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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</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/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="duality">Duality</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/duality">duality</a></strong></p> <ul> <li> <p>abstract duality: <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton duality</a></p> </li> <li> <p>concrete duality: <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a>, <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a>, <a class="existingWikiWord" href="/nlab/show/fully+dualizable+object">fully dualizable object</a>, <a class="existingWikiWord" href="/nlab/show/dualizing+object">dualizing object</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a></li> </ul> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p>between <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>/<a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></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/Stone+duality">Stone duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Langlands+duality">Langlands duality</a>, <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+duality">geometric Langlands duality</a>, <a class="existingWikiWord" href="/nlab/show/quantum+geometric+Langlands+duality">quantum geometric Langlands duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontryagin+duality">Pontryagin duality</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartier+duality">Cartier duality</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a> for <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul+duality">Koszul duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+duality">Grothendieck duality</a></p> </li> </ul> <p><strong>In QFT and String theory</strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/duality+in+physics">duality in physics</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Seiberg+duality">Seiberg duality</a>, <a class="existingWikiWord" href="/nlab/show/AGT+conjecture">AGT conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electro-magnetic+duality">electro-magnetic duality</a>, <a class="existingWikiWord" href="/nlab/show/Montonen-Olive+duality">Montonen-Olive duality</a>, <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+duality">geometric Langlands duality</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT">AdS/CFT duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </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='#statement'>Statement</a></li> <ul> <li><a href='#ForPermutationRepresentations'>For <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>-Sets</a></li> <ul> <li><a href='#quick_proof'>Quick Proof</a></li> <li><a href='#longwinded_proof'>Long-winded Proof</a></li> </ul> <li><a href='#ForVModules'>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-modules</a></li> <ul> <li><a href='#ForAlgebraModules'>For algebra modules</a></li> <li><a href='#for_linear_group_representations'>For linear group representations</a></li> <li><a href='#Coalgebras'>For coalgebra comodules</a></li> </ul> <li><a href='#for_lie_groupoids'>For Lie groupoids</a></li> <li><a href='#for_geometric_stacks'>For geometric stacks</a></li> <li><a href='#InHigherCategoryTheory'>In higher category theory</a></li> <ul> <li><a href='#ForInfinityPermutations'>For permutation <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>-representations</a></li> <li><a href='#InfinityGaloisTheory'><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>-Galois theory</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><a class="existingWikiWord" href="/nlab/show/Tannaka">Tannaka</a> duality or <em>Tannaka <a class="existingWikiWord" href="/nlab/show/reconstruction+theorem">reconstruction theorem</a>s</em> are statements of the form:</p> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a symmetry object (e.g. a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a>), <a class="existingWikiWord" href="/nlab/show/representation">represented</a> on objects in a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, one may <em>reconstruct</em> <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> from knowledge of the <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a>s of the forgetful functor – the <strong><a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></strong> –</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msub><mi>Rep</mi> <mi>D</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex"> F : Rep_D(A) \to D </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Rep</mi> <mi>D</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep_D(A)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/representation">representation</a>s 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> on <a class="existingWikiWord" href="/nlab/show/object">object</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> that remembers these underlying objects. In a generalization, called mixed Tannakian formalism, not a single <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a>, but a family of <a class="existingWikiWord" href="/nlab/show/fiber+functors">fiber functors</a> over different bases is needed for a reconstruction.</p> <p>There is a general-abstract and a concrete aspect to this. The general abstract one says that an algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is reconstructible from the <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> on the category of <em>all</em> its modules. The concrete one says that in nice cases it is reconstructible from the category of <em>dualizable</em> (finite dimensional) modules, even if it is itself not finite dimensional.