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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="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> <h4 id="tale_morphisms">Étale morphisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism">étale morphism</a></strong> (<a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>)</p> <p>generally in <strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_aaa8c720d35345b38f171647a1e2b8f35af14dd3_1"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale</a> (<a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+topos">étale topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+homotopy">étale homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+cohomology">étale cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%E2%88%9E-groupoid">étale ∞-groupoid</a></p> </li> </ul> <p>in <strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+homeomorphism">local homeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+space">étale space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+groupoid">étale groupoid</a></p> </li> </ul> <p>in <strong><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, (<a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/immersion+of+smooth+manifolds">immersion of smooth manifolds</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+Lie+groupoid">étale Lie groupoid</a></p> </li> </ul> <p>in <strong><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale morphism of schemes</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism+of+schemes">formally étale morphism of schemes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a>, <a class="existingWikiWord" href="/nlab/show/pro-%C3%A9tale+site">pro-étale site</a>, <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stack">Deligne-Mumford stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%84%93-adic+cohomology">ℓ-adic cohomology</a></p> </li> </ul></div></div> <h4 id="topos_theory_2"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#EtaleToposOfAScheme'>Étale topos of a scheme</a></li> <li><a href='#GeneralAbstract'>Étale topos of a differentially cohesive object</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#SheafConditionAndExamples'>Sheaf condition and examples of étale sheaves</a></li> <li><a href='#BaseChange'>Base change and sheaf cohomology</a></li> <li><a href='#QuasiCoherentModules'>Quasi-coherent modules</a></li> <li><a href='#relation_to_zariski_topos'>Relation to Zariski topos</a></li> <li><a href='#as_a_classifying_topos'>As a classifying topos</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#etale_topos_of_a_schemes'>Etale topos of a schemes</a></li> <li><a href='#etale_topos_of_a_differentially_cohesive_object'>Etale topos of a differentially cohesive object</a></li> </ul> </ul> </div> <h2 id="definition">Definition</h2> <p>In the context of the <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> of <a class="existingWikiWord" href="/nlab/show/schemes">schemes</a> there is a traditional notion of <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale morphism of schemes</a> and an <em>étale topos</em> is a <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> on the <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a> of a <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a>, consisting of <a class="existingWikiWord" href="/nlab/show/covers">covers</a> by such <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale morphisms</a>. This traditional notion we discuss in</p> <ul> <li><a href="#EtaleToposOfAScheme">Étale topos of a scheme</a>.</li> </ul> <p>More abstractly, given that <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphisms+of+schemes">étale morphisms of schemes</a> may be characterized as modal morphisms with respect to an <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a>, one can consider étale toposes in every context of <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a>. This we discuss in</p> <ul> <li><a href="#GeneralAbstract">Étale topos of a differentially cohesive object</a></li> </ul> <h3 id="EtaleToposOfAScheme">Étale topos of a scheme</h3> <p>An <strong>étale topos</strong> is the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> over an <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a>, hence over a site whose “open subsets” are <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphisms">étale morphisms</a> into the base <a class="existingWikiWord" href="/nlab/show/space">space</a>. The intrinsic <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of an étale <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is <em><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+cohomology">étale cohomology</a></em>.</p> <p>More generally there is the pro-étale topos over a <a class="existingWikiWord" href="/nlab/show/pro-%C3%A9tale+site">pro-étale site</a>, which is a bit better behaved. In particular the intrinsic <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of a pro-étale <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> includes the <a class="existingWikiWord" href="/nlab/show/Weil+cohomology+theory">Weil cohomology theory</a> <a class="existingWikiWord" href="/nlab/show/%E2%84%93-adic+cohomology">ℓ-adic cohomology</a>.</p> <p>Generally, given that an <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale morphism of schemes</a> is a <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> subject to a size constraint on its <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> – for an actual <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale morphism of schemes</a> the fibers are <a class="existingWikiWord" href="/nlab/show/finite+sets">finite sets</a> in the suitable sense (formal duals to <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+algebras">étale algebras</a>) while for a <a class="existingWikiWord" href="/nlab/show/pro-%C3%A9tale+morphism+of+schemes">pro-étale morphism of schemes</a> they are <a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a> of such fibers – in a suitable ambient context (“<a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a>”) one can drop all finiteness conditions and consider just opens given by <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphisms">formally étale morphisms</a> as encoded by an <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a>. This we discuss <a href="#GeneralAbstract">below</a>.</p> <h3 id="GeneralAbstract">Étale topos of a differentially cohesive object</h3> <p>We discuss how in <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th}</annotation></semantics></math> every object <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> canonically induces its étale topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{\mathbf{H}_{th}}(X)</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}_{th}</annotation></semantics></math> any object in a <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesive</a> <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, we formulate</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> denoted <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">\mathcal{X}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_\infty(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a> over <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>, or rather of formally étale maps into <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> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/structure+%28%E2%88%9E%2C1%29-sheaf">structure (∞,1)-sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{X}</annotation></semantics></math> 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>.