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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Tannaka duality for geometric stacks </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/2294/#Item_7" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" 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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="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="higher_geometry">Higher geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <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/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <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> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#setup'>Setup</a></li> <ul> <li><a href='#ringed_toposes'>Ringed toposes</a></li> <li><a href='#abelian_tensor_categories'>Abelian tensor categories</a></li> <li><a href='#GeometricStack'>Geometric stacks</a></li> </ul> <li><a href='#tannaka_duality_for_geometric_stacks'>Tannaka duality for geometric stacks</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Under mild conditions, a given <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>T</mi><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C \subset T Alg^{op}</annotation></semantics></math> of formal duals of <a class="existingWikiWord" href="/nlab/show/algebra+over+a+Lawvere+theory">algebras over an algebraic theory</a> admits <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> exhibited by an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</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><mi>𝒪</mi><mo>⊣</mo><mi>Spec</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>T</mi><msup><mi>Alg</mi> <mi>Δ</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mover><munder><mo>→</mo><mi>Spec</mi></munder><mover><mo>←</mo><mi>𝒪</mi></mover></mover><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\mathcal{O} \dashv Spec) : (T Alg^{\Delta})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Sh_{(\infty,1)}(C) </annotation></semantics></math></div> <p>as described at <a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a> Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(X)</annotation></semantics></math> is 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>-algebra of functions 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 entry describes for certain <a class="existingWikiWord" href="/nlab/show/algebraic+stacks">algebraic stacks</a> an analog of this situation where the 1-algebras are replaced by 2-algebras in the form of commutative algebra objects in the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a>: abelian <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+categories">symmetric monoidal categories</a>, and where the function algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(X)</annotation></semantics></math> are replaced with category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">QC(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a>.</p> <p>The replacement of the 1-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(X)</annotation></semantics></math> by the 2-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">QC(X)</annotation></semantics></math> is the starting point for what is called <a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a>.</p> <h2 id="setup">Setup</h2> <h3 id="ringed_toposes">Ringed toposes</h3> <p>All <a class="existingWikiWord" href="/nlab/show/topos">topos</a>es that we consider are <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a>es. A <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S, \mathcal{O}_S)</annotation></semantics></math> is a topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> equipped with a ring object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_S</annotation></semantics></math> – a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of rings – called the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> – on whatever <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> on. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>S</mi></msub><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}_S Mod</annotation></semantics></math> for the category of <a class="existingWikiWord" href="/nlab/show/module">module</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> (sheaves of modules) over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_S</annotation></semantics></math>.</p> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RngdTopos</mi></mrow><annotation encoding="application/x-tex">RngdTopos</annotation></semantics></math> for the category of ringed toposes.</p> <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> or more generally an <a class="existingWikiWord" href="/nlab/show/algebraic+stack">algebraic stack</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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">Sh(X_{et})</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/little+etale+topos">little etale topos</a>.