</p> <p>More precisely, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriching category</a> (a <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with all <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s). Then</p> <ol> <li> <p>for</p> <ul> <li> <p><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> a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> of <em>all</em> <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>-<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>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>A</mi><mi>Mod</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">F : A Mod \to V</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful</a> <em><a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></em> ;</p> </li> </ul> <p><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> can be reconstructed as the object of <a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched endomorphisms</a> of <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>, given by the <a class="existingWikiWord" href="/nlab/show/end">end</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≃</mo><mi>End</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>N</mi><mo>∈</mo><mi>A</mi><mi>Mod</mi></mrow></msub><mi>V</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \simeq End(F) := \int_{N \in A Mod} V(F(N), F(N)) \,. </annotation></semantics></math></div> <p>This is just the <a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a> in a slight disguise.</p> </li> <li> <p>In good cases, this <a class="existingWikiWord" href="/nlab/show/end">end</a> is computed already by restriction to the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mi>Mod</mi> <mi>dual</mi></msub></mrow><annotation encoding="application/x-tex">A Mod_{dual}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/dual+object">dualizable modules</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>N</mi><mo>∈</mo><mi>A</mi><msub><mi>Mod</mi> <mi>dual</mi></msub></mrow></msub><mi>V</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq \int_{N \in A Mod_{dual}} V(F(N), F(N)) \,. </annotation></semantics></math></div></li> </ol> <h2 id="statement">Statement</h2> <p>So far the following examples concern the abstract algebraic aspect of Tannaka duality only, which is narrated here as a consequence of the <a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a> in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>. Some of the Tannaka duality theorems involve subtle harmonic analysis.</p> <h3 id="ForPermutationRepresentations">For <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>-Sets</h3> <p>A simple case of Tannaka duality is that of <a class="existingWikiWord" href="/nlab/show/G-sets">G-sets</a> of a <a class="existingWikiWord" href="/nlab/show/group">group</a>, i.e. representations on a <a class="existingWikiWord" href="/nlab/show/set">set</a>. In this case, Tannaka duality follows entirely from repeated application of the ordinary <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Tannaka duality for <a class="existingWikiWord" href="/nlab/show/G-sets">G-sets</a>)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/group">group</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Set</mi></mrow><annotation encoding="application/x-tex">G Set</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/G-sets">G-sets</a> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mi>Set</mi><mo>⟶</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> F \colon G Set \longrightarrow Set </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> that sends a <a class="existingWikiWord" href="/nlab/show/G-set">G-set</a> to its underlying <a class="existingWikiWord" href="/nlab/show/set">set</a>.</p> <p>Then there is a canonical <a class="existingWikiWord" href="/nlab/show/group+homomorphism">group-</a><a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Aut(F) \;\simeq\; G \,. </annotation></semantics></math></div> <p>identifying the <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of <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> (the group of <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a> from <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> to itself) with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="quick_proof">Quick Proof</h6> <p>With a bit of evident abuse of notation, the proof is a one-line sequence of applications of the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>: we show <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">End(F) \cong G</annotation></semantics></math>, i.e., each endomorphism on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is invertible, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">End(F) = Aut(F) \cong G</annotation></semantics></math>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≔</mo><msup><mi>Set</mi> <mi>G</mi></msup><mo>=</mo><mi>G</mi><mi>Set</mi></mrow><annotation encoding="application/x-tex">C \coloneqq Set^G = G Set</annotation></semantics></math>. Observe that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F \colon C \to Set</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mi>C</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F = C(G, -)</annotation></semantics></math>. Then the argument is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>Set</mi> <mi>C</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>≅</mo><msup><mi>Set</mi> <mi>C</mi></msup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>C</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>G</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> End(F) = Set^C(F, F) \cong Set^C(C(G, -), C(G, -)) \cong C(G, G) \cong G. </annotation></semantics></math></div> <p>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>” here is used in multiple senses, but each sense is deducible from context.</p> </div> <div class="proof"> <h6 id="longwinded_proof">Long-winded Proof</h6> <p>We repeat the same proof, but with more notational details on what the entities involved in each step are precisely.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> of the <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Set</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>Func</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G Set \;=\; Func(\mathbf{B}G^{op}, Set) \,. </annotation></semantics></math></div> <p>The canonical inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">i : {*} \to \mathbf{B}G</annotation></semantics></math> induces the <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo><mo>:</mo><mi>G</mi><mi>Set</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> Func(i,Set) : G Set \to Set </annotation></semantics></math></div> <p>which evaluates a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\rho : \mathbf{B}G^{op} \to Set</annotation></semantics></math> on the unique object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>. By the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> this is the same as homming out of the functor <a class="existingWikiWord" href="/nlab/show/representable+functor">represented by</a> that unique object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>Y</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Func(i,Set) = Hom_{PSh(\mathbf{B}G)}(Y_{\mathbf{B}G} {*}, -) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y_{\mathbf{B}G} : \mathbf{B}G \to PSh(\mathbf{B}G)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>.