</p> </li> </ol> <p>The resulting structure is essentially that discussed (<a href="#Lurie">Lurie, Structured Spaces</a>) if we regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th}</annotation></semantics></math> equipped with its formally étale morphisms, (<a href="differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh">def.</a>), as a (<a class="existingWikiWord" href="/nlab/show/large+category">large</a>) <a class="existingWikiWord" href="/nlab/show/geometry+for+structured+%28%E2%88%9E%2C1%29-toposes">geometry for structured (∞,1)-toposes</a>.</p> <p>One way to motivate this is to consider structure sheaves of flat differential forms. To that end, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H}_{th})</annotation></semantics></math> a differential cohesive <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> with <a href="cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology">de Rham coefficient object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\flat_{dR}\mathbf{B}G</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}_{th}</annotation></semantics></math> any differential homotopy type, the product projection</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><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \times \flat_{dR} \mathbf{B}G \to X </annotation></semantics></math></div> <p>regarded as an object of the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}</annotation></semantics></math> <em>almost</em> qualifies as a “bundle of flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-valued differential forms” over <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>: for <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> an cover (a <a class="existingWikiWord" href="/nlab/show/1-epimorphism">1-epimorphism</a>) regarded in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}</annotation></semantics></math>, 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>-plot of this product projection is 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>-plot 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> together with a flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-valued de Rham cocycle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>This is indeed what the sections of a corresponding bundle of differential forms over <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> are supposed to look like – but only <em>if</em> <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> is sufficiently <em>spread out</em> over <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>, hence sufficiently <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+map">étale</a>. Because, on the extreme, if <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> is the point (the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>), then there should be no non-trivial section of differential forms relative to <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> over <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>, but the above product projection instead reproduces all the sections of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\flat_{dR} \mathbf{B}G</annotation></semantics></math>.</p> <p>In order to obtain the correct cotangent-like bundle from the product with the de Rham coefficient object, it needs to be <em>restricted</em> to plots out of suficiently étale maps into <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>. In order to correctly test differential form data, “suitable” here should be “formally”, namely infinitesimally. Hence the restriction should be along the full inclusion</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><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup><mo>↪</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> (\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X} </annotation></semantics></math></div> <p>of the formally étale maps (see <a href="differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh">def.</a>) into <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>. Since on formally étale covers the sections should be those given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\flat_{dR}\mathbf{B}G</annotation></semantics></math>, one finds that the corresponding “cotangent bundle” must be the <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflection</a> along this inclusion. The following proposition establishes that this coreflection indeed exists.</p> <div class="num_defn" id="EtaleSlice"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}_{th}</annotation></semantics></math> any object, write</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><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup><mo>↪</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> (\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X} </annotation></semantics></math></div> <p>for the full <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-category">sub-(∞,1)-category</a> of the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> over <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> on those maps into <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> which are formally étale, (see <a href="differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh">def.</a>).</p> <p>We also write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>FEt</mi> <mstyle mathvariant="bold"><mi>X</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">FEt_{\mathbf{X}}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{\mathbf{H}}(X)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}^{fet}</annotation></semantics></math>.</p> </div> <div class="num_prop" id="EtalificationIsCoreflection"> <h6 id="proposition">Proposition</h6> <p>The inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math> of def. <a class="maruku-ref" href="#EtaleSlice"></a> is both <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective</a> as well as <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective</a>, hence it fits into an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> of the form</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><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup><mover><mover><munder><mo>←</mo><mi>Et</mi></munder><mover><mo>↪</mo><mi>ι</mi></mover></mover><mover><mo>←</mo><mi>L</mi></mover></mover><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{H}_{th})_{/X}^{fet} \stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{\iota}{\hookrightarrow}}{\underset{Et}{\leftarrow}}} (\mathbf{H}_{th})_{/X} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the general discussion at <em><a class="existingWikiWord" href="/nlab/show/reflective+factorization+system">reflective factorization system</a></em>, the reflection is given by sending