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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Tannaka duality} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory} [[!include category theory - contents]] \hypertarget{duality}{}\paragraph*{{Duality}}\label{duality} [[!include duality - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak \noindent\hyperlink{ForPermutationRepresentations}{For $G$-Sets}\dotfill \pageref*{ForPermutationRepresentations} \linebreak \noindent\hyperlink{ForVModules}{For $V$-modules}\dotfill \pageref*{ForVModules} \linebreak \noindent\hyperlink{ForAlgebraModules}{For algebra modules}\dotfill \pageref*{ForAlgebraModules} \linebreak \noindent\hyperlink{for_linear_group_representations}{For linear group representations}\dotfill \pageref*{for_linear_group_representations} \linebreak \noindent\hyperlink{Coalgebras}{For coalgebra comodules}\dotfill \pageref*{Coalgebras} \linebreak \noindent\hyperlink{for_lie_groupoids}{For Lie groupoids}\dotfill \pageref*{for_lie_groupoids} \linebreak \noindent\hyperlink{for_geometric_stacks}{For geometric stacks}\dotfill \pageref*{for_geometric_stacks} \linebreak \noindent\hyperlink{InHigherCategoryTheory}{In higher category theory}\dotfill \pageref*{InHigherCategoryTheory} \linebreak \noindent\hyperlink{ForInfinityPermutations}{For permutation $\infty$-representations}\dotfill \pageref*{ForInfinityPermutations} \linebreak \noindent\hyperlink{InfinityGaloisTheory}{$\infty$-Galois theory}\dotfill \pageref*{InfinityGaloisTheory} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} [[Tannaka]] duality or \emph{Tannaka [[reconstruction theorem]]s} are statements of the form: if $A$ is a symmetry object (e.g. a [[locally compact topological group]], [[Hopf algebra]]), [[representation|represented]] on objects in a [[category]] $D$, one may \emph{reconstruct} $A$ from knowledge of the [[endomorphism]]s of the forgetful functor -- the \textbf{[[fiber functor]]} -- \begin{displaymath} F : Rep_D(A) \to D \end{displaymath} from the [[category]] $Rep_D(A)$ of [[representation]]s of $A$ on [[object]]s of $D$ that remembers these underlying objects. In a generalization, called mixed Tannakian formalism, not a single [[fiber functor]], but a family of [[fiber functors]] over different bases is needed for a reconstruction. There is a general-abstract and a concrete aspect to this. The general abstract one says that an algebra $A$ is reconstructible from the [[fiber functor]] on the category of \emph{all} its modules. The concrete one says that in nice cases it is reconstructible from the category of \emph{dualizable} (finite dimensional) modules, even if it is itself not finite dimensional. More precisely, let $V$ be any [[enriched category theory|enriching category]] (a [[locally small category|locally small]] [[closed monoidal category|closed]] [[symmetric monoidal category]] with all [[limit]]s). Then \begin{enumerate}% \item for \begin{itemize}% \item $A$ a [[monoid]] in $V$; \item $A Mod$ the $V$-[[enriched category]] of \emph{all} $A$-[[module]]s in $V$; \item $F : A Mod \to V$ the [[forgetful functor|forgetful]] \emph{[[fiber functor]]} ; \end{itemize} $A$ can be reconstructed as the object of [[enriched functor category|enriched endomorphisms]] of $F$, given by the [[end]] \begin{displaymath} A \simeq End(F) := \int_{N \in A Mod} V(F(N), F(N)) \,. \end{displaymath} This is just the [[enriched Yoneda lemma]] in a slight disguise. \item In good cases, this [[end]] is computed already by