abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
Tannaka duality or Tannaka reconstruction theorems are statements of the form:
if $A$ is a symmetry object (e.g. a locally compact topological group, Hopf algebra), represented on objects in a category $D$, one may reconstruct $A$ from knowledge of the endomorphisms of the forgetful functor – the fiber functor –
from the category $Rep_D(A)$ of representations of $A$ on objects 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 all its modules. The concrete one says that in nice cases it is reconstructible from the category of dualizable (finite dimensional) modules, even if it is itself not finite dimensional.
More precisely, let $V$ be any enriching category (a locally small closed symmetric monoidal category with all limits). Then
for
$A$ a monoid in $V$;
$A Mod$ the $V$-enriched category of all $A$-modules in $V$;
$F : A Mod \to V$ the forgetful fiber functor ;
$A$ can be reconstructed as the object of enriched endomorphisms of $F$, given by the end
This is just the enriched Yoneda lemma in a slight disguise.
In good cases, this end is computed already by restriction to the full subcategory $A Mod_{dual}$ of dualizable modules
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.
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.
(Tannaka duality for G-sets)
Let $G$ be a group, write $G Set$ for the category of G-sets and
for the forgetful functor that sends a G-set to its underlying set.
Then there is a canonical group-isomorphism
identifying the automorphism group of $F$ (the group of natural isomorphisms from $F$ to itself) with $G$.
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 $F = C(G, -)$. Then the argument is
The “$G$” here is used in multiple senses, but each sense is deducible from context.
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
The canonical inclusion $i : {*} \to \mathbf{B}G$ induces the fiber functor
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 represented by that unique object
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 $PSh(PSh(\mathbf{B}G)^{op})$
So applying the Yoneda lemma twice, we find that
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:
(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
Let $V$ be a (locally small) 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 $V$-enriched Tannaka duality theorem states that we can reconstruct a monoid $A$ as the monoid of endomorphisms of the forgetful $V$-functor from $A Mod$ to $V$:
(Tannaka duality for $V$-modules over $V$-algebras)
For $A$ a monoid in $V$ with delooping $V$-enriched category $\mathbf{B}A$, and with
the enriched functor category that encodes the $V$-modules 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 naturally isomorphic to $A$
We can repeat the argument given above for permutation representations, this time employing the enriched Yoneda lemma.
Indeed, we may identify $F$ with the $V$-functor
given by precomposition along the $V$-functor $[\bullet]\colon\mathbf{pt}_V\to\mathbf{B}A$ picking the unique object $\bullet$ of $\mathbf{B}A$.
Now we note that the functor $[\bullet]^*$ (given by sending an $A$-module $M\colon\mathbf{B}A\to V$) to its evaluation $M(\bullet)$ at the unique object $\bullet$ of $\mathbf{B}A$) is $V$-naturally isomorphic to the $V$-functor $\mathbf{Nat}_V(h_\bullet,-)$, since
by the enriched Yoneda lemma. So in summary we have $F\cong[\bullet]^*\cong\mathbf{Nat}_V(h_\bullet,-)$.
We can then compute $End(F)$ as follows:
where we have applied the enriched Yoneda lemma twice.
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
The case of permutation representations is re-obtained by setting $V =$ Set.
As before, the same proof actually shows the following more general statement
(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 isomorphisms
From this statement of Tannaka duality in $V$-enriched category theory now various special cases of interest follow, by simply choosing suitable enrichement categories $V$.
The general case of Tannaka duality for $V$-modules described above restricts to the classical case of Tannaka duality for linear representations by setting $V :=$ Vect, the category of vector spaces over some fixed ground field.
In this case the above says
(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
in Vect.
Additional structure on the algebra $A$ corresponds to addition structure on its category of modules as indicated in the following table:
Tannaka duality for categories of modules over monoids/associative algebras
monoid/associative algebra | category of modules |
---|---|
$A$ | $Mod_A$ |
$R$-algebra | $Mod_R$-2-module |
sesquialgebra | 2-ring = monoidal presentable category with colimit-preserving tensor product |
bialgebra | strict 2-ring: monoidal category with fiber functor |
Hopf algebra | rigid monoidal category with fiber functor |
hopfish algebra (correct version) | rigid monoidal category (without fiber functor) |
weak Hopf algebra | fusion category with generalized fiber functor |
quasitriangular bialgebra | braided monoidal category with fiber functor |
triangular bialgebra | symmetric monoidal category with fiber functor |
quasitriangular Hopf algebra (quantum group) | rigid braided monoidal category with fiber functor |
triangular Hopf algebra | rigid symmetric monoidal category with fiber functor |
supercommutative Hopf algebra (supergroup) | rigid symmetric monoidal category with fiber functor and Schur smallness |
form Drinfeld double | form Drinfeld center |
trialgebra | Hopf monoidal category |
2-Tannaka duality for module categories over monoidal categories
monoidal category | 2-category of module categories |
---|---|
$A$ | $Mod_A$ |
$R$-2-algebra | $Mod_R$-3-module |
Hopf monoidal category | monoidal 2-category (with some duality and strictness structure) |
3-Tannaka duality for module 2-categories over monoidal 2-categories
monoidal 2-category | 3-category of module 2-categories |
---|---|
$A$ | $Mod_A$ |
$R$-3-algebra | $Mod_R$-4-module |
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 representations of $G$ is
and we obtain
(Tannaka duality for linear group representations)
There is a natural isomorphism
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
and
it is shown that $A$ is recovered as the coend
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
in $Vect^{op}$. This is equivalently the coend
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.
