# Generalized Tannakian Duality?

By the classical theory of Tannakian duality we know that every $k$-linear rigid abelian tensor category ($k$ a field) which has a fibre functor to $\mathrm{Vec}_k$ (finite dim. vector spaces over $k$) is equivalent to the category of representations of an affine group scheme over $k$ (or equivalently the category of comodules over a commutative Hopf algebra in $\mathrm{Vec}_k$ ).

I'm interested in the more general case where one has a "good" tensor category $C$ with a fibre functor to another "good" tensor category $D$ (instead of $\mathrm{Vec}_k$ ). What happens in this case? Is $C$ equivalent to the category of comodules over a Hopf algebra in $D$? If yes, how this Hopf algebra can be constructed?

As an example consider this two special cases:

1- $\mathrm{PHS}_\mathbb{Q} \to \mathrm{Vec}^\bullet_\mathbb{Q}$: weight grading functor,

2- $\mathrm{MHS}_\mathbb{Q} \to \mathrm{PHS}_\mathbb{Q}$: sum of graded quotients functor.

Majid has such theorems. The target category $\mathcal{D}$ is assumed to be braided monoidal (with duals if required). One good place to look is chapter 9.4.2 in his book "Foundations of Quantum Groups".

If you care about non-strict monoidal categories $\mathcal{C}$, another good place is his paper "Tannaka-Krein Theorem for Quasi-Hopf Algebras and other Results". He doesn't do the quasi-Hopf algebra case in the generality of braided monoidal categories but I am convinced this generalizes.

One cannot always have an equivalence of categories of $\mathcal{C}$ with the obtained module category of modules over a (quasi)Hopf or bialgebra $B$ in $\mathcal{D}$ but there is a universal choice, denote by $B-\mathbb{Mod}(\mathcal{D})$. It is important to note that that one needs to assume representability of the functores $Nat(-\otimes F^{\otimes n},F^{\otimes n})\colon \mathcal{D}\to \mathbb{Set}$ by $B^{\otimes n}$ in a compatible way. I guess this is somehow a case of Barr-Beck (with extra structure) as hinted in a comment, but I don't know exactly how. In this case, there is a functor $\mathcal{C}\to B-\mathbb{Mod}(\mathcal{D})$ over $\mathcal{D}$. This functor is universal in the sense that whenever $B'$ is another bialgebra with the same property, then there exists a map $B'\to B$ and the diagram with the functor it induces commutes.

You can take a look at Lawrence Breen's survey in the Motives proceedings: Tannakian categories [1]. In general tensor functors $C \to D$ induce a morphisms between the corresponding gerbes.

If you have a “usual” fibre functor from $D$ to $\mathrm{Vec}_{k}$, you get a morphism of the associated Tannakian groups. In your first example that would be a morphism $\mathbb{G}\mathrm{m} \to \mathrm{MT}$. (Here $\mathrm{MT}$ is the proalgebraic Mumford–Tate group corresponding to all of $\mathrm{PHS}_{\mathbb{Q}}$. It's the limit over all “usual” MT-groups.)

I'm not sure if this satisfies your needs. After all, it is not exactly what you suggest you are looking for. So, alternatively, you might want to look at Deligne's Catégories tannakiennes [2]. In it he does some algebraic geometry internal to a Tannakian category. Maybe you can use this in your setup, so that $C$ gives rise to an affine group scheme in $D$. But I really didn't think about the details; this is just a vague suggestion.

[1] Tannakian categories, in Motives, Proc. Symp. Pure Math. 55 part 1, 337-376 (1994)

[2] Catégories tannakiennes in Grothendieck Festschrift vol II. Progress in Mathematics. 87 ( Birkhäuser Boston 1990) pp. 111–195. You can find it at the IAS website: http://publications.ias.edu/sites/default/files/60_categoriestanna.pdf

This paper is devoted to the generalization of Tannakian formalism for fiber functors over more general tensor categories: $F:\cal{C}\to \cal{D}$. One can see easily that for representing $F$ as a forgetful functor, the existence of a "section" $s:\cal{D}\to \cal{C}$ is necessary.

Corollary 5.3 in the paper (which is an immediate corollary of a theorem in Deligne's "Categories Tannakiennes") shows that under the condition of the existence of the section, the required Hopf algebra (or equivalently "affine group scheme") exists if we assume that $\cal{C}$ and $\cal{D}$ are tensor categories over a field with characteristic zero and $F$ is also a tensor functor.

There is also another approach in the paper which gives a similar result but assumes $\cal{C}$ and $\cal{D}$ are $k$-tensor categories over a field $k$, and $\cal{D}$ is semisimple, the existence of section for $F$ and some other technical conditions. But there is no restriction on $k$.