nLab
Tannaka duality

Contents

Idea

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

if AA is a symmetry object (e.g. a locally compact topological group, Hopf algebra), represented on objects in a category DD, one may reconstruct AA from knowledge of the endomorphisms of the forgetful functor – the fiber functor

F:Rep D(A)D F : Rep_D(A) \to D

from the category Rep D(A)Rep_D(A) of representations of AA on objects of DD 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 AA 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 VV be any enriching category (a locally small closed symmetric monoidal category with all limits). Then

  1. for

    AA can be reconstructed as the object of enriched endomorphisms of FF, given by the end

    AEnd(F):= NAModV(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 a slight disguise.

  2. In good cases, this end is computed already by restriction to the full subcategory AMod dualA Mod_{dual} of dualizable modules

    NAMod dualV(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 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.

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

Then there is a canonical isomorphism of groups

Aut(F)G. Aut(F) \simeq G \,.

Here Aut(F)Aut(F) denotes the group of invertible natural transformations from FF 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 End(F)GEnd(F) \cong G, i.e., each endomorphism on FF is invertible, so End(F)=Aut(F)GEnd(F) = Aut(F) \cong G.

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

End(F)=Set C(F,F)Set C(C(G,),C(G,))C(G,G)G. End(F) = Set^C(F, F) \cong Set^C(C(G, -), C(G, -)) \cong C(G, G) \cong G.

The “GG” 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\mathbf{B}G be the delooping groupoid of the group GG. Then

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

The canonical inclusion i:*BGi : {*} \to \mathbf{B}G induces the fiber functor

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

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

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

where Y BG:BGPSh(BG)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(BG)SetFunc(i,Set) : PSh(\mathbf{B}G) \to Set is itself a representable functor in the presheaf category PSh(PSh(BG) op)PSh(PSh(\mathbf{B}G)^{op})

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

So applying the Yoneda lemma twice, we find that

Aut PSh(PSh(BG) op)Func(i,Set) =Aut PSh(PSh(BG) op)Y PSh(BG) opY BG* Aut PSh(BG) op)Y BG* Aut BG* G. \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 GG was assumed to be a group, but only that GG 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\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:

Theorem

(Tannaka duality for permutation representations of categories)

Let CC be a locally small category and Rep Set(C):=Func(C,Set)Rep_{Set}(C) := Func(C,Set) the functor category. For every object cCc \in C let F c:CSetF_c : C \to Set be the fiber-functor that evaluates at cc.

Then we have a natural isomorphism

Hom(F c,F c)Hom C(c,c). Hom(F_c,F_{c'}) \simeq Hom_C(c,c') \,.

For VV-modules

Let VV be a (locally small) closed symmetric monoidal category, so that VV 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=V = Set of a statement verbatim the same in VV-enriched category theory, with the ordinary functor category replaced everywhere by the VV-enriched functor category:

Then the statement says:

Theorem

(Tannaka duality for VV-modules over VV-algebras)

For AA a monoid in VV with delooping VV-enriched category BA\mathbf{B}A, and with

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

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

End(AModFV)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

End(F)= NAModV(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=V = Set.

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

Theorem

(Tannaka duality for VV-modules over VV-algebroids)

Let CC be a VV-enriched category (a “VV-algebroid”). Write CMod:=[C,V]C Mod := [C,V] for the VV-enriched functor category. For every object cCc \in C write F c:CModVF_c : C Mod \to V for the fiber functor that evaluates at CC. Then we have natural isomorphisms

hom(F c,F c)C(c,c). hom(F_c, F_{c'}) \simeq C(c,c') \,.

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

For algebra modules

The general case of Tannaka duality for VV-modules described above restricts to the classical case of Tannaka duality for linear representations by setting V:=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 AA an algebra and AModA Mod its category of modules, and for F:AModVectF : A Mod \to Vect the fiber functor that sends a module to its underlying vector space, we have a natural isomorphism

End(AModVect)A End( A Mod \to Vect ) \simeq A

in Vect.

Additional structure on the algebra AA 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 algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

For linear group representations

Still for the special case V=VectV = Vect, let now GG be a group and let the algeba in question specifically be its group algebra A=k[G]A = k[G] . Then the category of linear representations of GG is

Rep(G)k[G]Mod Rep(G) \simeq k[G] Mod

and we obtain

Corollary

(Tannaka duality for linear group representations)

There is a natural isomorphism

End(Rep(G)Vect)k[G]. End(Rep(G) \to Vect) \simeq k[G] \,.