See Tannaka duality for Lie groupoids.
See Tannaka duality for geometric stacks.
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.
By applying the $(\infty,1)$-Yoneda lemma verbatim four times in a row as above for permutation representations, we obtain the following statement for ∞-permutation representations.
(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 ∞-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
As before, this holds immediately even for representations of (∞,1)-categories
(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
in ∞Grpd.
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 ∞-stacks 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
For $x \in X$ there is a natural weak homotopy equivalence
In particular we have natural isomorphisms of homotopy groups
More on this is at cohesive (∞,1)-topos – structures in the section Galois theory in a cohesive (∞,1)-topos
André Joyal, Ross Street, An introduction to Tannaka duality and quantum groups, pdf
B.J. Day, Enriched Tannaka reconstruction, J. Pure Appl. Algebra 108 (1996) 17-22, doi
The following paper shortens Deligne’s proof
Deligne’s proof in turn fills the gap in the seminal work with the same title
A revival in algebraic geometry related to the theory of mixed motives was marked by
Analogous discussion for symmetric monoidal stable (infinity,1)-categories includes
James Wallbridge, Tannaka duality over ring spectra (arXiv:1204.5787)
Isamu Iwanari, Tannaka duality and stable infinity-categories (arXiv:1409.3321)
Jacob Lurie, Spectral Algebraic Geometry, Chap. 9
Ulbrich made a major contribution at the coalgebra and Hopf algebra level
This Hopf-direction has been advanced by many authors including
S. L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted $SU(N)$ groups, Inventiones Mathematicae 93, No. 1, 35-76, doi
Shahn Majid, Foundations of quantum group theory, chapter 9
Phung Ho Hai, Tannaka-Krein duality for Hopf algebroids, Israel J. Math. 167 (1):193–225 (2008) math.QA/0206113
Volodymyr V. Lyubashenko, Modular transformations and tensor categories, J. Pure Appl. Algebra 98 (1995) 279–327 doi; Squared Hopf algebras and reconstruction theorems, Proc. Workshop “Quantum Groups and Quantum Spaces” (Warszawa), Banach Center Publ. 40, Inst. Math. Polish Acad. Sci. (1997) 111–137, q-alg/9605035; Squared Hopf algebras, Mem. Amer. Math. Soc. 142 (677):x 180, 1999; Алгебры Хопфа и вектор-симметрии, УМН, 41:5(251) (1986), 185–186, pdf, transl. as: Hopf algebras and vector symmetries, Russian Math. Surveys 41(5):153154, 1986.
A. Bruguières, Théorie tannakienne non commutative, Comm. Algebra 22, 5817–5860, 1994
K. Szlachanyi, Fiber functors, monoidal sites and Tannaka duality for bialgebroids, arxiv/0907.1578
B. Day, R. Street, Quantum categories, star autonomy, and quantum groupoids, in “Galois theory, Hopf algebras, and semiabelian categories”, Fields Inst. Comm. 43 (2004) 187-225
Daniel Schäppi, The formal theory of Tannaka duality, arxiv/1112.5213, superseding earlier Tannaka duality for comonoids in cosmoi, arXiv:0911.0977
A generalization of several classical reconstruction theorems with nontrivial functional analysis is in
Categorically oriented notes were written also by Pareigis, emphasising on using Coend in dual picture. His works can be found here but the most important is the chapter 3 of his online book
A very neat Tannaka theorem for stacks is proved in
Jacob Lurie, Tannaka duality for geometric stacks, (arXiv:math.AG/0412266)
Jacob Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems
Bertrand Toen, Higher Tannaka duality, MSRI 2002 (talk, video)
Moshe Kamensky, Model theory and the Tannakian formalism, arXiv:0908.0604; Tannakian formalism over fields with operators, arxiv/1111.7285
H. Fukuyama, I. Iwanari, Monoidal infinity category of complexes from Tannakian viewpoint, arxiv/1004.3087
remark of Ben-Zvi on Tannaka reconstruction for monoidal categories
David Kazhdan, Michael Larsen, Yakov Varshavsky, The Tannakian formalism and the Langlands conjectures, arxiv/1006.3864
The classical articles are
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
Tannaka duality for dg-categories is studied in
J.P.Pridham, Tannaka duality for enhanced triangulated categories, arxiv/1309.0637
MathOverflow, Does the Tannaka-Krein theorem come from an equivalence of 2-categories?