</p> <p>But this way we see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Func(i,Set) : PSh(\mathbf{B}G) \to Set</annotation></semantics></math> is itself a representable functor in the presheaf category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(PSh(\mathbf{B}G)^{op})</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Y</mi> <mstyle mathvariant="bold"><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></mstyle></msub><msub><mi>Y</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Func(i,Set) = Y_{\mathbf{PSh(\mathbf{B}G)^{op}}} Y_{\mathbf{B}G} * \,. </annotation></semantics></math></div> <p>So applying the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> twice, we find 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>Aut</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow></msub><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>Aut</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow></msub><msub><mi>Y</mi> <mstyle mathvariant="bold"><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></mstyle></msub><msub><mi>Y</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Aut</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><msub><mi>Y</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Aut</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Aut_{PSh(PSh(\mathbf{B}G)^{op})} Func(i,Set) & = Aut_{PSh(PSh(\mathbf{B}G)^{op})} Y_{\mathbf{PSh(\mathbf{B}G)^{op}}} Y_{\mathbf{B}G} * \\ & \simeq Aut_{PSh(\mathbf{B}G)^{op}} Y_{\mathbf{B}G} * \\ & \simeq Aut_{\mathbf{B}G} * \\ & \simeq G \,. \end{aligned} </annotation></semantics></math></div></div> <p>Notice that the proof in no way used the fact that <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> was assumed to be a <a class="existingWikiWord" href="/nlab/show/group">group</a>, but only that <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> is a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>. So the statement holds just as well for arbitrary monoids.</p> <p>But moreover, as the long-winded proof above makes manifest, even more abstractly the proof really only depended on the fact that the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>. It need not have a single object for the proof to go through verbatim. Therefore we immediately obtain the following much more general statement of Tannaka duality for permutation representations of categories:</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p><strong>(Tannaka duality for permutation representations of categories)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Rep</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep_{Set}(C) := Func(C,Set)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>. For every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>c</mi></msub><mo>:</mo><msub><mi>Rep</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F_c : Rep_{Set}(C) \to Set</annotation></semantics></math> be the fiber-functor that evaluates at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>.</p> <p>Then we have a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>c</mi></msub><mo>,</mo><msub><mi>F</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom(F_c,F_{c'}) \simeq Hom_C(c,c') \,. </annotation></semantics></math></div></div> <h3 id="ForVModules">For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-modules</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a (<a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a>) <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is enriched in itself via its <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>.</p> <p>Observe that the setup, statement and proof of Tannaka duality for permutation representations given above is the special case for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">V = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of a statement verbatim the same in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>, with the ordinary <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> replaced everywhere by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a>:</p> <p>Then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched Tannaka duality theorem states that we can reconstruct a monoid <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> as the monoid of endomorphisms of the forgetful <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>:</p> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p><strong>(Tannaka duality for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-modules over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-algebras)</strong></p> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math>, and with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> A Mod := [\mathbf{B}A,V] </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a> that encodes the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>s 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>, we have that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a> algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>F</mi><mo>,</mo><mi>F</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">End(F) := [F,F]</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Rep</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">F : Rep(A) \to V</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally isomorphic</a> to <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mover><mo>→</mo><mi>F</mi></mover><mi>V</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> End(A Mod \stackrel{F}{\to} V) \simeq A \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We can repeat the argument given above for permutation representations, this time employing the <a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a>.