</p> <div class="un_def" id="localWRTEtaleTopology"> <h6 id="definition">Definition</h6> <p>A ringed topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S,\mathcal{O}_S)</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/locally+ringed+topos">locally ringed topos</a></strong> with respect to the <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+topology">étale topology</a> if for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">U \in S</annotation></semantics></math> and every family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>Spec</mi><msub><mi>R</mi> <mi>i</mi></msub><mo>→</mo><mi>Spec</mi><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Spec R_i \to Spec \mathcal{O}_S(U)\}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism">étale morphism</a>s such that</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>S</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><msub><mi>R</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> \mathcal{O}_S(U) \to \prod_i R_i </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/faithfully+flat">faithfully flat</a>, there exists morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>i</mi></msub><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">E_i \to E</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and factorizations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>R</mi> <mi>i</mi></msub><mo>→</mo><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}_S(U) \to R_i \to \mathcal{O}_S(E_i)</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></munder><msub><mi>E</mi> <mi>i</mi></msub><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> \coprod_i E_i \to E </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>.</p> </div> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S, \mathcal{O}_S)</annotation></semantics></math> is local for the étale topology precisely if the <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}_S(x)</annotation></semantics></math> at every <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">x : Set \to S</annotation></semantics></math> is a strictly <a class="existingWikiWord" href="/nlab/show/Henselian+ring">Henselian</a> <a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>.</p> </div> <p>This is (<a href="#Lurie">Lurie, remark 4.4</a>).</p> <div class="un_prop"> <h6 id="example">Example</h6> <ul> <li>The <a class="existingWikiWord" href="/nlab/show/little+%C3%A9tale+topos">little étale topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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">Sh(X_{et})</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stack">Deligne-Mumford stack</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> is locally ringed with respect to the étale topology.</li> </ul> </div> <h3 id="abelian_tensor_categories">Abelian tensor categories</h3> <div class="un_def" id="AbelianTensorCategory"> <h6 id="definition_2">Definition</h6> <p>An <strong>abelian tensor category</strong> (for the purposes of the present discusission) is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, \otimes)</annotation></semantics></math> such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>;</p> </li> <li> <p>for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">x \in C</annotation></semantics></math> the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>x</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">(-) \otimes x : C\to C</annotation></semantics></math> is additive and right-<a class="existingWikiWord" href="/nlab/show/exact+functor">exact</a>: it commutes with finite <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s.</p> </li> </ul> <p>A <strong>complete abelian tensor category</strong> is an abelian tensor category such that</p> <ul> <li> <p>it satisfies the axiom AB5 at <a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">(-) \otimes x</annotation></semantics></math> commutes with <em>all small</em> colimits.</p> <p>(equivalently, we have a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a>).</p> </li> </ul> <p>An abelian tensor category is called <strong>tame</strong> if for any <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>M</mi><mo>′</mo><mo>→</mo><mi>M</mi><mo>→</mo><mi>M</mi><mo>″</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to M'\to M \to M''\to 0 </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">M''</annotation></semantics></math> a <em>flat object</em> (such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>x</mi><mo>⊗</mo><mi>M</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">x \mapsto x \otimes M''</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a>) and any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">N \in C</annotation></semantics></math> also the induced sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>M</mi><mo>′</mo><mo>⊗</mo><mi>N</mi><mo>→</mo><mi>M</mi><mo>⊗</mo><mi>N</mi><mo>→</mo><mi>M</mi><mo>″</mo><mo>⊗</mo><mi>N</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to M'\otimes N \to M\otimes N \to M''\otimes N \to 0 </annotation></semantics></math></div> <p>is exact.</p> </div> <p>This appears as (<a href="#Lurie">Lurie, def. 5.2</a>) together with the paragraph below remark 5.3.</p> <div class="un_def" id="TCAbTens"> <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>C</mi><mo>,</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C,D</annotation></semantics></math> two <a href="#AbelianTensorCategory">complete abelian tensor categories</a> write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Func</mi> <mo>⊗</mo></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Func_\otimes(C,D) \subset Func(C,D) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/core">core</a> of the <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> on those <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s that</p> <ul> <li> <p>are <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a>s;</p> </li> <li> <p>commute with all small <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s (which implies they are <a class="existingWikiWord" href="/nlab/show/additive+functor">additive</a> and <a class="existingWikiWord" href="/nlab/show/exact+functor">right exact</a>)</p> </li> <li> <p>preserve flat objects and short exact sequences whose last object is flat.