a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon Y \to X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times_{\mathbf{\Pi}_{inf}(X)} \mathbf{\Pi}_{inf}(Y) \to Y</annotation></semantics></math> and the reflection unit is the left horizontal morphism in</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> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><msub><mo>×</mo> <mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></msub><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Y &\to& X \times_{\mathbf{\Pi}_{inf}(Y)} \mathbf{\Pi}_{inf}(Y) &\to& \mathbf{\Pi}_{inf}(Y) \\ & \searrow & \downarrow^{} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ && X &\to& \mathbf{\Pi}_{inf}(X) } \,. </annotation></semantics></math></div> <p>Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}^{fet}</annotation></semantics></math>, being a reflective subcategory of a <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a>, is (as discussed there) itself locally presentable. Hence by the <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a> it is now sufficient to show that the inclusion preserves all small <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimits">(∞,1)-colimits</a> in order to conclude that it also has a right <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a>.</p> <p>So consider any <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>→</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup></mrow><annotation encoding="application/x-tex">I \to (\mathbf{H}_{th})_{/X}^{fet}</annotation></semantics></math> out of a <a class="existingWikiWord" href="/nlab/show/small+%28%E2%88%9E%2C1%29-category">small (∞,1)-category</a>. Since the inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}^{fet}</annotation></semantics></math> is full, it is sufficient to show that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-colimit over this diagram taken in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}</annotation></semantics></math> lands again in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}^{fet}</annotation></semantics></math> in order to have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-colimits are preserved by the inclusion. Moreover, colimits in a slice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th}</annotation></semantics></math> are computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th}</annotation></semantics></math> itself (this is discussed at <em><a href="overcategory#LimitsAndColimits">slice category - Colimits</a></em>).</p> <p>Therefore we are reduced to showing that the square</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><munder><mi>lim</mi><mrow><msub><mo>→</mo> <mi>i</mi></msub></mrow></munder><msub><mi>Y</mi> <mi>i</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><munder><mi>lim</mi><mrow><msub><mo>→</mo> <mi>i</mi></msub></mrow></munder><msub><mi>Y</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{\to_i}{\lim} Y_i &\to& \mathbf{\Pi}_{inf} \underset{\to_i}{\lim} Y_i \\ \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_{inf}(X) } </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> square. But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}_{inf}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> it commutes with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-colimit on objects and hence this diagram is equivalent to</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><munder><mi>lim</mi><mrow><msub><mo>→</mo> <mi>i</mi></msub></mrow></munder><msub><mi>Y</mi> <mi>i</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><munder><mi>lim</mi><mrow><msub><mo>→</mo> <mi>i</mi></msub></mrow></munder><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><msub><mi>Y</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{\to_i}{\lim} Y_i &\to& \underset{\to_i}{\lim} \mathbf{\Pi}_{inf} Y_i \\ \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_{inf}(X) } \,. </annotation></semantics></math></div> <p>This diagram is now indeed an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> by the fact that we have <a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th}</annotation></semantics></math>, hence that on the left the component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Y_i</annotation></semantics></math> for each <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> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><msub><mi>Y</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X)</annotation></semantics></math>, by assumption that we are taking an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-colimit over formally étale morphisms.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The <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>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}^{fet}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and the canonical inclusion into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By prop. <a class="maruku-ref" href="#EtalificationIsCoreflection"></a> the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup><mo>↪</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28infinity%2C1%29-category">reflective</a> with reflector given by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo>−</mo><mi>equivalences</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo>−</mo><mi>closed</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{\Pi}_{inf}-equivalences , \mathbf{\Pi}_{inf}-closed)</annotation></semantics></math> factorization system. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}_{inf}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> and hence in particular preserves <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullbacks">(∞,1)-pullbacks</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}_{inf}</annotation></semantics></math>-equivalences are stable under pullbacks. By the discussion at <em><a class="existingWikiWord" href="/nlab/show/stable+factorization+system">stable factorization system</a></em> this is the case precisely if the corresponding reflector preserves <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limits">finite (∞,1)-limits</a>. Hence the embedding is a <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> which exhibits a <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-topos">sub-(∞,1)-topos</a> inclusion.</p> </div> <div class="num_defn" id="TheStructureSheafOfX"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}_{th}</annotation></semantics></math> we speak of</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>Sh</mi> <mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup></mrow><annotation encoding="application/x-tex"> \mathcal{X} \coloneqq Sh_{\mathbf{H}_{th}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{fet} </annotation></semantics></math></div> <p>also as the (<a class="existingWikiWord" href="/nlab/show/petit+%28%E2%88%9E%2C1%29-topos">petit</a>) <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> 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>, or the <strong>étale topos</strong> 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>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow></mover><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mover><mo>→</mo><mi>Et</mi></mover><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup></mrow><annotation encoding="application/x-tex"> \mathcal{O}_X \colon \mathbf{H}_{th} \stackrel{(-) \times X}{\to} (\mathbf{H}_{th})_{/X} \stackrel{Et}{\to} (\mathbf{H}_{th})_{/X}^{fet} </annotation></semantics></math></div> <p>for the composite <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> that sends any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">A \in \mathbf{H}_{th}</annotation></semantics></math> to the etalification, prop. <a class="maruku-ref" href="#EtalificationIsCoreflection"></a>, of the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \times X \to X</annotation></semantics></math>.