</p> <p>Indeed, we may identify <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> with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-functor</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><mo>•</mo><msup><mo stretchy="false">]</mo> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><msub><mstyle mathvariant="bold"><mi>Fun</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><msub><munder><mrow><msub><mstyle mathvariant="bold"><mi>Fun</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>pt</mi></mstyle> <mi>V</mi></msub><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder> <mrow><mo>≅</mo><mi>V</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[\bullet]^*\colon\mathbf{Fun}_V(\mathbf{B}A,V)\to\underbrace{\mathbf{Fun}_V(\mathbf{pt}_V,V)}_{\cong V}</annotation></semantics></math></div> <p>given by precomposition along the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>•</mo><mo stretchy="false">]</mo><mo lspace="verythinmathspace">:</mo><msub><mstyle mathvariant="bold"><mi>pt</mi></mstyle> <mi>V</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">[\bullet]\colon\mathbf{pt}_V\to\mathbf{B}A</annotation></semantics></math> picking the unique object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math>.</p> <p>Now we note that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>•</mo><msup><mo stretchy="false">]</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">[\bullet]^*</annotation></semantics></math> (given by sending an <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>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">M\colon\mathbf{B}A\to V</annotation></semantics></math>) to its evaluation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mo>•</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(\bullet)</annotation></semantics></math> at the unique object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math>) is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-naturally isomorphic to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Nat</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mo>•</mo></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Nat}_V(h_\bullet,-)</annotation></semantics></math>, since</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Nat</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mo>•</mo></msub><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>M</mi><mo stretchy="false">(</mo><mo>•</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Nat}_V(h_\bullet,M)\cong M(\bullet)</annotation></semantics></math></div> <p>by the enriched Yoneda lemma. So in summary we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>≅</mo><mo stretchy="false">[</mo><mo>•</mo><msup><mo stretchy="false">]</mo> <mo>*</mo></msup><mo>≅</mo><msub><mstyle mathvariant="bold"><mi>Nat</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mo>•</mo></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F\cong[\bullet]^*\cong\mathbf{Nat}_V(h_\bullet,-)</annotation></semantics></math>.</p> <p>We can then compute <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">End(F)</annotation></semantics></math> as follows:</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><mi>End</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>=</mo><mi mathvariant="normal">def</mi></mover><msub><mstyle mathvariant="bold"><mi>Nat</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo><msub><mstyle mathvariant="bold"><mi>Nat</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>Nat</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mo>•</mo></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mi>Nat</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mo>•</mo></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo><msub><mstyle mathvariant="bold"><mi>Nat</mi></mstyle> <mi>V</mi></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>h</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≅</mo><msub><mstyle mathvariant="bold"><mi>Hom</mi></mstyle> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow></msub><mo stretchy="false">(</mo><mo>•</mo><mo>,</mo><mo>•</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mo>=</mo><mi mathvariant="normal">def</mi></mover><mi>A</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} End(F) &\overset{\mathrm{def}}{=} \mathbf{Nat}_V(F,F) \\ &\cong \mathbf{Nat}_V(\mathbf{Nat}_V(h_{\bullet},-),\mathbf{Nat}_V(h_{\bullet},-)) \\ &\cong \mathbf{Nat}_V(h_{\bullet},h_{\bullet}) \\ &\cong \mathbf{Hom}_{\mathbf{B}A}(\bullet,\bullet) \\ &\overset{\mathrm{def}}{=} A, \end{aligned} </annotation></semantics></math></div> <p>where we have applied the <a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a> twice.</p> </div> <p>Notice that the <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a> object here is taken in the sense of enriched category theory, as described at <a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a>. It is given by the <a class="existingWikiWord" href="/nlab/show/end">end</a> expression</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>N</mi><mo>∈</mo><mi>A</mi><mi>Mod</mi></mrow></msub><mi>V</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> End(F) = \int_{N \in A Mod} V(F(N), F(N)) \,. </annotation></semantics></math></div> <p>The case of permutation representations is re-obtained by setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">V = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <p>As before, the same proof actually shows the following more general statement</p> <div class="num_theorem"> <h6 id="theorem_4">Theorem</h6> <p><strong>(Tannaka duality for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-modules over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-algebroids)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> (a “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/algebroid">algebroid</a>”). Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mi>Mod</mi><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C Mod := [C,V]</annotation></semantics></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a>. For every <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><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>c</mi></msub><mo>:</mo><mi>C</mi><mi>Mod</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">F_c : C Mod \to V </annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> that evaluates at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Then we have <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>c</mi></msub><mo>,</mo><msub><mi>F</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> hom(F_c, F_{c'}) \simeq C(c,c') \,. </annotation></semantics></math></div></div> <p>From this statement of Tannaka duality in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched category theory now various special cases of interest follow, by simply choosing suitable enrichment categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> <h4 id="ForAlgebraModules">For algebra modules</h4> <p>The general case of Tannaka duality for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-modules described <a href="#ForVModules">above</a> restricts to the classical case of Tannaka duality for linear representations by setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>:</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">V :=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>, the category of <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>s over some fixed <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a>.