restriction to the [[full subcategory]] $A Mod_{dual}$ of [[dual object|dualizable modules]] \begin{displaymath} \cdots \simeq \int_{N \in A Mod_{dual}} V(F(N), F(N)) \,. \end{displaymath} \end{enumerate} \hypertarget{statement}{}\subsection*{{Statement}}\label{statement} So far the following examples concern the abstract algebraic aspect of Tannaka duality only, which is narrated here as a consequence of the [[enriched Yoneda lemma]] in [[enriched category theory]]. Some of the Tannaka duality theorems involve subtle harmonic analysis. \hypertarget{ForPermutationRepresentations}{}\subsubsection*{{For $G$-Sets}}\label{ForPermutationRepresentations} A simple case of Tannaka duality is that of [[G-sets]] of a [[group]], i.e. representations on a [[set]]. In this case, Tannaka duality follows entirely from repeated application of the ordinary [[Yoneda lemma]]. \begin{theorem} \label{}\hypertarget{}{} \textbf{(Tannaka duality for [[G-sets]])} Let $G$ be a [[group]], write $G Set$ for the [[category]] of [[G-sets]] and \begin{displaymath} F \colon G Set \longrightarrow Set \end{displaymath} for the [[forgetful functor]] that sends a [[G-set]] to its underlying [[set]]. Then there is a canonical [[group homomorphism|group-]][[isomorphism]] \begin{displaymath} Aut(F) \;\simeq\; G \,. \end{displaymath} identifying the [[automorphism group]] of $F$ (the group of [[natural isomorphisms]] from $F$ to itself) with $G$. \end{theorem} \begin{proof} With a bit of evident abuse of notation, the proof is a one-line sequence of applications of the [[Yoneda lemma]]: we show $End(F) \cong G$, i.e., each endomorphism on $F$ is invertible, so $End(F) = Aut(F) \cong G$. Write $C \coloneqq Set^G = G Set$. Observe that the functor $F \colon C \to Set$ is the [[representable functor|representable]] $F = C(G, -)$. Then the argument is \begin{displaymath} End(F) = Set^C(F, F) \cong Set^C(C(G, -), C(G, -)) \cong C(G, G) \cong G. \end{displaymath} The ``$G$'' here is used in multiple senses, but each sense is deducible from context. \end{proof} \begin{proof} We repeat the same proof, but with more notational details on what the entities involved in each step are precisely. Let $\mathbf{B}G$ be the [[delooping]] [[groupoid]] of the [[group]] $G$. Then \begin{displaymath} G Set \;=\; Func(\mathbf{B}G^{op}, Set) \,. \end{displaymath} The canonical inclusion $i : {*} \to \mathbf{B}G$ induces the [[fiber functor]] \begin{displaymath} Func(i,Set) : G Set \to Set \end{displaymath} which evaluates a functor $\rho : \mathbf{B}G^{op} \to Set$ on the unique object of $\mathbf{B}G$. By the [[Yoneda lemma]] this is the same as homming out of the functor [[representable functor|represented by]] that unique object \begin{displaymath} Func(i,Set) = Hom_{PSh(\mathbf{B}G)}(Y_{\mathbf{B}G} {*}, -) \,, \end{displaymath} where $Y_{\mathbf{B}G} : \mathbf{B}G \to PSh(\mathbf{B}G)$ is the [[Yoneda embedding]]. But this way we see that $Func(i,Set) : PSh(\mathbf{B}G) \to Set$ is itself a representable functor in the presheaf category $PSh(PSh(\mathbf{B}G)^{op})$ \begin{displaymath} Func(i,Set) = Y_{\mathbf{PSh(\mathbf{B}G)^{op}}} Y_{\mathbf{B}G} * \,. \end{displaymath} So applying the [[Yoneda lemma]] twice, we find that \begin{displaymath} \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} \end{displaymath} \end{proof} Notice that the proof in no way used the fact that $G$ was assumed to be a [[group]], but only that $G$ is a [[monoid]]. So the statement holds just as well for arbitrary monoids. But moreover, as the long-winded proof above makes manifest, even more abstractly the proof really only depended on the fact that the [[delooping]] $\mathbf{B}G$ is a [[small category]]. 