</p> <p>We call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_X</annotation></semantics></math> the <strong><a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a></strong> 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>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">X, A \in \mathbf{H}_{th}</annotation></semantics></math> and for <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> a <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th}</annotation></semantics></math> (hence like an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> 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>), we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><msub><mi>Sh</mi> <mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><msub><mi>Sh</mi> <mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>Et</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo>×</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathcal{O}_{X}(A)(U) & \coloneqq Sh_{\mathbf{H}_{th}}(X)( U , \mathcal{O}_{X}(A) ) \\ & \coloneqq Sh_{\mathbf{H}_{th}}(X)( U , Et(X \times A) ) \\ & \simeq (\mathbf{H}_{th})_{/X}(U, X \times A) \\ & \simeq \mathbf{H}_{th}(U,A) \\ & \simeq A(U) \end{aligned} \,, </annotation></semantics></math></div> <p>where we used the <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">∞-adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ι</mi><mo>⊣</mo><mi>Et</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\iota \dashv Et)</annotation></semantics></math> of prop. <a class="maruku-ref" href="#EtalificationIsCoreflection"></a> and the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a>.</p> <p>This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}_{X}(A)</annotation></semantics></math> behaves as the <em>sheaf of <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>-valued functions over <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></em>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{X}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> to the forgetful functor</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> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup><mo>↪</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mover><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder></mover><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex"> Sh_{\mathbf{H}}(X) \simeq (\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X} \stackrel{\underset{X}{\sum}}{\to} \mathbf{H}_{th} </annotation></semantics></math></div> <p>it preserves <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limits">(∞,1)-limits</a>. Therefore this is a <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> which exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{\mathbf{H}_{th}}(X)</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th}</annotation></semantics></math> regarded as a (large) <a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a>, with the formally étale morphisms being the “admissible morphisms”.</p> </div> <div class="num_example" id="CotangentBundle"> <h6 id="example">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H}_{th})</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th}</annotation></semantics></math> for the corresponding de Rham coefficient object.</p> <p>Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>Sh</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O}_X(\flat_{dR}\mathbf{B}G) \in Sh_{\mathbf{H}}(X) </annotation></semantics></math></div> <p>we may call the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-valued flat cotangent sheaf</strong> 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>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">U \in \mathbf{H}_{th}</annotation></semantics></math> a test object (say an object in a <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a> of definition, under the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>) a formally étale 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> is like an <a class="existingWikiWord" href="/nlab/show/open+map">open map</a>/open embedding. Regarded as an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}^{fet}</annotation></semantics></math> we may consider the sections over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of the cotangent bundle as defined above, which in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th}</annotation></semantics></math> are diagrams</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>U</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ U &&\to&& \mathcal{O}_X(\flat_{dR}\mathbf{B}G) \\ & \searrow && \swarrow \\ && X } \,. </annotation></semantics></math></div> <p>By the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Et</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Et(-)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, such diagrams are in bijection to diagrams</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>U</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ U &&\to&& X \times \flat_{dR} \mathbf{B}G \\ & \searrow && \swarrow \\ && X } </annotation></semantics></math></div> <p>where we are now simply including on the left the formally étale map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U \to X)</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>fet</mi></msubsup><mo>↪</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}</annotation></semantics></math>.</p> <p>In other words, the sections of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-valued flat cotangent sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}_X(\flat_{dR}\mathbf{B}G)</annotation></semantics></math> are just the sections of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times \flat_{dR}\mathbf{B}G \to X</annotation></semantics></math> itself, only that the <em>domain</em> of the section is constrained to be a formally é patch 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>.