</p> <p>In this case the above says</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p><strong>(Tannaka duality for linear modules)</strong></p> <p>For <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> an <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> its category of <a class="existingWikiWord" href="/nlab/show/modules">modules</a>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>A</mi><mi>Mod</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">F : A Mod \to Vect</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> that sends a module to its underlying vector space, we have a natural isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>→</mo><mi>Vect</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> End( A Mod \to Vect ) \simeq A </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>.</p> </div> <p>Additional structure on the algebra <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> corresponds to addition structure on its <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> as indicated in the following table:</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> for <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> over <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a></th></tr></thead><tbody><tr><td style="text-align: left;"><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></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-module">2-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sesquialgebra">sesquialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a> = <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/presentable+category">presentable category</a> with <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>-preserving <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a></td><td style="text-align: left;">strict <a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a>: <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/hopfish+algebra">hopfish algebra</a> (correct version)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> (without fiber functor)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebra">weak Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> with generalized <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+bialgebra">quasitriangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+bialgebra">triangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+Hopf+algebra">quasitriangular Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebra">triangular Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superalgebra">supercommutative</a> <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> and Schur smallness</td></tr> <tr><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+double">Drinfeld double</a></td><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+center">Drinfeld center</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trialgebra">trialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td></tr> </tbody></table> <p><strong>2-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+categories">module categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></th><th><a class="existingWikiWord" href="/nlab/show/2-category+of+module+categories">2-category of module categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><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></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-algebra">2-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-module">3-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> (with some duality and strictness structure)</td></tr> </tbody></table> <p><strong>3-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+2-categories">module 2-categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+2-categories">monoidal 2-categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></th><th><a class="existingWikiWord" href="/nlab/show/3-category+of+module+2-categories">3-category of module 2-categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><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></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-algebra">3-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/4-module">4-module</a></td></tr> </tbody></table> </div> <h4 id="for_linear_group_representations">For linear group representations</h4> <p>Still for the special case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">V = Vect</annotation></semantics></math>, let now <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> be a <a class="existingWikiWord" href="/nlab/show/group">group</a> and let the algebra in question specifically be its <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">A = k[G]</annotation></semantics></math> . Then the category of linear <a class="existingWikiWord" href="/nlab/show/representation">representation</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> Rep(G) \simeq k[G] Mod </annotation></semantics></math></div> <p>and we obtain</p> <div class="num_cor"> <h6 id="corollary_2">Corollary</h6> <p><strong>(Tannaka duality for linear group representations)</strong></p> <p>There is a natural isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Vect</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> End(Rep(G) \to Vect) \simeq k[G] \,. </annotation></semantics></math></div></div> <h4 id="Coalgebras">For coalgebra comodules</h4> <p>If for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> we choose not <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> but its <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Vect</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Vect^{op}</annotation></semantics></math>, then a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> object <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msup><mi>Mod</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A