nLab
Tannaka duality

Idea

Tannaka duality or Tannaka reconstruction theorems are statements of the form:

for $A$ some algebraic structure, represented on objects in a category $D$, one may reconstruct $A$ from knowledge of the endomorphisms of the forgetful functor – the fiber functor –

$$F:{\mathrm{Rep}}_D(A)\to D$$F : Rep_D(A) \to D

from the category ${\mathrm{Rep}}_D(A)$ of representations of $A$ on objects of $D$ that remembers these underlying objects.

There is a general-abstract and concrete aspect to this, and a concrete one. 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 dualizble (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

1. we have in full generality that:

   for

   • $A$ a monoid in $V$;

   • $A\mathrm{Mod}$ the $V$-enriched category of all $A$-modules in $V$;

   • $F:A\mathrm{Mod}\to V$ the forgetful fiber functor ;

   we have that $A$ is reobtained as the enriched endomorphisms of $F$, given by the end

   $$A\simeq \mathrm{End}(F):={\int }_{N\in A\mathrm{Mod}}V(F(N),F(N)).$$A \simeq End(F) := \int_{N \in A Mod} V(F(N), F(N)) \,.

   This is just the enriched Yoneda lemma in slight disguise.

2. In good cases this end is computed already by restriction to the full subcategory $A{\mathrm{Mod}}_{\mathrm{dual}}$ of dualizable modules

   $$\cdots \simeq {\int }_{N\in A{\mathrm{Mod}}_{\mathrm{dual}}}V(F(N),F(N)).$$\cdots \simeq \int_{N \in A Mod_{dual}} V(F(N), F(N)) \,.

Statement

So far the following examples concern the abstract aspect of Tannaka duality only, which is narrated here as a consequence of the enriched Yoneda lemma in enriched category theory.

For permutation representations

A simple case of Tannaka duality is that of permutation representations of a group, i.e. representations on a set. In this case, Tannaka duality follows entirely from repeated application of the ordinary Yoneda lemma.

Theorem

(Tannaka duality for permutation representations)

Let $G$ be a group, ${\mathrm{Rep}}_{\mathrm{Set}}(G)$ the category of its permutation representations and $F:{\mathrm{Rep}}_{\mathrm{Set}}(G)\to \mathrm{Set}$ the forgetful functor that sends a representation to its underlying set.

Then there is a canonical isomorphism of groups

$$\mathrm{Aut}(F)\simeq G.$$Aut(F) \simeq G \,.

Here $\mathrm{Aut}(F)$ denotes the group of invertbile natural transformations from $F$ to itself.

Quick Proof

With a bit of evident abuse of notation, the proof is a one-line sequence of applications of the Yoneda lemma: we show $\mathrm{End}(F)\cong G$, i.e., each endomorphism on $F$ is invertible, so $\mathrm{End}(F)=\mathrm{Aut}(F)\cong G$.

Write $C:={\mathrm{Set}}^G={\mathrm{Rep}}_{\mathrm{Set}}(G)$. Observe that the functor $F:C\to \mathrm{Set}$ is the representable $F=C(G,-)$. Then the argument is

$$\mathrm{End}(F)={\mathrm{Set}}^C(F,F)\cong {\mathrm{Set}}^C(C(G,-),C(G,-))\cong C(G,G)\cong G.$$End(F) = Set^C(F, F) \cong Set^C(C(G, -), C(G, -)) \cong C(G, G) \cong G.

The ”$G$” here is used in multiple senses, but each sense is deducible from context.

Long-winded Proof

We repeat the same proof, but with more notational details on what the entities involved in each step are precisely.

Let $BG$ be the delooping groupoid of the group $G$. Then

$${\mathrm{Rep}}_{\mathrm{Set}}(G):=\mathrm{Func}(B{G}^{\mathrm{op}},\mathrm{Set}).$$Rep_{Set}(G) := Func(\mathbf{B}G^{op}, Set) \,.