</p> <p>But then by the very nature of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\flat_{dR}\mathbf{B}G</annotation></semantics></math> it follows that the flat sections of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-valued cotangent bundle 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> are indeed nothing but the flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-valued differential forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_prop" id="StructuredPetitToposesAreLocallyContractible"> <h6 id="proposition_3">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}_{th}</annotation></semantics></math> an object in a differentially cohesive <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, then its petit structured <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{\mathbf{H}_{th}}(X)</annotation></semantics></math>, according to def. <a class="maruku-ref" href="#TheStructureSheafOfX"></a>, is <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected</a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>We need to check that the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi><mover><mo>⟶</mo><mi>Disc</mi></mover><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow></mover><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mover><mo>⟶</mo><mi>L</mi></mover><msub><mi>Sh</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \infty Grpd \stackrel{Disc}{\longrightarrow} \mathbf{H}_{th} \stackrel{(-) \times X}{\longrightarrow} (\mathbf{H}_{th})_{/X} \stackrel{L}{\longrightarrow} Sh_{\mathbf{H}}(X) </annotation></semantics></math></div> <p>preserves <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limits">(∞,1)-limits</a>, so that it has a further <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is the reflector from prop. <a class="maruku-ref" href="#EtalificationIsCoreflection"></a>. Inspection shows that this composite sends an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">A \in \infty Grpd</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}_{inf}(Disc(A)) \times X \to X</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><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{\Pi}_{inf}(Disc(A)) \times X &\longrightarrow& \mathbf{\Pi}_{inf}(Disc(A) \times X) & \simeq \mathbf{\Pi}_{inf}(Disc(A)) \times \mathbf{\Pi}_{inf}(X) \\ \downarrow &{}^{(pb)}& \downarrow \\ X &\longrightarrow& \mathbf{\Pi}_{inf}(X) } \,. </annotation></semantics></math></div> <p>By the discussion at <a href="slice+infinity-category#LimitsAndColimits">slice (∞,1)-category – Limits and colimits</a> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> in the slice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\mathbf{H}_{th})_{/X}</annotation></semantics></math> is computed as an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> with the slice <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> adjoined. By <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjointness</a> of the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Sh_{\mathbf{H}}(X) \hookrightarrow (\mathbf{H}_{th})_{/X}</annotation></semantics></math> the same is then true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow> <mi>et</mi></msubsup></mrow><annotation encoding="application/x-tex">Sh_{\mathbf{H}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{et}</annotation></semantics></math>.</p> <p>Now for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo lspace="verythinmathspace">:</mo><mi>J</mi><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">A \colon J \to \infty Grpd</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>, it is taken to the diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>↦</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">j \mapsto \mathbf{\Pi}_{inf}(Disc(A_j)) \times X \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{\mathbf{H}}(X)</annotation></semantics></math> and so its <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>-limit is computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> over the diagram locally 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>X</mi><mo>×</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>j</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>≃</mo><mrow><mtable><mtr><mtd><mi>X</mi><mo>×</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>j</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \times \mathbf{\Pi}_{inf}(Disc(A_{j})) &&\longrightarrow&& X \times \mathbf{\Pi}_{inf}(Disc(A_{j'})) \\ & \searrow && \swarrow \\ && X } \simeq \array{ X \times \mathbf{\Pi}_{inf}(Disc(A_{j})) &&\longrightarrow&& X \times \mathbf{\Pi}_{inf}(Disc(A_{j'})) \\ & \searrow && \swarrow \\ && X \times \ast } \,. </annotation></semantics></math></div> <p>Since <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>-limits commute with each other this limit is the product of</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>←</mo></munder> <mi>j</mi></msub><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))</annotation></semantics></math></p> </li> <li> <p><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>J</mi><mo>⋆</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}{\lim}_{J \star \Delta^0} X</annotation></semantics></math> (over the co-coned diagram constant on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>).</p> </li> </ol> <p>For the first of these, since the <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}_{inf}</annotation></semantics></math> is in particular a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> (with <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> the <a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a>), and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi></mrow><annotation encoding="application/x-tex">Disc</annotation></semantics></math> is also <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> by <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a>, we have a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>←</mo></munder> <mi>j</mi></msub><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>Disc</mi><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>←</mo></munder> <mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j)) \simeq \mathbf{\Pi}_{inf}(Disc(\underset{\leftarrow}{\lim}_j(A_j))) \,. </annotation></semantics></math></div> <p>For the second, the <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>-limit over an <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>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>⋆</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">J \star \Delta^0</annotation></semantics></math> of a functor constant on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><munder><mi>lim</mi><mo>←</mo></munder> <mrow><mi>J</mi><mo>⋆</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow></msub><mi>X</mi></mtd> <mtd><mo>≃</mo><msub><munder><mi>lim</mi><mo>←</mo></munder> <mrow><mi>J</mi><mo>⋆</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow></msub><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">[</mo><msub><munder><mi>lim</mi><mo>→</mo></munder> <mrow><mi>J</mi><mo>⋆</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow></msub><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">[</mo><mrow><mo stretchy="false">|</mo><mrow><mi>J</mi><mo>⋆</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow><mo stretchy="false">|</mo></mrow><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \underset{\leftarrow}{\lim}_{J \star \Delta^0} X & \simeq \underset{\leftarrow}{\lim}_{J \star \Delta^0} [\ast, X] \\ & \simeq [\underset{\rightarrow}{\lim}_{J \star \Delta^0} \ast, X] \\ & \simeq [{\vert {J \star \Delta^0}\vert}, X] \\ & \simeq [\ast, X] \simeq X \end{aligned} \,, </annotation></semantics></math></div> <p>where the last line follows since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mi>J</mi><mo>⋆</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow></mrow><annotation encoding="application/x-tex">{J \star \Delta^0}</annotation></semantics></math> has a terminal object and hence contractible geometric realization.