Mod^{op}</annotation></semantics></math>, rather) is the category of comodules over this coalgebra. Again we have a forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>A</mi><mi>Mod</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">F : A Mod \to Vect</annotation></semantics></math></p> <p>In</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>An introduction to Tannaka duality and quantum groups</em>, <a href="http://www.math.mq.edu.au/~street/CT90Como.pdf">pdf</a></li> </ul> <p>(<a href="http://www.maths.mq.edu.au/~street/CT90Como.pdf#page=40">proposition 5, page 40</a>)</p> <p>and</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em><a class="existingWikiWord" href="/nlab/show/Cat%C3%A9gories+Tannakiennes">Catégories Tannakiennes</a></em></li> </ul> <p>it is shown that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is recovered as the <a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∫</mo> <mrow><mi>N</mi><mo>∈</mo><mi>A</mi><msub><mi>Mod</mi> <mi>fin</mi></msub></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> \int^{N \in A Mod_{fin}} F(N) \otimes F(N)^* </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>, where the coend ranges over finite dimensional modules.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> itself is finite dimensional then this is yet again just a special case of the enriched Yoneda lemma for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-modules, for the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><msup><mi>FinDimVect</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">V = FinDimVect^{op}</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/opposite">opposite</a> of the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/FinDimVect">FinDimVect</a> of <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+spaces">finite-dimensional vector spaces</a>): this general statement says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is recovered as the <a class="existingWikiWord" href="/nlab/show/end">end</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><msub><mo>∫</mo> <mrow><mi>N</mi><mo>∈</mo><mi>A</mi><msub><mi>Mod</mi> <mi>fin</mi></msub></mrow></msub><mi>V</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A = \int_{N \in A Mod_{fin}} V(F(N), F(N)) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Vect</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Vect^{op}</annotation></semantics></math>. This is equivalently the <a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msup><mo>∫</mo> <mrow><mi>N</mi><mo>∈</mo><mi>A</mi><mi>Mod</mi></mrow></msup><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq \int^{N \in A Mod}( Vect(F(N), F(N))) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math>. Finally using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FinDimVect</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>V</mi><mo>⊗</mo><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">FinDimVect(V,W) \simeq V\otimes W^*</annotation></semantics></math> the above coend expression follows.</p> <p>As before, more work is required to show that even for <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> itself not finite dimensional, it is still recovered in terms of the above (co)end over just its finite dimensional modules.</p> <h3 id="for_lie_groupoids">For Lie groupoids</h3> <p>See <a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+Lie+groupoids">Tannaka duality for Lie groupoids</a>.</p> <h3 id="for_geometric_stacks">For geometric stacks</h3> <p>See <a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a>.</p> <h3 id="InHigherCategoryTheory">In higher category theory</h3> <p>In as far as the proof of Tannaka duality only depends on the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, the statement immediately generalizes to <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> whenever a higher generalization of the Yoneda lemma is available.</p> <p>This is notably the case for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> theory, where we have the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a>.</p> <h4 id="ForInfinityPermutations">For permutation <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>-representations</h4> <p>By applying 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>-Yoneda lemma verbatim four times in a row as above <a href="#ForPermutationRepresentations">for permutation representations</a>, we obtain the following statement for <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-permutation+representations">∞-permutation representations</a>.</p> <div class="num_theorem"> <h6 id="theorem_5">Theorem</h6> <p><strong>(Tannaka duality for <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>-permutation representations)</strong></p> <p>Let <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> be an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Rep</mi> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Func</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep_{\infty Grpd}(G) := Func(\mathbf{B}G, \infty Grpd)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-permutation+representations">∞-permutation representations</a>, the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a> from its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> to <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msub><mi>Rep</mi> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">F : Rep_{\infty Grpd}(G) \to \infty Grpd</annotation></semantics></math> be the fiber functor that remembers the underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid. Then there is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalence in a quasi-category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><msub><mi>Rep</mi> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grp</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> End(Rep_{\infty Grpd}(G) \to \infty Grp) \simeq G \,. </annotation></semantics></math></div></div> <p>As before, this holds immediately even for representations of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a></p> <div class="num_theorem"> <h6 id="theorem_6">Theorem</h6> <p><strong>(Tannaka duality for <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>-permutation representations)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Rep</mi> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep_{\infty Grpd}(C) := Func(C,\infty Grpd)</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/object">object</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>c</mi></msub><mo>:</mo><msub><mi>Rep</mi> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">F_c : Rep_{\infty Grpd}(C) \to \infty Grpd</annotation></semantics></math> for the corresponding fiber functor.