The canonical inclusion $i:*\to BG$ induces the fiber functor

$$\mathrm{Func}(i,\mathrm{Set}):{\mathrm{Rep}}_{\mathrm{Set}}(G)\to \mathrm{Set}$$Func(i,Set) : Rep_{Set}(G) \to Set

which evaluates a functor $\rho :B{G}^{\mathrm{op}}\to \mathrm{Set}$ on the unique object of $BG$. By the Yoneda lemma this is the same as homming out of the functor represented by that unique object

$$\mathrm{Func}(i,\mathrm{Set})={\mathrm{Hom}}_{\mathrm{PSh}(BG)}(Y_{BG})*,-),$$Func(i,Set) = Hom_{PSh(\mathbf{B}G)}(Y_{\mathbf{B}G}) {*}, -) \,,

where $Y_{BG}:BG\to \mathrm{PSh}(BG)$ is the Yoneda embedding.

But this way we see that $\mathrm{Func}(i,\mathrm{Set}):\mathrm{PSh}(BG)\to \mathrm{Set}$ is itself a representable functor in the presheaf category $\mathrm{PSh}(\mathrm{PSh}(BG{)}^{\mathrm{op}})$

$$\mathrm{Func}(i,\mathrm{Set})=Y_{\mathrm{PSh}(BG{)}^{\mathrm{op}}}{Y}_{BG}*.$$Func(i,Set) = Y_{\mathbf{PSh(\mathbf{B}G)^{op}}} Y_{\mathbf{B}G} * \,.

So applying the Yoneda lemma twice, we find that

$$\begin{array}{rl}{\mathrm{Aut}}_{\mathrm{PSh}(\mathrm{PSh}(BG{)}^{\mathrm{op}})}\mathrm{Func}(i,\mathrm{Set})& ={\mathrm{Aut}}_{\mathrm{PSh}(\mathrm{PSh}(BG{)}^{\mathrm{op}})}{Y}_{\mathrm{PSh}(BG{)}^{\mathrm{op}}}{Y}_{BG}*\\ & \simeq {\mathrm{Aut}}_{\mathrm{PSh}(BG{)}^{\mathrm{op}})}{Y}_{BG}*\\ & \simeq {\mathrm{Aut}}_{BG}*\\ & \simeq G.\end{array}$$\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}

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 $BG$ 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:

Theorem

(Tannaka duality for permutation representations of categories)

Let $C$ be a locally small category and ${\mathrm{Rep}}_{\mathrm{Set}}(C):=\mathrm{Func}(C,\mathrm{Set})$ the functor category. For every object $c\in C$ let $F_c:C\to \mathrm{Set}$ be the fiber-functor that evaluates at $c$.

Then we have a natural isomorphism

$$\mathrm{Hom}(F_c,F_{c\prime })\simeq {\mathrm{Hom}}_C(c,c\prime ).$$Hom(F_c,F_{c'}) \simeq Hom_C(c,c') \,.

For $V$-modules

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 statement says:

Theorem

(Tannaka duality for $V$-modules over $V$-algebras)

For $A$ a monoid in $V$ with delooping $V$-enriched category $BA$, and with

$$A\mathrm{Mod}:=[BA,V]$$A Mod := [\mathbf{B}A,V]

the enriched functor category that encodes the $V$-modules of $A$, we have that the $V$-enriched endomorphism algebra $\mathrm{End}(F):=[F,F]$ of the $V$-enriched functor $F:\mathrm{Rep}(A)\to V$ is naturally isomorphic to $V$

$$\mathrm{End}(A\mathrm{Mod}\stackrel{F}{\to }V)\simeq A.$$End(A Mod \stackrel{F}{\to} V) \simeq A \,.

Proof

Apply the enriched Yoneda lemma verbatim as for the statement about permutation representations as above.

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

$$\mathrm{End}(F)={\int }_{N\in A\mathrm{Mod}}V(F(N),F(N)).$$End(F) = \int_{N \in A Mod} V(F(N), F(N)) \,.

The case of permutation representations is re-obtained by setting $V=$ Set.

As before, the same proof actually shows the following more general statement

Theorem

(Tannaka duality for $V$-modules over $V$-algebroids)

Let $C$ be a $V$-enriched category (a ”$V$-algebroid”). Write $C\mathrm{Mod}:=[C,V]$ for the $V$-enriched functor category. For every object $c\in C$ write $F_c:C\mathrm{Mod}\to V$ for the fiber functor that evaluates at $C$. Then we have natural isomorphisms

$$\mathrm{hom}(F_c,F_{c\prime })\simeq C(c,c\prime ).$$hom(F_c, F_{c'}) \simeq C(c,c') \,.