</p> <p>In conclusion this shows that <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>-limits are preserved by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo>∘</mo><mi>Disc</mi></mrow><annotation encoding="application/x-tex">L \circ (-)\times X\circ Disc</annotation></semantics></math>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="SheafConditionAndExamples">Sheaf condition and examples of étale sheaves</h3> <div class="num_prop" id="EtaleDescentDetectedOnOpenImmersionCovers"> <h6 id="proposition_4">Proposition</h6> <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 <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a>, and <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><msub><mi>X</mi> <mi>et</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in PSh(X_{et})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a>, for checking the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> condition it is sufficient to check <a class="existingWikiWord" href="/nlab/show/descent">descent</a> on the following two kinds of <a class="existingWikiWord" href="/nlab/show/covers">covers</a> in the <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a></p> <ol> <li> <p>jointly surjective collections of <a class="existingWikiWord" href="/nlab/show/open+immersions+of+schemes">open immersions of schemes</a>;</p> </li> <li> <p>single surjective/<a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale</a> morphisms between <a class="existingWikiWord" href="/nlab/show/affine+schemes">affine schemes</a></p> </li> </ol> <p>(all over <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> <p>(<a href="#Tamme">Tamme, II Lemma (3.1.1)</a>, <a href="#Milne">Milne, prop. 6.6</a>)</p> <div class="proof"> <h6 id="proof_sketch">Proof (sketch)</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/covers">covers</a> by standard <a class="existingWikiWord" href="/nlab/show/open+immersions+of+schemes">open immersions of schemes</a> in the <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a> are also <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphisms+of+schemes">étale morphisms of schemes</a> and étale covers, we may take any étale cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Y</mi> <mi>i</mi></msub><mo>→</mo><mi>Y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Y_i \to Y\}</annotation></semantics></math> over <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>, find an Zariski cover <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 \to X\}</annotation></semantics></math> 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>, pull back the original cover to that and in turn cover the pullbacks themselves by Zariski covers. The result is still a cover and is so by a collection of <a class="existingWikiWord" href="/nlab/show/open+immersions+of+schemes">open immersions of schemes</a>. Now using compactness assumptions we find finite subcovers of all these covers. This makes their <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> be a single morphisms of affines.</p> </div> <div class="num_prop" id="XSchemesRepresentSheaves"> <h6 id="proposition_5">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Z \to X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> over a <a class="existingWikiWord" href="/nlab/show/scheme">scheme</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>, the induced <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on the <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a></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><msub><mi>U</mi> <mi>Y</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (U_Y \to X) \mapsto Hom_X(U_Y, Z) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>.</p> </div> <p>This is due to (<a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a>, <a class="existingWikiWord" href="/nlab/show/SGA">SGA</a>1 exp. XIII 5.3) A review is in (<a href="#Tamme">Tamme, II theorem (3.1.2)</a>, <a href="#Milne">Milne, 6.2</a>).</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By prop. <a class="maruku-ref" href="#EtaleDescentDetectedOnOpenImmersionCovers"></a> we are reduced to showing that the represented presheaf satisfies <a class="existingWikiWord" href="/nlab/show/descent">descent</a> along collections of open immersions and along surjective maps of affines. For the first this is clear (it is <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a>-descent). For the second case of a <a class="existingWikiWord" href="/nlab/show/faithfully+flat">faithfully flat</a> cover of affines <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(B) \to Spec(A)</annotation></semantics></math> it follows with the exactness of the correspomnding <a class="existingWikiWord" href="/nlab/show/Amitsur+complex">Amitsur complex</a>. See there for details.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>This map from <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>-schemes to sheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>et</mi></msub></mrow><annotation encoding="application/x-tex">X_{et}</annotation></semantics></math> is not injective, different <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>-schemes may represent the same sheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>et</mi></msub></mrow><annotation encoding="application/x-tex">X_{et}</annotation></semantics></math>. Unique representatives are given by <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale schemess</a> over <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> <p>(e.g. <a href="#Tamme">Tamme, II theorem 3.1</a>)</p> <p>We consider some examples of <a class="existingWikiWord" href="/nlab/show/sheaves+of+abelian+groups">sheaves of abelian groups</a> induced by prop. <a class="maruku-ref" href="#XSchemesRepresentSheaves"></a> from <a class="existingWikiWord" href="/nlab/show/group+schemes">group schemes</a> over <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 class="num_example"> <h6 id="example_2">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/additive+group">additive group</a> over <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> is the <a class="existingWikiWord" href="/nlab/show/group+scheme">group scheme</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝔾</mi> <mi>a</mi></msub><mo>≔</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>t</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow></msub><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{G}_a \coloneqq Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, the corresponding sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝔾</mi> <mi>a</mi></msub><msub><mo stretchy="false">)</mo> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">(\mathbb{G}_a)_X</annotation></semantics></math> is given by the assignment</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">(</mo><msub><mi>𝔾</mi> <mi>a</mi></msub><msub><mo