</p> <p>Then there is a natural equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>c</mi></msub><mo>,</mo><msub><mi>F</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> hom(F_c, F_{c'}) \simeq C(c,c') </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> </div> <h4 id="InfinityGaloisTheory"><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>-Galois theory</h4> <p>As a special case of this, we obtain a statement about <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>-Galois theory. For details and background see <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy groups in an (∞,1)-topos</a>. In that context one finds for a <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">LConst(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>s 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 equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Rep</mi> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep_{\infty Grpd}(\Pi(X))</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</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>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> a point, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub><mo>:</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">F_x : LConst(X) \to \infty Grpd</annotation></semantics></math> for the corresponding fiber functor.</p> <p>Then we have</p> <div class="num_theorem"> <h6 id="theorem_7">Theorem</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>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> there is a natural <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>F</mi> <mi>x</mi></msub></mrow></mover><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Aut</mi> <mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> End(LConst(X) \stackrel{F_x}{\to} \infty Grpd) \simeq \mathbf{B} Aut_{\Pi(X)}(x) \,. </annotation></semantics></math></div> <p>In particular we have <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mi>End</mi><mo stretchy="false">(</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>F</mi> <mi>x</mi></msub></mrow></mover><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_n End(LConst(X) \stackrel{F_x}{\to} \infty Grpd) \simeq \pi_n(X,x) \,. </annotation></semantics></math></div></div> <p>More on this is at <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos+--+structures">cohesive (∞,1)-topos – structures</a> in the section <a href="http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#Homotopy">Galois theory in a cohesive (∞,1)-topos</a></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannakian+category">Tannakian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne%27s+theorem+on+tensor+categories">Deligne's theorem on tensor categories</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>An introduction to Tannaka duality and quantum groups</em>, <a href="http://www.math.mq.edu.au/~street/CT90Como.pdf">pdf</a></p> </li> <li> <p>B.J. Day, <em>Enriched Tannaka reconstruction</em>, J. Pure Appl. Algebra <strong>108</strong> (1996) 17-22, <a href="http://dx.doi.org/10.1016/0022-4049(95)00039-9">doi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em><a class="existingWikiWord" href="/nlab/show/Cat%C3%A9gories+Tannakiennes">Catégories Tannakiennes</a></em></p> </li> </ul> <p>The following paper shortens Deligne’s proof</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Rosenberg">Alexander L. Rosenberg</a>, <em>The existence of fiber functors</em>, The Gelfand Mathematical Seminars, 1996–1999, 145–154, Birkhäuser, Boston 2000.</li> </ul> <p>Deligne’s proof in turn fills the gap in the seminal work with the same title</p> <ul> <li>N. Saavedra Rivano, “Catégories Tannakiennes.” <em>Bulletin de la Société Mathématique de France</em> 100 (1972): 417-430. <a href="https://eudml.org/doc/87193">EuDML</a></li> </ul> <p>A revival in algebraic geometry related to the theory of mixed motives was marked by</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/P.+Deligne">P. Deligne</a>, <a class="existingWikiWord" href="/nlab/show/J.+Milne">J. Milne</a>, <em>Tannakian categories</em>, Springer Lecture Notes in Math. <strong>900</strong>, 1982, pp. 101-228, retyped <a href="http://jmilne.org/math/xnotes/tc.pdf">pdf</a></li> </ul> <p>Analogous discussion for <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28infinity%2C1%29-categories">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-categories">stable (infinity,1)-categories</a> includes</p> <ul> <li id="Wallbridge12"> <p><a class="existingWikiWord" href="/nlab/show/James+Wallbridge">James Wallbridge</a>, <em>Tannaka duality over ring spectra</em> (<a href="https://arxiv.org/abs/1204.5787">arXiv:1204.5787</a>)</p> </li> <li id="Iwanari14"> <p><a class="existingWikiWord" href="/nlab/show/Isamu+Iwanari">Isamu Iwanari</a>, <em>Tannaka duality and stable infinity-categories</em> (<a href="http://arxiv.org/abs/1409.3321">arXiv:1409.3321</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/Spectral+Algebraic+Geometry">Spectral Algebraic Geometry</a>, Chap. 9</p> </li> </ul> <p>Ulbrich made a major contribution at the coalgebra and Hopf algebra level</p> <ul> <li>K-H. Ulbrich, <em>On Hopf algebras and rigid monoidal categories</em>, in special volume, Hopf algebras, Israel J. Math. <strong>72</strong> (1990), no. 1-2, 252–256, <a href="http://dx.doi.org/10.1007/BF02764622">doi</a></li> </ul> <p>This Hopf-direction has been advanced by many authors including</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/S.+L.