh(S)[W^{-1}]</annotation></semantics></math>, which in turn is the category with the same objects as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S)</annotation></semantics></math> and with morphisms given by spans out of hypercovers in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</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>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>colim</mi> <mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>W</mi></mrow></msub><mi>A</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PSh(S)[W^{-1}](X,A) = colim_{(Y \to X) \in W} A(Y) \,. </annotation></semantics></math></div> <p>So we have</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>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover><mover><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mover></mover></mtd> <mtd><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo>↘</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo>⇓</mo> <mo>≃</mo></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { Sh(S) &&\stackrel{\stackrel{f_*}{\to}}{\stackrel{f^*}{\leftarrow}}& PSh(S) \\ & \searrow_{\simeq}&\Downarrow^{\simeq}& \downarrow \\ && PSh(S)[W^{-1}] \,. } </annotation></semantics></math></div> <p>and deduce</p> <ul> <li> <p>by <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>PSh</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\bar A(U) = PSh_S(U, \bar A)</annotation></semantics></math>;</p> </li> <li> <p>by the <a class="existingWikiWord" href="/nlab/show/adjoint+functor">hom-adjunction</a> this is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msub><mi>Sh</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mover><mi>U</mi><mo stretchy="false">¯</mo></mover><mo>,</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cdots \simeq Sh_S(\bar U, \bar A)</annotation></semantics></math>;</p> </li> <li> <p>by the equivalence just mentioned this is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msub><mi>PSh</mi> <mi>S</mi></msub><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cdots \simeq PSh_S[W^{-1}](U,A)</annotation></semantics></math>.</p> </li> </ul> </div> <div class="un_remark"> <h6 id="remark_covers_versus_hypercovers">Remark: covers versus hypercovers</h6> <p>For checking the sheaf condition the <a class="existingWikiWord" href="/nlab/show/dense+monomorphism">dense monomorphism</a>s, i.e. the ordinary <a class="existingWikiWord" href="/nlab/show/cover">cover</a>s are already sufficient. But for <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> one really needs the <a class="existingWikiWord" href="/nlab/show/local+isomorphism">local isomorphism</a>s, i.e. the <a class="existingWikiWord" href="/nlab/show/hypercover">hypercover</a>s. If one takes the colimit in the sheafification prescription above only over <a class="existingWikiWord" href="/nlab/show/cover">cover</a>s, one obtains instead of sheafification the <a class="existingWikiWord" href="/nlab/show/plus+construction+on+presheaves">plus-construction</a>.</p> </div> <div class="un_definition"> <h6 id="definition_plusconstruction">Definition: plus-construction</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in PSh(S)</annotation></semantics></math> a presheaf, the <strong><a class="existingWikiWord" href="/nlab/show/plus+construction+on+presheaves">plus-construction</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the presheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>+</mo></msup><mo>:</mo><mi>X</mi><mo>↦</mo><msub><mi>colim</mi> <mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>↪</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>W</mi></mrow></msub><mi>A</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A^+ : X \mapsto colim_{(Y \hookrightarrow X) \in W } A(Y) </annotation></semantics></math></div> <p>where the colimit is over all <a class="existingWikiWord" href="/nlab/show/dense+monomorphism">dense monomorphism</a>s (instead of over all <a class="existingWikiWord" href="/nlab/show/local+isomorphism">local isomorphism</a>s as for <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar A</annotation></semantics></math>).</p> </div> <div class="un_remark"> <h6 id="remark_plusconstruction_versus_sheafification">Remark: plus-construction versus sheafification</h6> <p>In general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">A^+</annotation></semantics></math> is not yet a sheaf. It is however in general closer to being a sheaf than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is, namely it is a <a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a>.</p> <p>But applying the plus-construction twice yields the desired sheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>+</mo></msup><msup><mo stretchy="false">)</mo> <mo>+</mo></msup><mo>=</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (A^+)^+ = \bar A \,. </annotation></semantics></math></div> <p>This is essentially due to the fact that in the context of ordinary sheaves discussed here, all <a class="existingWikiWord" href="/nlab/show/hypercover">hypercover</a>s are already of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>colim</mi><mo stretchy="false">(</mo><mi>W</mi><mover><mo>→</mo><mo>→</mo></mover><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> colim(W \stackrel{\to}{\to} U) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>→</mo><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>U</mi></mrow><annotation encoding="application/x-tex">W \to U \times_X U</annotation></semantics></math> a cover. For higher <a class="existingWikiWord" href="/nlab/show/stack">stack</a>s the hypercover is in general a longer simplicial object of covers and accordingly if one restricts to covers instead of using hypercovers one will need to use the plus-construction more and more often. Specifically, for stacks of <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>-groupoids one needs to apply the plus-construction <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> times; see <a class="existingWikiWord" href="/nlab/show/plus+construction+on+presheaves">plus construction on presheaves</a>.