stretchy="false">)</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>X</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>Hom</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>X</mi></msub><mo>,</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>t</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow></msub><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>X</mi></msub><mo>,</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>t</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>t</mi><mo stretchy="false">]</mo><mo>,</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>𝒪</mi> <mrow><msub><mi>U</mi> <mi>X</mi></msub></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>𝒪</mi> <mrow><msub><mi>U</mi> <mi>X</mi></msub></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} (\mathbb{G}_a)_X(U_X \to X) & = Hom_X(U_X, Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X) \\ & = Hom(U_X, Spec(\mathbb{Z}[t])) \\ & = Hom(\mathbb{Z}[t], \Gamma(U_X, \mathcal{O}_{U_X})) \\ & = \Gamma(U_X, \mathcal{O}_{U_X}) \end{aligned} \,. </annotation></semantics></math></div> <p>In other words, the sheaf represented by the <a class="existingWikiWord" href="/nlab/show/additive+group">additive group</a> is the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf">abelian sheaf</a> underlying the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> 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>.</p> </div> <p>Similarly one finds</p> <div class="num_example"> <h6 id="example_3">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a> over <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 class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝔾</mi> <mi>m</mi></msub><mo>≔</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>t</mi><mo>,</mo><msup><mi>t</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex"> \mathbb{G}_m \coloneqq Spec(\mathbb{Z}[t,t^{-1}]) \times_{Spec(\mathbb{Z})} X </annotation></semantics></math></div> <p>represents the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><msub><mo stretchy="false">)</mo> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">(\mathbb{G}_m)_X</annotation></semantics></math> given by</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><msub><mi>𝔾</mi> <mi>m</mi></msub><msub><mo stretchy="false">)</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>𝒪</mi> <mrow><msub><mi>U</mi> <mi>X</mi></msub></mrow></msub><msup><mo stretchy="false">)</mo> <mo>×</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbb{G}_m)_X(U_X) \mapsto \Gamma(U_X, \mathcal{O}_{U_X})^\times \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#Tamme">Tamme, II, 3</a>)</p> <h3 id="BaseChange">Base change and sheaf cohomology</h3> <div class="num_defn" id="BaseChangeOnSites"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/schemes">schemes</a>, there is induced a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> on the <a class="existingWikiWord" href="/nlab/show/categories">categories</a> underlying the <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</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> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Y</mi> <mi>et</mi></msub><mo>⟶</mo><msub><mi>X</mi> <mi>et</mi></msub></mrow><annotation encoding="application/x-tex"> f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et} </annotation></semantics></math></div> <p>given by sending an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>Y</mi></msub><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">U_Y \to Y</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>/<a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><msub><mi>U</mi> <mi>Y</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X) \,. </annotation></semantics></math></div></div> <div class="num_prop" id="DirectAndInverseImageAlongMapOfBases"> <h6 id="proposition_6">Proposition</h6> <p>The morphism in def. <a class="maruku-ref" href="#BaseChangeOnSites"></a> is a <a class="existingWikiWord" href="/nlab/show/morphism+of+sites">morphism of sites</a> and hence induces a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> between the étale toposes</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>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>et</mi></msub><mo stretchy="false">)</mo><mover><munder><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mover><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>Y</mi> <mi>et</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (f^\ast \dashv f_\ast) \;\colon\; Sh(X_{et}) \stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}} Sh(Y_{et}) \,. </annotation></semantics></math></div> <p>Here the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> is given on a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</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><msub><mi>X</mi> <mi>et</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{F} \in Sh(X_{et})</annotation></semantics></math> by</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><mi>ℱ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>ℱ</mi><mo stretchy="false">(</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ℱ</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_\ast \mathcal{F} \;\colon\; (U_Y \to Y) \mapsto \mathcal{F}(f^{-1}(U_Y)) = \mathcal{F}(X \times_X U_Y) </annotation></semantics></math></div> <p>while the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> is given on a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>∈</mo><msub><mi>Sh</mi> <mo stretchy="false">(</mo></msub><msub><mi>Y</mi> <mi>et</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{F} \in Sh_(Y_{et})</annotation></semantics></math> by</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><mi>ℱ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>X</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↦</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msub><mi>U</mi> <mi>X</mi></msub><mo>→</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow></munder></munder><mi>ℱ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^\ast \mathcal{F} \;\colon\; (U_X \to X) \mapsto \underset{\underset{U_X \to f^{-1}(U_Y)}{\longrightarrow}}{\lim} \mathcal{F}(U_Y) \,. </annotation></semantics></math></div></div> <p>By the discussion at <em><a href="morphism+of+sites#RelationToGeometricMorphisms">morphisms of sites – Relation to geometric morphisms</a></em>. See also for instance (<a href="#Tamme">Tamme I 1.4</a>).