For coalgebra comodules

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

In

(proposition 5, page 40)

and

it is shown that AA is recovered as the coend

NAMod finF(N)F(N) * \int^{N \in A Mod_{fin}} F(N) \otimes F(N)^*

in Vect, where the coend ranges over finite dimensional modules.

If AA itself is finite dimensional then this is yet again just a special case of the enriched Yoneda lemma for VV-modules, for the case V=FinVect opV = FinVect^{op}: this general statement says that AA is recovered as the end

A= NAMod finV(F(N),F(N)) A = \int_{N \in A Mod_{fin}} V(F(N), F(N))

in Vect opVect^{op}. This is equivalently the coend

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

in VectVect. Finally using that FinVect(V,W)VW *FinVect(V,W) \simeq V\otimes W^* the above coend expression follows.

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

For Lie groupoids

See Tannaka duality for Lie groupoids.

For geometric stacks

See Tannaka duality for geometric stacks.

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.

For permutation \infty-representations

By applying the (,1)(\infty,1)-Yoneda lemma verbatim four times in a row as above for permutation representations, we obtain the following statement for ∞-permutation representations.

Theorem

(Tannaka duality for \infty-permutation representations)

Let GG be an ∞-group and Rep Grpd(G):=Func(BG,Grpd)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 Grpd(G)GrpdF : 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

End(Rep Grpd(G)Grp)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 \infty-permutation representations)

Let cc be an (∞,1)-category and Rep Grpd(C):=Func(C,Grpd)Rep_{\infty Grpd}(C) := Func(C,\infty Grpd). For cCc \in C an object, write F c:Rep Grpd(C)GrpdF_c : Rep_{\infty Grpd}(C) \to \infty Grpd for the corresponding fiber functor.

Then there is a natural equivalence

hom(F c,F c)C(c,c) hom(F_c, F_{c'}) \simeq C(c,c')

in ∞Grpd.

\infty-Galois theory

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 XX that the ∞-groupoid LConst(X)LConst(X) of locally constant ∞-stacks on XX is equivalent to Rep Grpd(Π(X))Rep_{\infty Grpd}(\Pi(X)), where Π(X)\Pi(X) is the fundamental ∞-groupoid of XX. For xXx \in X a point, write F x:LConst(X)GrpdF_x : LConst(X) \to \infty Grpd for the corresponding fiber functor.

Then we have

Theorem

For xXx \in X there is a natural weak homotopy equivalence

End(LConst(X)F xGrpd)BAut Π(X)(x). End(LConst(X) \stackrel{F_x}{\to} \infty Grpd) \simeq \mathbf{B} Aut_{\Pi(X)}(x) \,.

In particular do we have natural isomorphisms of homotopy groups

π nEnd(LConst(X)F xGrpd)π n(X,x). \pi_n End(LConst(X) \stackrel{F_x}{\to} \infty Grpd) \simeq \pi_n(X,x) \,.

More on this is at cohesive (∞,1)-topos -- structures in the section Galois theory in a cohesive (∞,1)-topos

References

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.

Deligne’s proof in turn fills the gap in the seminal work with the same title

  • N. Saavedra Rivano, Catégories Tannakiennes, Springer LNM 265 (1972).

A revival in algebraic geometry related to the theory of mixed motives was marked by

  • P. Deligne, J. Milne, Tannakian categories, Springer Lecture Notes in Math. 900, 1982, pp. 101-228, retyped pdf

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, doi

This Hopf-direction has been advanced by many authors including

  • S. L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N)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, 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

  • 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

The classical articles are

  • Tadao Tannaka, Über den Dualitätssatz der nichtkommutativen topologischen Gruppen, Tohoku Math. J. 45 (1938), n. 1, 1–12 (project euclid has only Tohoku new series!), see Tannaka-Krein theorem.
  • N. Tatsuuma, A duality theorem for locally compact groups, J. Math., Kyoto Univ. 6 (1967), 187–293.
  • M.G. Krein, A principle of duality for bicompact groups and quadratic block algebras, Doklady AN SSSR 69 (1949), 725–728.
  • Eiichi Abe, Dualité de Tannaka des groupes algébriques, Tohoku Mathematical Journal. Volume 12, Number 2 (1960), 327-332.

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

Revised on September 5, 2013 18:00:33 by Zoran Škoda (161.53.130.104)