</p> </li> </ul> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>TCAbTens</mi></mrow><annotation encoding="application/x-tex"> TCAbTens </annotation></semantics></math></div> <p>for the (<a class="existingWikiWord" href="/nlab/show/strict+2-category">strict</a>) <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> of <a href="#AbelianTensorCategory">tame complete abelian tensor categories</a> with hom-<a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>s given by this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Func</mi> <mo>⊗</mo></msub></mrow><annotation encoding="application/x-tex">Func_\otimes</annotation></semantics></math>.</p> </div> <p>This appears as (<a href="#Lurie">Lurie, def 5.9</a>) together with the following remarks.</p> <div class="un_lemma"> <h6 id="example_2">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">k Mod</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> of <a class="existingWikiWord" href="/nlab/show/module">module</a>s</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S,\mathcal{O}_S)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a>. 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>S</mi></msub><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> \mathcal{O}_S Mod </annotation></semantics></math></div> <p>(the category of sheaves of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_S</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>s) is a tame <a href="#AbelianTensorCategory">complete abelian tensor category</a>.</p> </div> <p>This is (<a href="#Lurie">Lurie, example 5.7</a>).</p> <div class="un_lemma"> <h6 id="example_3">Example</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> an <a class="existingWikiWord" href="/nlab/show/algebraic+stack">algebraic stack</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> QC(X) </annotation></semantics></math></div> <p>for its category <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a>.</p> <p>This is a <a href="#AbelianTensorCategory">complete abelian tensor category</a></p> </div> <div class="un_lemma"> <h6 id="lemma">Lemma</h6> <p>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 a Noetherian geometric stack, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">QC(X)</annotation></semantics></math> is the category of <a class="existingWikiWord" href="/nlab/show/ind-object">ind-object</a>s of its full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Coh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Coh(X) \subset QC(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/coherent+sheaves">coherent sheaves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>Coh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> QC(X) \simeq Ind(Coh(X)) \,. </annotation></semantics></math></div></div> <p>This appears as (<a href="#Lurie">Lurie, lemma 3.9</a>).</p> <h3 id="GeometricStack">Geometric stacks</h3> <div class="un_def" id="geometricStack"> <h6 id="definition_4">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/geometric+stack">geometric stack</a></strong> is</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/algebraic+stack">algebraic stack</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> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">Spec \mathbb{Z}</annotation></semantics></math></p> </li> <li> <p>that is quasi-compact, in particular there is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Spec A \to X</annotation></semantics></math>;</p> </li> <li> <p>with affine and <a class="existingWikiWord" href="/nlab/show/representable+morphism+of+stacks">representable</a> diagonal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \to X \times X</annotation></semantics></math>.</p> </li> </ul> </div> <div class="un_prop"> <h6 id="example_4">Example</h6> <ul> <li> <p>A quasicompact <a class="existingWikiWord" href="/nlab/show/separated+scheme">separated scheme</a> is a geometric stack.</p> </li> <li> <p>The classifying stack of a <a class="existingWikiWord" href="/nlab/show/smooth+scheme">smooth</a> affine <a class="existingWikiWord" href="/nlab/show/group+object">group</a> <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> is a geometric stack.</p> </li> </ul> </div> <p>The geometricity condition on an algebraic stack implies that there are “enough” <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a> on it, as formalized by the following statement.</p> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>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 a <a href="#geometricStack">geometric stack</a> then the <a class="existingWikiWord" href="/nlab/show/bounded+chain+complex">bounded-below</a> <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent 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> is naturally <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of the left-bounded derived category of smooth-<a class="existingWikiWord" href="/nlab/show/etale+site">etale</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>-modules whose <a class="existingWikiWord" href="/nlab/show/chain+cohomology">chain cohomology</a> sheaves are quasicoherent.</p> </div> <p>This is (<a href="#Lurie">Lurie, theorem 3.8</a>).