nLab
Tannaka duality

Idea
For permutation representations
References

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 : {Rep}_{D} (A) \to D

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

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 Yoneda lemma.

Theorem

(Tannaka duality for permutation representations)

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

Then there is a canonical isomorphism of groups

Aut (F) ≃ G .

Here $Aut (F)$ 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 $End (F) ≅ G$ , i.e., each endomorphism on $F$ is invertible, so $End (F) = Aut (F) ≅ G$ .

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

End (F) = {Set}^{C} (F, F) ≅ {Set}^{C} (C (G, -), C (G, -)) ≅ C (G, G) ≅ G .

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

Long-winded Proof

Let $B G$ be the delooping groupoid. Then

{Rep}_{Set} (G) : = Func (B G^{op}, Set) .

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

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

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

Func (i, Set) = {Hom}_{PSh (B G)} (Y_{B G}) *, -),

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

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

Func (i, Set) = Y_{PSh (B G)^{op}} Y_{B G} * .

So applying the Yoneda lemma twice, we find that

\begin{aligned} {Aut}_{PSh (PSh (B G)^{op})} Func (i, Set) & = {Aut}_{PSh (PSh (B G)^{op})} Y_{PSh (B G)^{op}} Y_{B G} * \\ ≃ {Aut}_{PSh (B G)^{op})} Y_{B G} * \\ ≃ {Aut}_{B G} * \\ ≃ G . \end{aligned}

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.

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

nLab Tannaka duality

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