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: \begin{theorem} \label{}\hypertarget{}{} \textbf{(Tannaka duality for permutation representations of categories)} Let $C$ be a [[locally small category]] and $Rep_{Set}(C) := Func(C,Set)$ the [[functor category]]. For every object $c \in C$ let $F_c : Rep_{Set}(C) \to Set$ be the fiber-functor that evaluates at $c$. Then we have a [[natural isomorphism]] \begin{displaymath} Hom(F_c,F_{c'}) \simeq Hom_C(c,c') \,. \end{displaymath} \end{theorem} \hypertarget{ForVModules}{}\subsubsection*{{For $V$-modules}}\label{ForVModules} Let $V$ be a ([[locally small category|locally small]]) [[closed monoidal category|closed]] [[symmetric monoidal category]], so that $V$ is enriched in itself via its [[internal hom]]. Observe that the setup, statement and proof of Tannaka duality for permutation representations given above is the special case for $V =$ [[Set]] of a statement verbatim the same in $V$-[[enriched category theory]], with the ordinary [[functor category]] replaced everywhere by the $V$-[[enriched functor category]]: Then the statement says: \begin{theorem} \label{}\hypertarget{}{} \textbf{(Tannaka duality for $V$-modules over $V$-algebras)} For $A$ a [[monoid]] in $V$ with [[delooping]] $V$-[[enriched category]] $\mathbf{B}A$, and with \begin{displaymath} A Mod := [\mathbf{B}A,V] \end{displaymath} the [[enriched functor category]] that encodes the $V$-[[module]]s of $A$, we have that the $V$-enriched [[endomorphism]] algebra $End(F) := [F,F]$ of the $V$-[[enriched functor]] $F : Rep(A) \to V$ is [[natural isomorphism|naturally isomorphic]] to $V$ \begin{displaymath} End(A Mod \stackrel{F}{\to} V) \simeq A \,. \end{displaymath} \end{theorem} \begin{proof} Apply the [[enriched Yoneda lemma]] verbatim as for the statement about permutation representations as above. \end{proof} Notice that the [[endomorphism]] object here is taken in the sense of enriched category theory, as described at [[enriched functor category]]. It is given by the [[end]] expression \begin{displaymath} End(F) = \int_{N \in A Mod} V(F(N), F(N)) \,. \end{displaymath} The case of permutation representations is re-obtained by setting $V =$ [[Set]]. As before, the same proof actually shows the following more general statement \begin{theorem} \label{}\hypertarget{}{} \textbf{(Tannaka duality for $V$-modules over $V$-algebroids)} Let $C$ be a $V$-[[enriched category]] (a ``$V$-[[algebroid]]''). Write $C Mod := [C,V]$ for the $V$-[[enriched functor category]]. For every [[object]] $c \in C$ write $F_c : C Mod \to V$ for the [[fiber functor]] that evaluates at $C$. Then we have [[natural isomorphism]]s \begin{displaymath} hom(F_c, F_{c'}) \simeq C(c,c') \,. \end{displaymath} \end{theorem} From this statement of Tannaka duality in $V$-enriched category theory now various special cases of interest follow, by simply choosing suitable enrichement categories $V$. \hypertarget{ForAlgebraModules}{}\paragraph*{{For algebra modules}}\label{ForAlgebraModules} The general case of Tannaka duality for $V$-modules described \hyperlink{ForVModules}{above} restricts to the classical case of Tannaka duality for linear representations by setting $V :=$ [[Vect]], the category of [[vector space]]s over some fixed [[ground field]]. In this case the above says \begin{cor} \label{}\hypertarget{}{} \textbf{(Tannaka duality for linear modules)} For $A$ an [[algebra]] and $A Mod$ its category of [[modules]], and for $F : A Mod \to Vect$ the [[fiber functor]] that sends a module to its underlying vector space, we have a natural isomorphism \begin{displaymath} End( A Mod \to Vect ) \simeq A \end{displaymath} in [[Vect]]. \end{cor} Additional structure on the algebra $A$ corresponds to addition structure on its [[category of modules]] as indicated in the following table: [[!include structure on algebras and their module categories - table]] \hypertarget{for_linear_group_representations}{}\paragraph*{{For linear group representations}}\label{for_linear_group_representations} Still for the special case $V = Vect$, let now $G$ be a [[group]] and let the algeba in question specifically be its [[group algebra]] $A = k[G]$ . Then the category of linear [[representation]]s of $G$ is \begin{displaymath} Rep(G) \simeq k[G] Mod \end{displaymath} and we obtain \begin{cor} \label{}\hypertarget{}{} \textbf{(Tannaka duality for linear group representations)} There is a natural isomorphism \begin{displaymath} End(Rep(G) \to Vect) \simeq k[G] \,. \end{displaymath} \end{cor} \hypertarget{Coalgebras}{}\paragraph*{{For coalgebra comodules}}\label{Coalgebras} If for $V$ we choose not [[Vect]] but its [[opposite category]] $Vect^{op}$, then a [[monoid]] object $A$ in $V$ is a [[coalgebra]] and $A Mod$ (or $A Mod^{op}$, rather) is the category of comodules over this coalgebra. Again we have a forgetful functor $F : A Mod \to Vect$ In \begin{itemize}% \item [[André Joyal]], [[Ross Street]], \emph{An introduction to Tannaka duality and quantum groups}, \href{http://www.math.mq.edu.au/~street/CT90Como.pdf}{pdf} \end{itemize} (\href{http://www.maths.mq.edu.au/~street/CT90Como.pdf#page=40}{proposition 5, page 40}) and \begin{itemize}% \item [[Pierre Deligne]], \emph{[[Catégories Tannakiennes]]} \end{itemize} it is shown that $A$ is recovered as the [[coend]] \begin{displaymath} \int^{N \in A Mod_{fin}} F(N) \otimes F(N)^* \end{displaymath} in [[Vect]], where the coend ranges over finite dimensional modules. If $A$ itself is finite dimensional then this is yet again just a special case of the enriched Yoneda lemma for $V$-modules, for the case $V = FinVect^{op}$: this general statement says that $A$ is recovered as the [[end]] \begin{displaymath} A = \int_{N \in A Mod_{fin}} V(F(N), F(N)) \end{displaymath} in $Vect^{op}$. This is equivalently the [[coend]] \begin{displaymath} \cdots \simeq \int^{N \in A Mod}( Vect(F(N), F(N))) \end{displaymath} in $Vect$. Finally using that $FinVect(V,W) \simeq V\otimes W^*$ the above coend expression follows. As before, more work is required to show that even for $A$ itself not finite dimensional, it is still recovered in terms of the above (co)end over just its finite dimensional modules. \hypertarget{for_lie_groupoids}{}\subsubsection*{{For Lie groupoids}}\label{for_lie_groupoids} See [[Tannaka duality for Lie groupoids]]. \hypertarget{for_geometric_stacks}{}\subsubsection*{{For geometric stacks}}\label{for_geometric_stacks} See [[Tannaka duality for geometric stacks]]. \hypertarget{InHigherCategoryTheory}{}\subsubsection*{{In higher category theory}}\label{InHigherCategoryTheory} In as far as the proof of Tannaka duality only depends on the [[Yoneda lemma]], the statement immediately generalizes to [[higher category theory]] whenever a higher generalization of the Yoneda lemma is available. This is notably the case for [[(∞,1)-category]] theory, where we have the [[(∞,1)-Yoneda lemma]]. \hypertarget{ForInfinityPermutations}{}\paragraph*{{For permutation $\infty$-representations}}\label{ForInfinityPermutations} By applying the $(\infty,1)$-Yoneda lemma verbatim four times in a row as above \hyperlink{ForPermutationRepresentations}{for permutation representations}, we obtain the following statement for [[∞-permutation representations]]. \begin{theorem} \label{}\hypertarget{}{} \textbf{(Tannaka duality for $\infty$-permutation representations)} Let $G$ be an [[∞-group]] and $Rep_{\infty Grpd}(G) := Func(\mathbf{B}G, \infty Grpd)$ the [[category of representations|category of]] [[∞-permutation representations]], the [[(∞,1)-category of (∞,1)-functors]] from its [[delooping]] [[∞-groupoid]] to [[∞Grpd]]. Let $F : Rep_{\infty Grpd}(G) \to \infty Grpd$ be the fiber functor that remembers the underlying $\infty$-groupoid. Then there is an [[equivalence in a quasi-category]] \begin{displaymath} End(Rep_{\infty Grpd}(G) \to \infty Grp) \simeq G \,. \end{displaymath} \end{theorem} As before, this holds immediately even for representations of [[(∞,1)-categories]] \begin{theorem} \label{}\hypertarget{}{} \textbf{(Tannaka duality for $\infty$-permutation representations)} Let $c$ be an [[(∞,1)-category]] and $Rep_{\infty Grpd}(C) := Func(C,\infty Grpd)$. For $c \in C$ an [[object]], write $F_c : Rep_{\infty Grpd}(C) \to \infty Grpd$ for the corresponding fiber functor. Then there is a natural equivalence \begin{displaymath} hom(F_c, F_{c'}) \simeq C(c,c') \end{displaymath} in [[∞Grpd]]. \end{theorem} \hypertarget{InfinityGaloisTheory}{}\paragraph*{{$\infty$-Galois theory}}\label{InfinityGaloisTheory} As a special case of this, we obtain a statement about $\infty$-Galois theory. For details and background see [[homotopy groups in an (∞,1)-topos]]. In that context one finds for a [[locally contractible space]] $X$ that the [[∞-groupoid]] $LConst(X)$ of [[locally constant ∞-stack]]s on $X$ is equivalent to $Rep_{\infty Grpd}(\Pi(X))$, where $\Pi(X)$ is the [[fundamental ∞-groupoid]] of $X$. For $x \in X$ a point, write $F_x : LConst(X) \to \infty Grpd$ for the corresponding fiber functor. Then we have \begin{theorem} \label{}\hypertarget{}{} For $x \in X$ there is a natural [[weak homotopy equivalence]] \begin{displaymath} End(LConst(X) \stackrel{F_x}{\to} \infty Grpd) \simeq \mathbf{B} Aut_{\Pi(X)}(x) \,. \end{displaymath} In particular we have [[natural isomorphisms]] of [[homotopy group]]s \begin{displaymath} \pi_n End(LConst(X) \stackrel{F_x}{\to} \infty Grpd) \simeq \pi_n(X,x) \,. \end{displaymath} \end{theorem} More on this is at [[cohesive (∞,1)-topos -- structures]] in the section \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Tannakian category]] \item [[Deligne's theorem on tensor categories]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[André Joyal]], [[Ross Street]], \emph{An introduction to Tannaka duality and quantum groups}, \href{http://www.math.mq.edu.au/~street/CT90Como.pdf}{pdf} \item B.J. Day, \emph{Enriched Tannaka reconstruction}, J. Pure Appl. Algebra \textbf{108} (1996) 17-22, \item [[Pierre Deligne]], \emph{[[Catégories Tannakiennes]]} \end{itemize} The following paper shortens Deligne's proof \begin{itemize}% \item [[Alexander Rosenberg|Alexander L. Rosenberg]], \emph{The existence of fiber functors}, The Gelfand Mathematical Seminars, 1996--1999, 145--154, Birkh\"a{}user, Boston 2000. \end{itemize} Deligne's proof in turn fills the gap in the