From this statement of Tannaka duality in $V$-enriched category theory now various special cases of interest follow, by simply choosing suitable enrichement categories $V$.

For algebra modules

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

Corollary

(Tannaka duality for linear modules)

For $A$ an algebra and $A\mathrm{Mod}$ its category of modules, and for $F:A\mathrm{Mod}\to \mathrm{Vect}$ the fiber functor that sends a module to its underlying vector space, we have a natural isomorphism

$$\mathrm{End}(A\mathrm{Mod}\to \mathrm{Vect})\simeq A$$End( A Mod \to Vect ) \simeq A

in Vect.

For linear group representations

Still for the special case $V=\mathrm{Vect}$, 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

$$\mathrm{Rep}(G)\simeq k[G]\mathrm{Mod}$$Rep(G) \simeq k[G] Mod

and we obtain

Corollary

(Tannaka duality for linear group representations)

There is a natural isomorphism

$$\mathrm{End}(\mathrm{Rep}(G)\to \mathrm{Vect})\simeq k[G].$$End(Rep(G) \to Vect) \simeq k[G] \,.

For coalgebra comodules

If for $V$ we choose not Vect but its opposite category ${\mathrm{Vect}}^{\mathrm{op}}$, then a monoid object $A$ in $V$ is a coalgebra and $A\mathrm{Mod}$ (or $A{\mathrm{Mod}}^{\mathrm{op}}$, rather) is the category of comodules over this coalgebra. Again we have a forgetful functor $F:A\mathrm{Mod}\to \mathrm{Vect}$

In

• André Joyal, Ross Street, An introduction to Tannaka duality and quantum groups, pdf

(proposition 5, page 40)

and

• Pierre Deligne, Catégories Tannakiennes

it is shown that $A$ is recovered as the coend

$${\int }^{N\in A{\mathrm{Mod}}_{\mathrm{fin}}}F(N)\otimes F(N{)}^{*}$$\int^{N \in A Mod_{fin}} F(N) \otimes F(N)^*

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 $V={\mathrm{FinVect}}^{\mathrm{op}}$: this general statement says that $A$ is recovered as the end

$$A={\int }_{N\in A{\mathrm{Mod}}_{\mathrm{fin}}}V(F(N),F(N))$$A = \int_{N \in A Mod_{fin}} V(F(N), F(N))

in ${\mathrm{Vect}}^{\mathrm{op}}$. This is equivalently the coend

$$\cdots \simeq {\int }^{N\in A\mathrm{Mod}}(\mathrm{Vect}(F(N),F(N)))$$\cdots \simeq \int^{N \in A Mod}( Vect(F(N), F(N)))

in $\mathrm{Vect}$. Finally using that $\mathrm{FinVect}(V,W)\simeq V\otimes W^{*}$ the above coend expression follows.

As before, more work is required to show that ven for $A$ itself not finite dimensional, it is still recovered in terms of the above (co)end over just its finite dimensional modules.

In higher category theory

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 this verbatim four times in a row as above, we obtain the following statement for ”$\mathrm{\infty }$-permutation representations”.

Theorem

(Tannaka duality for $\mathrm{\infty }$-permutation representations)

Let $G$ be an ∞-group and ${\mathrm{Rep}}_{\mathrm{\infty }\mathrm{Grpd}}(G):=\mathrm{Func}(BG,\mathrm{\infty }\mathrm{Grpd})$ the (∞,1)-category of (∞,1)-functors from its delooping ∞-groupoid to ∞Grpd. Let $F:{\mathrm{Rep}}_{\mathrm{\infty }\mathrm{Grpd}}(G)\to \mathrm{\infty }\mathrm{Grpd}$ be the fiber functor that remembers the underlying $\mathrm{\infty }$-groupoid. Then we have an equivalence in a quasi-category

$$\mathrm{End}({\mathrm{Rep}}_{\mathrm{\infty }\mathrm{Grpd}}(G)\to \mathrm{\infty }\mathrm{Grp})\simeq G.$$End(Rep_{\infty Grpd}(G) \to \infty Grp) \simeq G \,.

