abstract duality: opposite category,
if is a symmetry object (e.g. a locally compact topological group, Hopf algebra), represented on objects in a category , one may reconstruct from knowledge of the endomorphisms of the forgetful functor – the fiber functor –
from the category of representations of on objects of 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 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.
This is just the enriched Yoneda lemma in a slight disguise.
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 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.
(Tannaka duality for permutation representations)
Here denotes the group of invertible natural transformations from to itself.
With a bit of evident abuse of notation, the proof is a one-line sequence of applications of the Yoneda lemma: we show , i.e., each endomorphism on is invertible, so .
Write . Observe that the functor is the representable . Then the argument is
The “” 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.
The canonical inclusion induces the fiber functor
where is the Yoneda embedding.
But this way we see that is itself a representable functor in the presheaf category
So applying the Yoneda lemma twice, we find that
But moreover, as the long-winded proof above makes manifest, even more abstractly the proof really only depended on the fact that the delooping 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)
Then we have a natural isomorphism
Observe that the setup, statement and proof of Tannaka duality for permutation representations given above is the special case for Set of a statement verbatim the same in -enriched category theory, with the ordinary functor category replaced everywhere by the -enriched functor category:
Then the statement says:
(Tannaka duality for -modules over -algebras)
Apply the enriched Yoneda lemma verbatim as for the statement about permutation representations as above.
The case of permutation representations is re-obtained by setting Set.
As before, the same proof actually shows the following more general statement
(Tannaka duality for -modules over -algebroids)
From this statement of Tannaka duality in -enriched category theory now various special cases of interest follow, by simply choosing suitable enrichement categories .
The general case of Tannaka duality for -modules described above restricts to the classical case of Tannaka duality for linear representations by setting Vect, the category of vector spaces over some fixed ground field.
In this case the above says
(Tannaka duality for linear modules)
Additional structure on the algebra corresponds to addition structure on its category of modules as indicated in the following table:
|monoid/associative algebra||category of modules|
|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|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|
and we obtain
(Tannaka duality for linear group representations)
There is a natural isomorphism
it is shown that is recovered as the coend
in Vect, where the coend ranges over finite dimensional modules.
If itself is finite dimensional then this is yet again just a special case of the enriched Yoneda lemma for -modules, for the case : this general statement says that is recovered as the end
in . This is equivalently the coend
in . Finally using that the above coend expression follows.
As before, more work is required to show that even for itself not finite dimensional, it is still recovered in terms of the above (co)end over just its finite dimensional modules.
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.
(Tannaka duality for -permutation representations)
Let be an ∞-group and the category of ∞-permutation representations, the (∞,1)-category of (∞,1)-functors from its delooping ∞-groupoid to ∞Grpd. Let be the fiber functor that remembers the underlying -groupoid. Then there is an equivalence in a quasi-category
As before, this holds immediately even for representations of (∞,1)-categories
(Tannaka duality for -permutation representations)
Then there is a natural equivalence
As a special case of this, we obtain a statement about -Galois theory. For details and background see homotopy groups in an (∞,1)-topos. In that context one finds for a locally contractible space that the ∞-groupoid of locally constant ∞-stacks on is equivalent to , where is the fundamental ∞-groupoid of . For a point, write for the corresponding fiber functor.
Then we have
For there is a natural weak homotopy equivalence
B.J. Day, Enriched Tannaka reconstruction, J. Pure Appl. Algebra 108 (1996) 17-22, doi
The following paper shortens the 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
Ulbrich made a major contribution at the coalgebra and Hopf algebra level
This Hopf-direction has been advanced by many authors including
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
A generalization of several classical reconstruction theorems with nontrivial functional analysis is in
A very neat Tannaka theorem for stacks is proved in
H. Fukuyama, I. Iwanari, Monoidal infinity category of complexes from Tannakian viewpoint, arxiv/1004.3087
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