</p> <h2 id="tannaka_duality_for_geometric_stacks">Tannaka duality for geometric stacks</h2> <div class="un_theorem"> <h6 id="theorem_2">Theorem</h6> <p>Let <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> be a <a href="#GeometricStack">geometric stack</a>.</p> <p>Then for every ring <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> there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>RngdTopos</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Spec</mi><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>et</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>et</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mo>⊗</mo></msub><mo stretchy="false">(</mo><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> RngdTopos(Sh((Spec A)_{et}),Sh(X_{et})) \simeq Hom_\otimes(QC(X), A Mod) </annotation></semantics></math></div> <p>hence (by the <a class="existingWikiWord" href="/nlab/show/2-Yoneda+lemma">2-Yoneda lemma</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>Spec</mi><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mo>⊗</mo></msub><mo stretchy="false">(</mo><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>QC</mi><mo stretchy="false">(</mo><mi>Spec</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X(Spec A) \simeq Hom_\otimes(QC(X), QC(Spec A)) \,. </annotation></semantics></math></div> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S, \mathcal{O}_S)</annotation></semantics></math> any <a href="#localWRTEtaleTopology">etale-locally</a> <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>RngdTopos</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>et</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mo>⊗</mo></msub><mo stretchy="false">(</mo><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>𝒪</mi> <mi>S</mi></msub><mi>Mod</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> RngdTopos(S,Sh(X_{et})) \simeq Hom_\otimes(QC(X), \mathcal{O}_S Mod) \,. </annotation></semantics></math></div></div> <p>This is (<a href="#Lurie">Lurie, theorem 5.11</a>) in view of (<a href="#Lurie">Lurie, remark 4.5</a>).</p> <div class="un_remark"> <h6 id="remark">Remark</h6> <p>It follows that forming <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a> constitutes a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+%28infinity%2C1%29-functor">full and faithful (2,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo>:</mo><mi>GeomStacks</mi><mo>→</mo><msup><mi>TCAbTens</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> QC : GeomStacks \to TCAbTens^{op} </annotation></semantics></math></div> <p>from geometric stacks to <a href="#TCAbTens">tame complete abelian tensor categories</a>.</p> <p>This statement justifies thinking of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">QC(X)</annotation></semantics></math> as being the “2-algebra” of functions 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>. This perspective is the basis for <a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/analytification">analytification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-algebraic+geometry">2-algebraic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+a+tensor+triangulated+category">spectrum of a tensor triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+symmetric+monoidal+stable+%28%E2%88%9E%2C1%29-category">prime spectrum of a symmetric monoidal stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+Lie+groupoids">Tannaka duality for Lie groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bondal-Orlov+reconstruction+theorem">Bondal-Orlov reconstruction theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a></p> </li> </ul> <h2 id="references">References</h2> <p>The above material is taken from</p> <ul> <li id="Lurie"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>Tannaka duality for geometric stacks</em>, (<a href="http://arxiv.org/abs/math/0412266">arXiv:math.AG/0412266</a>)</li> </ul> <p>The generalization to geometric stacks in the context of <a class="existingWikiWord" href="/nlab/show/Spectral+Schemes">Spectral Schemes</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Quasi-Coherent+Sheaves+and+Tannaka+Duality+Theorems">Quasi-Coherent Sheaves and Tannaka Duality Theorems</a></em></li> </ul> <p>Related discussion is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martin+Brandenburg">Martin Brandenburg</a>, <a class="existingWikiWord" href="/nlab/show/Alexandru+Chirvasitu">Alexandru Chirvasitu</a>, <em>Tensor functors between categories of quasi-coherent sheaves</em> (<a href="http://arxiv.org/abs/1202.5147">arXiv:abs/1202.5147</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 9, 2014 at 21:03:51. See the <a href="/nlab/history/Tannaka+duality+for+geometric+stacks" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Tannaka+duality+for+geometric+stacks" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2294/#Item_7">Discuss</a><span class="backintime"><a href="/nlab/revision/Tannaka+duality+for+geometric+stacks/11" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Tannaka+duality+for+geometric+stacks" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Tannaka+duality+for+geometric+stacks" accesskey="S" class="navlink" id="history" rel="nofollow">History (11 revisions)</a> <a href="/nlab/show/Tannaka+duality+for+geometric+stacks/cite" style="color: black">Cite</a> <a href="/nlab/print/Tannaka+duality+for+geometric+stacks" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Tannaka+duality+for+geometric+stacks" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>