+Woronowicz">S. L. Woronowicz</a>, <em>Tannaka-Krein duality for compact matrix pseudogroups. Twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(N)</annotation></semantics></math> groups</em>, Inventiones Mathematicae <strong>93</strong>, No. 1, 35-76, <a href="http://dx.doi.org/10.1007/BF01393687">doi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shahn+Majid">Shahn Majid</a>, <em>Foundations of quantum group theory</em>, chapter 9</p> </li> <li> <p>Phung Ho Hai, <em>Tannaka-Krein duality for Hopf algebroids</em>, Israel J. Math. <strong>167</strong> (1):193–225 (2008) <a href="http://arxiv.org/abs/math/0206113">math.QA/0206113</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Volodymyr+Lyubashenko">Volodymyr V. Lyubashenko</a>, <em>Modular transformations and tensor categories</em>, J. Pure Appl. Algebra <strong>98</strong> (1995) 279–327 <a href="https://doi.org/10.1016/0022-4049(94)00045-K">doi</a>; <em>Squared Hopf algebras and reconstruction theorems</em>, Proc. Workshop “Quantum Groups and Quantum Spaces” (Warszawa), Banach Center Publ. <strong>40</strong>, Inst. Math. Polish Acad. Sci. (1997) 111–137, <a href="http://arxiv.org/abs/q-alg/9605035">q-alg/9605035</a>; <em>Squared Hopf algebras</em>, Mem. Amer. Math. Soc. <strong>142</strong> (677):x 180, 1999; Алгебры Хопфа и вектор-симметрии, УМН, 41:5(251) (1986), 185–186, <a href="http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=2247&what=fullt&option_lang=rus">pdf</a>, transl. as: <em>Hopf algebras and vector symmetries</em>, Russian Math. Surveys 41(5):153154, 1986.</p> </li> <li> <p>A. Bruguières, <em>Théorie tannakienne non commutative</em>, Comm. Algebra <strong>22</strong>, 5817–5860, 1994</p> </li> <li> <p>K. Szlachanyi, <em>Fiber functors, monoidal sites and Tannaka duality for bialgebroids</em>, <a href="http://arxiv.org/abs/0907.1578">arxiv/0907.1578</a></p> </li> <li> <p>B. Day, R. Street, <em>Quantum categories, star autonomy, and quantum groupoids</em>, in “Galois theory, Hopf algebras, and semiabelian categories”, Fields Inst. Comm. <strong>43</strong> (2004) 187-225</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Sch%C3%A4ppi">Daniel Schäppi</a>, <em>The formal theory of Tannaka duality</em>, <a href="http://arxiv.org/abs/1112.5213">arxiv/1112.5213</a>, superseding earlier <em>Tannaka duality for comonoids in cosmoi</em>, <a href="http://arxiv.org/abs/0911.0977">arXiv:0911.0977</a></p> </li> </ul> <p>A generalization of several classical reconstruction theorems with nontrivial <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Rosenberg">Alexander L. Rosenberg</a>, <em><a class="existingWikiWord" href="/nlab/show/Reconstruction+of+Groups">Reconstruction of groups</a></em>, Selecta Math. (N.S.) <strong>9</strong>, 1 (2003), 101–118, <a href="http://dx.doi.org/10.1007/s00029-003-0322-x">doi</a>.</li> </ul> <p>Categorically oriented notes were written also by Pareigis, emphasising on using <a class="existingWikiWord" href="/nlab/show/coend">Coend</a> in dual picture. His works can be found <a href="http://www.mathematik.uni-muenchen.de/~pareigis/pa_schft.html">here</a> but the most important is the chapter 3 of his online book</p> <ul> <li>Bodo Pareigis, <em>Quantum groups and noncommutative geometry</em>, Chapter 3: Representation theory, reconstruction and Tannaka duality, <a href="http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/98SS/Quantum_Groups/LN3.PDF">pdf</a></li> </ul> <p>A very neat Tannaka theorem for stacks is proved in</p> <ul> <li> <p><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>)</p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bertrand+Toen">Bertrand Toen</a>, <a href="http://www.msri.org/publications/ln/msri/2002/hodgetheory/toen/2/index.html">Higher Tannaka duality</a>, MSRI 2002 (talk, video)</p> </li> <li> <p>Moshe Kamensky, <em>Model theory and the Tannakian formalism</em>, <a href="http://arxiv.org/abs/0908.0604">arXiv:0908.0604</a>; <em>Tannakian formalism over fields with operators</em>, <a href="http://arxiv.org/abs/1111.7285">arxiv/1111.7285</a></p> </li> <li> <p>H. Fukuyama, I. Iwanari, <em>Monoidal infinity category of complexes from Tannakian viewpoint</em>, <a href="http://arxiv.org/abs/1004.3087">arxiv/1004.3087</a></p> </li> <li> <p>remark of <a href="http://mathoverflow.net/questions/3446/tannakian-formalism/3467#3467">Ben-Zvi on Tannaka reconstruction for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Kazhdan">David Kazhdan</a>, Michael Larsen, Yakov Varshavsky, <em>The Tannakian formalism and the Langlands conjectures</em>, <a href="http://arxiv.org/abs/1006.3864">arxiv/1006.3864</a></p> </li> </ul> <p>The classical articles are</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tadao+Tannaka">Tadao Tannaka</a>, <em>Über den Dualitätssatz der nichtkommutativen topologischen Gruppen</em>, Tohoku Math. J. 45 (1938), n. 1, 1–12 (project euclid has only Tohoku new series!), see <a class="existingWikiWord" href="/nlab/show/Tannaka-Krein+theorem">Tannaka-Krein theorem</a>.</p> </li> <li> <p>N. Tatsuuma, <em>A duality theorem for locally compact groups</em>, J. Math., Kyoto Univ. 6 (1967), 187–293.</p> </li> <li> <p>Nobuhiko Tatsuuma, <em>Duality theorem for locally compact groups and some related topics</em>, Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), 387–408. Colloq. Internat. CNRS, 274, Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1979. ISBN: 2-222-02441-2.</p> </li> <li> <p>M.G. Krein, <em>A principle of duality for bicompact groups and quadratic block algebras</em>, Doklady AN SSSR <strong>69</strong> (1949), 725–728.</p> </li> <li> <p>Eiichi Abe, <em>Dualité de Tannaka des groupes algébriques</em>, Tohoku Mathematical Journal. Volume 12, Number 2 (1960), 327-332.</p> </li> </ul> <p>The Tannaka-type reconstruction in quantum field theory see <a class="existingWikiWord" href="/nlab/show/Doplicher-Roberts+reconstruction+theorem">Doplicher-Roberts reconstruction theorem</a>.</p> <p>Tannaka duality in the context of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/James+Wallbridge">James Wallbridge</a>, <em>Higher Tannaka duality</em>, PhD thesis, Adelaide/Toulouse (2011) (<a href="http://digital.library.adelaide.edu.au/dspace/bitstream/2440/69436/1/02whole.pdf">Adelaide University repository</a>, <a href="http://arxiv.org/abs/1204.5787">arXiv:1204.5787</a>)</li> </ul> <p>Tannaka duality for dg-categories is studied in</p> <ul> <li> <p>J.P.Pridham, <em>Tannaka duality for enhanced triangulated categories</em>, <a href="http://arxiv.org/abs/1309.0637">arxiv/1309.0637</a></p> </li> <li> <p>MathOverflow, <a href="http://mathoverflow.net/questions/30453/does-the-tannaka-krein-theorem-come-from-an-equivalence-of-2-categories">Does the Tannaka-Krein theorem come from an equivalence of 2-categories?</a></p> </li> </ul> <p>See also</p> <ul> <li>Lukas Rollier. <em>Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures</em> (2024). (<a href="https://arxiv.org/abs/2405.03364">arXiv:2405.03364</a>).</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 7, 2024 at 07:57:54. 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