</p> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>et</mi></msub></mrow><annotation encoding="application/x-tex">X_{et}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>et</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{D}(X_{et})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of the <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>et</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ab(Sh(X_{et}))</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>q</mi></msup><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">R^q f_\ast</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> functor of def. <a class="maruku-ref" href="#DirectAndInverseImageAlongMapOfBases"></a> sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>∈</mo><mi>Ab</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>et</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{F} \in Ab(Sh(X_{et}))</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> of the <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></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><msub><mi>U</mi> <mi>Y</mi></msub><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>↦</mo><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><msub><mi>U</mi> <mi>Y</mi></msub><mo>,</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (U_Y \to Y) \mapsto H^q(X \times_Y U_Y, \mathcal{F}) \,, </annotation></semantics></math></div> <p>where on the right we have the degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> <a class="existingWikiWord" href="/nlab/show/cohomology+group">group</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the given <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> (<a class="existingWikiWord" href="/nlab/show/%C3%A9tale+cohomology">étale cohomology</a>).</p> </div> <p>By the discussion at <em><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></em>. See e.g. (<a href="#Tamme">Tamme, II (1.3.4)</a>, <a href="#Milne">Milne prop. 12.1</a>).</p> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub><mover><mo>←</mo><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mover><msub><mi>O</mi> <mi>Y</mi></msub><mover><mo>←</mo><mrow><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mover><msub><mi>O</mi> <mi>Z</mi></msub></mrow><annotation encoding="application/x-tex">O_X \stackrel{f^{-1}}{\leftarrow} O_Y \stackrel{g^{-1}}{\leftarrow} O_Z</annotation></semantics></math> two composable <a class="existingWikiWord" href="/nlab/show/morphisms+of+sites">morphisms of sites</a>, the <a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a> for the corresponding <a class="existingWikiWord" href="/nlab/show/direct+images">direct images</a> exists and 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><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>=</mo><msup><mi>R</mi> <mi>p</mi></msup><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><msup><mi>R</mi> <mi>q</mi></msup><msub><mi>g</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⇒</mo><msup><mi>E</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo>=</mo><msup><mi>R</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>g</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{p,q}_2 = R^p f_\ast(R^q g_\ast(\mathcal{F})) \Rightarrow E^{p+q} = R^{p+q}(g f)_\ast(\mathcal{F}) \,. </annotation></semantics></math></div> <p>For the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>Z</mi></msub><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">S_Z = \ast</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">g^{-1}</annotation></semantics></math> includes an <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>Y</mi></msub><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">U_Y \to Y</annotation></semantics></math> this yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>=</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><mo>,</mo><msup><mi>R</mi> <mi>q</mi></msup><msub><mi>f</mi> <mo>*</mo></msub><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⇒</mo><msup><mi>E</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo>=</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi><mo>,</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{p,q}_2 = H^p(U_Y, R^q f_\ast \mathcal{F}) \Rightarrow E^{p+q} = H^{p+q}(U_Y \times_Y X , \mathcal{F}) \,. </annotation></semantics></math></div></div> <h3 id="QuasiCoherentModules">Quasi-coherent modules</h3> <div class="num_prop"> <h6 id="proposition_8">Proposition</h6> <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 <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> and <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> a <a class="existingWikiWord" href="/nlab/show/quasicoherent+module">quasicoherent module</a> over its <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_X</annotation></semantics></math>, then this induces an <a class="existingWikiWord" href="/nlab/show/abelian+sheaf">abelian sheaf</a> on the <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>et</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>X</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><mo>,</mo><mi>N</mi><msub><mo>⊗</mo> <mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow></msub><msub><mi>𝒪</mi> <mrow><msub><mi>U</mi> <mi>X</mi></msub></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_{et} \;\colon\; (U_X \to X) \mapsto \Gamma(U_Y, N \otimes_{\mathcal{O}_X} \mathcal{O}_{U_X}) \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#Tamme">Tamme, II 3.2.1</a>)</p> <h3 id="relation_to_zariski_topos">Relation to Zariski topos</h3> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>A <a class="existingWikiWord" href="/nlab/show/cover">cover</a> in the <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a> on <a class="existingWikiWord" href="/nlab/show/schemes">schemes</a> is an <a class="existingWikiWord" href="/nlab/show/open+immersion+of+schemes">open immersion of schemes</a> and hence is in particular an <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale morphism of schemes</a>. Hence the <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a> is finer than the <a class="existingWikiWord" href="/nlab/show/Zariski+site">Zariski site</a> and so every étale <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> is a Zariski sheaf, but not necessarily conversely. Expressed in a different way, the étale topos is a <a class="existingWikiWord" href="/nlab/show/subtopos">subtopos</a> of the Zariski topos.</p> </div> <p>For more see at <em><a href="etale+cohomology#RelationZariskiEtaleCohomology">étale cohomology – Properties – Relation to Zariski cohomology</a></em>.</p> <h3 id="as_a_classifying_topos">As a classifying topos</h3> <p>The étale topos over the big étale site of <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a> is the <a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a> for <a class="existingWikiWord" href="/nlab/show/strict+local+rings">strict local rings</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/basics+of+%C3%A9tale+cohomology">basics of étale cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+homotopy">étale homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+cohomology">étale cohomology</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="etale_topos_of_a_schemes">Etale topos of a schemes</h3> <ul id="Tamme"> <li><a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Tamme">Günter Tamme</a>, section II 1 of <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+%C3%89tale+Cohomology">Introduction to Étale Cohomology</a></em></li> </ul> <ul id="Milne"> <li><a class="existingWikiWord" href="/nlab/show/James+Milne">James Milne</a>, section 7 of <em><a class="existingWikiWord" href="/nlab/show/Lectures+on+%C3%89tale+Cohomology">Lectures on Étale Cohomology</a></em></li> </ul> <h3 id="etale_topos_of_a_differentially_cohesive_object">Etale topos of a differentially cohesive object</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em> (<a href="http://arxiv.org/abs/1310.7930">arXiv:1310.7930</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 11, 2022 at 10:58:33. 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