As before, this holds immediately even for representations of (∞,1)-categories

Theorem

(Tannaka duality for $\mathrm{\infty }$-permutation representations)

Let $c$ be an (∞,1)-category and ${\mathrm{Rep}}_{\mathrm{\infty }\mathrm{Grpd}}(C):=\mathrm{Func}(C,\mathrm{\infty }\mathrm{Grpd})$. For $c\in C$ an object, write $F_c:{\mathrm{Rep}}_{\mathrm{\infty }\mathrm{Grpd}}(C)\to \mathrm{\infty }\mathrm{Grpd}$ for the corresponding fiber functor.

Then we have a natural equivalence

$$\mathrm{hom}(F_c,F_{c\prime })\simeq C(c,c\prime )$$hom(F_c, F_{c'}) \simeq C(c,c')

in ∞Grpd.

$\mathrm{\infty }$-Galois theory

As a special case of this, we obtain a statement about $\mathrm{\infty }$-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 $\mathrm{LConst}(X)$ of locally constant ∞-stacks on $X$ is equivalent to ${\mathrm{Rep}}_{\mathrm{\infty }\mathrm{Grpd}}(\Pi (X))$, where $\Pi (X)$ is the fundamental ∞-groupoid of $X$. For $x\in X$ a point, write $F_x:\mathrm{LConst}(X)\to \mathrm{\infty }\mathrm{Grpd}$ for the corresponding fiber functor.

Then we have

Theorem

For $x\in X$ there is a natural weak homotopy equivalence

$$\mathrm{End}(\mathrm{LConst}(X)\stackrel{F_x}{\to }\mathrm{\infty }\mathrm{Grpd})\simeq B{\mathrm{Aut}}_{\Pi (X)}(x).$$End(LConst(X) \stackrel{F_x}{\to} \infty Grpd) \simeq \mathbf{B} Aut_{\Pi(X)}(x) \,.

In particular do we have naturall isomorphisms of homotopy groups

$${\pi }_n\mathrm{End}(\mathrm{LConst}(X)\stackrel{F_x}{\to }\mathrm{\infty }\mathrm{Grpd})\simeq {\pi }_n(X,x).$$\pi_n End(LConst(X) \stackrel{F_x}{\to} \infty Grpd) \simeq \pi_n(X,x) \,.

References

• André Joyal, Ross Street, An introduction to Tannaka duality and quantum groups, pdf

• Pierre Deligne, Catégories Tannakiennes

The following paper shortens the Deligne’s proof

• Alexander L. Rosenberg, The existence of fiber functors, The Gelfand Mathematical Seminars, 1996–1999, 145–154, Birkhäuser, Boston 2000.

Ulbrich made a major contribution at the coalgebra and Hopf algebra level

• K-H. Ulbrich, On Hopf algebras and rigid monoidal categories, in special volume, Hopf algebras, Israel J. Math. 72 (1990), no. 1-2, 252–256.

This Hopf-direction has been advanced by many authors including Shahn Majid and

• Phung Ho Hai, Tannaka-Krein duality for Hopf algebroids, Israel J. Math. 167 (1):193–225 (2008) math.QA/0206113

• Volodymyr V. Lyubashenko, 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, arXiv:q-alg/9605035; Squared Hopf algebras, Mem. Amer. Math. Soc. 142 (677):x 180, 1999.

• K. Szlachanyi, Fiber functors, monoidal sites and Tannaka duality for bialgebroids, arxiv/0907.1578

More on Tannak duality for Hopf algebras is in

• Daniel Schäppi?, Tannak duality for comonoids in cosmoi (arXiv:0911.0977)

A generalization of several classical reconstruction theorems with nontrivial functional analysis is in

• Alexander L. Rosenberg, Reconstruction of groups, Selecta Math. (N.S.) 9, 1 (2003), 101–118, doi.

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

• Bodo Pareigis, Quantum groups and noncommutative geometry, Chapter 3: Representation theory, reconstruction and Tannaka duality, pdf

A very neat Tannaka theorem for stacks is proved in

• Jacob Lurie, Tannaka duality for geometric stacks, math.AG/0412266

• Bertrand Toen, Higher Tannaka duality, MSRI 2002 (talk, video)

• Moshe Kamensky, Model theory and the Tannakian formalism, arXiv:0908.0604.

• 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