seminal work with the same title \begin{itemize}% \item N. Saavedra Rivano, ``Cat\'e{}gories Tannakiennes.'' \emph{Bulletin de la Soci\'e{}t\'e{} Math\'e{}matique de France} 100 (1972): 417-430. \href{https://eudml.org/doc/87193}{EuDML} \end{itemize} A revival in algebraic geometry related to the theory of mixed motives was marked by \begin{itemize}% \item [[P. Deligne]], [[J. Milne]], \emph{Tannakian categories}, Springer Lecture Notes in Math. \textbf{900}, 1982, pp. 101-228, retyped \href{http://jmilne.org/math/xnotes/tc.pdf}{pdf} \end{itemize} Analogous discussion for [[symmetric monoidal (infinity,1)-categories|symmetric monoidal]] [[stable (infinity,1)-categories]] includes \begin{itemize}% \item [[James Wallbridge]], \emph{Tannaka duality over ring spectra} (\href{https://arxiv.org/abs/1204.5787}{arXiv:1204.5787}) \item [[Isamu Iwanari]], \emph{Tannaka duality and stable infinity-categories} (\href{http://arxiv.org/abs/1409.3321}{arXiv:1409.3321}) \item [[Jacob Lurie]], [[Spectral Algebraic Geometry]], Chap. 9 \end{itemize} Ulbrich made a major contribution at the coalgebra and Hopf algebra level \begin{itemize}% \item K-H. Ulbrich, \emph{On Hopf algebras and rigid monoidal categories}, in special volume, Hopf algebras, Israel J. Math. \textbf{72} (1990), no. 1-2, 252--256, \href{http://dx.doi.org/10.1007/BF02764622}{doi} \end{itemize} This Hopf-direction has been advanced by many authors including \begin{itemize}% \item [[S. L. Woronowicz]], \emph{Tannaka-Krein duality for compact matrix pseudogroups. Twisted $SU(N)$ groups}, Inventiones Mathematicae \textbf{93}, No. 1, 35-76, \href{http://dx.doi.org/10.1007/BF01393687}{doi} \item [[Shahn Majid]], \emph{Foundations of quantum group theory}, chapter 9 \item Phung Ho Hai, \emph{Tannaka-Krein duality for Hopf algebroids}, Israel J. Math. \textbf{167} (1):193--225 (2008) \href{http://arxiv.org/abs/math/0206113}{math.QA/0206113} \item [[Volodymyr Lyubashenko|Volodymyr V. Lyubashenko]], \emph{Modular transformations and tensor categories}, J. Pure Appl. Algebra \textbf{98} (1995) 279–327 ; \emph{Squared Hopf algebras and reconstruction theorems}, Proc. Workshop ``Quantum Groups and Quantum Spaces'' (Warszawa), Banach Center Publ. \textbf{40}, Inst. Math. Polish Acad. Sci. (1997) 111--137, \href{http://arxiv.org/abs/q-alg/9605035}{q-alg/9605035}; \emph{Squared Hopf algebras}, Mem. Amer. Math. Soc. \textbf{142} (677):x 180, 1999; -, , 41:5(251) (1986), 185--186, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=2247&what=fullt&option_lang=rus}{pdf}, transl. as: \emph{Hopf algebras and vector symmetries}, Russian Math. Surveys 41(5):153154, 1986. \item A. Brugui\`e{}res, \emph{Th\'e{}orie tannakienne non commutative}, Comm. Algebra \textbf{22}, 5817--5860, 1994 \item K. Szlachanyi, \emph{Fiber functors, monoidal sites and Tannaka duality for bialgebroids}, \href{http://arxiv.org/abs/0907.1578}{arxiv/0907.1578} \item B. Day, R. Street, \emph{Quantum categories, star autonomy, and quantum groupoids}, in ``Galois theory, Hopf algebras, and semiabelian categories'', Fields Inst. Comm. \textbf{43} (2004) 187-225 \item [[Daniel Schäppi]], \emph{The formal theory of Tannaka duality}, \href{http://arxiv.org/abs/1112.5213}{arxiv/1112.5213}, superseding earlier \emph{Tannaka duality for comonoids in cosmoi}, \href{http://arxiv.org/abs/0911.0977}{arXiv:0911.0977} \end{itemize} A generalization of several classical reconstruction theorems with nontrivial [[functional analysis]] is in \begin{itemize}% \item [[Alexander Rosenberg|Alexander L. Rosenberg]], \emph{[[Reconstruction of Groups|Reconstruction of groups]]}, Selecta Math. (N.S.) \textbf{9}, 1 (2003), 101--118, \href{http://dx.doi.org/10.1007/s00029-003-0322-x}{doi}. \end{itemize} Categorically oriented notes were written also by Pareigis, emphasising on using [[coend|Coend]] in dual picture. His works can be found \href{http://www.mathematik.uni-muenchen.de/~pareigis/pa_schft.html}{here} but the most important is the chapter 3 of his online book \begin{itemize}% \item Bodo Pareigis, \emph{Quantum groups and noncommutative geometry}, Chapter 3: Representation theory, reconstruction and Tannaka duality, \href{http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/98SS/Quantum_Groups/LN3.PDF}{pdf} \end{itemize} A very neat Tannaka theorem for stacks is proved in \begin{itemize}% \item [[Jacob Lurie]], \emph{Tannaka duality for geometric stacks}, (\href{http://arxiv.org/abs/math/0412266}{arXiv:math.AG/0412266}) \item [[Jacob Lurie]], \emph{[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]} \item [[Bertrand Toen]], \href{http://www.msri.org/publications/ln/msri/2002/hodgetheory/toen/2/index.html}{Higher Tannaka duality}, MSRI 2002 (talk, video) \item Moshe Kamensky, \emph{Model theory and the Tannakian formalism}, \href{http://arxiv.org/abs/0908.0604}{arXiv:0908.0604}; \emph{Tannakian formalism over fields with operators}, \href{http://arxiv.org/abs/1111.7285}{arxiv/1111.7285} \item H. Fukuyama, I. Iwanari, \emph{Monoidal infinity category of complexes from Tannakian viewpoint}, \href{http://arxiv.org/abs/1004.3087}{arxiv/1004.3087} \item remark of \href{http://mathoverflow.net/questions/3446/tannakian-formalism/3467#3467}{Ben-Zvi on Tannaka reconstruction for monoidal categories} \item [[David Kazhdan]], Michael Larsen, Yakov Varshavsky, \emph{The Tannakian formalism and the Langlands conjectures}, \href{http://arxiv.org/abs/1006.3864}{arxiv/1006.3864} \end{itemize} The classical articles are \begin{itemize}% \item [[Tadao Tannaka]], \emph{\"U{}ber den Dualit\"a{}tssatz der nichtkommutativen topologischen Gruppen}, Tohoku Math. J. 45 (1938), n. 1, 1--12 (project euclid has only Tohoku new series!), see [[Tannaka-Krein theorem]]. \item N. Tatsuuma, \emph{A duality theorem for locally compact groups}, J. Math., Kyoto Univ. 6 (1967), 187--293. \item M.G. Krein, \emph{A principle of duality for bicompact groups and quadratic block algebras}, Doklady AN SSSR \textbf{69} (1949), 725--728. \item Eiichi Abe, \emph{Dualit\'e{} de Tannaka des groupes alg\'e{}briques}, Tohoku Mathematical Journal. Volume 12, Number 2 (1960), 327-332. \end{itemize} The Tannaka-type reconstruction in quantum field theory see [[Doplicher-Roberts reconstruction theorem]]. Tannaka duality in the context of [[(∞,1)-category theory]] is discussed in \begin{itemize}% \item [[James Wallbridge]], \emph{Higher Tannaka duality}, PhD thesis, Adelaide/Toulouse (2011) (\href{http://digital.library.adelaide.edu.au/dspace/bitstream/2440/69436/1/02whole.pdf}{Adelaide University repository}, \href{http://arxiv.org/abs/1204.5787}{arXiv:1204.5787}) \end{itemize} Tannaka duality for dg-categories is studied in \begin{itemize}% \item J.P.Pridham, \emph{Tannaka duality for enhanced triangulated categories}, \href{http://arxiv.org/abs/1309.0637}{arxiv/1309.0637} \item MathOverflow, \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?} \end{itemize} [[!redirects Tannaka reconstruction]] [[!redirects Tannaka reconstruction theorem]] \end{document}