</p> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">n=\infty</annotation></semantics></math>, even a countable sequence of applications does not suffice in general, but a sufficiently long transfinite sequence does. In this case, using hypercovers instead actually produces a different answer, namely the reflection into the <a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a> of the sheaf <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>-topos.</p> </div> <h2 id="examples">Examples</h2> <p>The archetypical example of sheaves are <em>sheaves of <a class="existingWikiWord" href="/nlab/show/function">function</a>s</em>:</p> <ul> <li> <p>for <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> a topological space, <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">\mathbb{C}</annotation></semantics></math> a topological space and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(X)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/site">site</a> of open subsets of <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>, the assignment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↦</mo><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \mapsto C(U,\mathbb{C})</annotation></semantics></math> of continuous functions from <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> to <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">\mathbb{C}</annotation></semantics></math> for every open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> is a sheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(X)</annotation></semantics></math>.</p> </li> <li> <p>for <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> a complex manifold and <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">\mathbb{C}</annotation></semantics></math> a complex manifold, the assignment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↦</mo><msub><mi>C</mi> <mi>hol</mi></msub><mrow><mi>X</mi><mo>,</mo><mi>ℂ</mi></mrow></mrow><annotation encoding="application/x-tex">U \mapsto C_{hol}{X,\mathbb{C}}</annotation></semantics></math> of holomorphic functions in a sheaf.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-sheaf">(0,1)-sheaf</a> / <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> / <a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a> / <strong>sheaf</strong> / <a class="existingWikiWord" href="/nlab/show/cosheaf">cosheaf</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf">abelian sheaf</a>, <a class="existingWikiWord" href="/nlab/show/sheaf+of+abelian+groups">sheaf of abelian groups</a>, <a class="existingWikiWord" href="/nlab/show/sheaf+of+modules">sheaf of modules</a>, <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaf">quasicoherent sheaf</a>, <a class="existingWikiWord" href="/nlab/show/sheaf+of+meromorphic+functions">sheaf of meromorphic functions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+sheaf">locally constant sheaf</a>, <a class="existingWikiWord" href="/nlab/show/constructible+sheaf">constructible sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+with+transfer">sheaf with transfer</a></p> </li> </ul> </li> <li> <p><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/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sheaf+of+spectra">sheaf of spectra</a></li> </ul> </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/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/soft+sheaf">soft sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fine+sheaf">fine sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flabby+sheaf">flabby sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent">descent</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cover">cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+descent">cohomological descent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+morphism">descent morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadic+descent">monadic descent</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Sweedler+coring">Sweedler coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+monadic+descent">higher monadic descent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+in+noncommutative+algebraic+geometry">descent in noncommutative algebraic geometry</a></p> </li> </ul> </li> </ul> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/homotopy+level">homotopy level</a></th><th><a class="existingWikiWord" href="/nlab/show/n-truncated+object+in+an+%28infinity%2C1%29-category">n-truncation</a></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+theory">higher topos theory</a></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">h-level 0</td><td style="text-align: left;">(-2)-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28-2%29-groupoid">(-2)-groupoid</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a>/<a class="existingWikiWord" href="/nlab/show/contractible+type">contractible type</a></td></tr> <tr><td style="text-align: left;">h-level 1</td><td style="text-align: left;">(-1)-truncated</td><td style="text-align: left;">contractible-if-<a class="existingWikiWord" href="/nlab/show/inhabited+space">inhabited</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28-1%29-groupoid">(-1)-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-sheaf">(0,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/ideal">ideal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a>/<a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a></td></tr> <tr><td style="text-align: left;">h-level 2</td><td style="text-align: left;">0-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+0-type">homotopy 0-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-groupoid">0-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;">h-level 3</td><td style="text-align: left;">1-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/1-groupoid">1-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/stack">stack</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-groupoid">h-groupoid</a></td></tr> <tr><td style="text-align: left;">h-level 4</td><td style="text-align: left;">2-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+2-type">homotopy 2-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></td><td style="text-align: left;">(3,1)-sheaf/2-stack</td><td style="text-align: left;">h-2-groupoid</td></tr> <tr><td style="text-align: left;">h-level 5</td><td style="text-align: left;">3-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+3-type">homotopy 3-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></td><td style="text-align: left;">(4,1)-sheaf/3-stack</td><td style="text-align: left;">h-3-groupoid</td></tr> <tr><td style="text-align: left;">h-level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n+2</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28n%2B1%2C1%29-sheaf">(n+1,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/n-stack">n-stack</a></td><td style="text-align: left;">h-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoid</td></tr> <tr><td style="text-align: left;">h-level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td><td style="text-align: left;">untruncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a></td><td style="text-align: left;">h-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid</td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>The original definition is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean+Leray">Jean Leray</a>, <em>L’anneau d’homologie d’une représentation</em>. Comptes rendus hebdomadaires des séances de l’Académie des Sciences 222 (1946), 1366–1368. <a class="existingWikiWord" href="/nlab/files/Leray-0.pdf" title="PDF">PDF</a></li> </ul> <p>Subsequent development by Leray, incorporating ideas of <a class="existingWikiWord" href="/nlab/show/Henri+Cartan">Henri Cartan</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean+Leray">Jean Leray</a>, <em>L’anneau spectral et l’anneau filtré d’homologie d’un espace localement compact et d’une application continue</em>, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 29 (1950), 1–80, 81–139.</li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Henri+Cartan">Henri Cartan</a>‘s account of the theory:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Henri+Cartan">Henri Cartan</a>, <em>Faisceaux sur un espace topologique. I, II</em>, Séminaire Henri Cartan, Exposés 14, 15. <p>numdam: <a href="http://www.numdam.org/item/SHC_1950-1951__3__A14_0/">I</a>, <a href="http://www.numdam.org/item/SHC_1950-1951__3__A15_0/">II</a>.</p> </li> </ul> <p>It refers to a previous exposition of the theory in Exposés 12–17 of the first year (1948/1949), which apparently are not scanned, unlike Exposés 1–11.</p> <p>Further references:</p> <p>Section C2 in</p> <ul id="Johnstone"> <li><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a></em></li> </ul> <ul> <li><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em></li> <li><a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, <em>Methods of homological algebra</em></li> </ul> <p>A concise and contemporary overview can be found in</p> <ul> <li id="CentazzoVitale04">C. Centazzo, <a class="existingWikiWord" href="/nlab/show/Enrico+Vitale">E. M. Vitale</a>, <em>Sheaf theory</em> , pp.311-358 in Pedicchio, Tholen (eds.), <em>Categorical Foundations</em> , Cambridge UP 2004. (<a href="https://perso.uclouvain.be/enrico.vitale/chapter7.pdf">draft</a>)</li> </ul> <p>With motivation from <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, leading over in the last chapter to the notion of <a class="existingWikiWord" href="/nlab/show/stack">stack</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>, Grundlehren der Mathematischen Wissenschaften <strong>332</strong>, Springer (2006)</li> </ul> <p>Lecture notes:</p> <ul> <li id="Warner12"> <p><a class="existingWikiWord" href="/nlab/show/Garth+Warner">Garth Warner</a>: <em>Fibrations and Sheaves</em>, EPrint Collection, University of Washington (2012) [<a href="http://hdl.handle.net/1773/20977">hdl:1773/20977</a>, <a href="https://sites.math.washington.edu//~warner/Warner_FIBRATIONS%20AND%20SHEAVES.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Warner-FibrationsAndSheaves.pdf" title="pdf">pdf</a>]</p> </li> <li id="Schapira23"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, <em>An Introduction to Categories and Sheaves</em>, lecture notes (2023) [<a href="https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/CatShv.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Schapira-Sheaves.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>A quick pedagogical introduction with an eye towards the generalization to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Dugger">Dan Dugger</a>, <em>Sheaves and homotopy theory</em>, <a href="http://ncatlab.org/nlab/files/cech.pdf">pdf</a></li> </ul> <p>Classics of sheaf theory on topological spaces are</p> <ul> <li> <p>Roger Godement, <em>Topologie algébrique et théorie des faisceaux</em>, Hermann, 1958, 283 p. <a href="http://books.google.fr/books/about/Topologie_alg%C3%A9brique_et_th%C3%A9orie_des_fa.html?id=JVrvAAAAMAAJ">gBooks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A.+Grothendieck">A. Grothendieck</a>, <a class="existingWikiWord" href="/nlab/show/Tohoku">Tohoku</a></p> </li> </ul> <p>Recently, an improvement in understanding the interplay of derived functors (inverse image and proper direct image) in sheaf theory on topological spaces has been exhibited in</p> <ul> <li>Olaf M. Schnuerer, Wolfgang Sergel, <em>Proper base change for separated locally proper maps</em>, <a href="http://arxiv.org/abs/1404.7630">arxiv/1404.7630</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 10, 2025 at 05:25:06. 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