Details in applying the Barr-Beck monadicity theorem to Tannakian reconstruction

The Barr-Beck monadicity theorem gives necessary and sufficient conditions for a category $\mathcal{C}$ to be equivalent to a category of (co)algebras over a (co)monad. A functor $F:\mathcal{C}\to\mathcal{D}$ is said to be comonadic if it has a right adjoint $G$ and the "enhanced functor" $F': \mathcal{C}\to\mathsf{Coalg}_{FG} (\mathcal{D})$ into colagebras over the comonad $FG$ on $\mathcal{D}$ is an equivalence. The comonadic version of Barr-Beck says:

A functor $F:\mathcal{C}\to\mathcal{D}$ is comonadic if and only if:

1. F has a right adjoint;
2. F reflects isomorphisms;
3. Every $F$-cosplit pair in $\mathcal{C}$ admits an equaliser in $\mathcal{C}$ and it is preserved by $F$.

I have recently learned that this substantially simplifies the proof of Tannaka duality for a (neutral) Tannakian category $\mathcal{C}$ with fiber functor $\omega:\mathcal{C}\to\mathsf{Vecf}_k$ to be equivalent to the category of finite-dimensional representations of an affine group scheme $G$. Namely, the Barr-Beck theorem applied to $\omega$ establishes that $\mathcal{C}$ is equivalent to the category of comodules over a coalgebra $A$. Then the rigid tensor structure on $\mathcal{C}$ can be used to show that $A$ is a commutative Hopf algebra and hence is the coordinate ring of an affine group scheme.

I would like some tips towards the Barr-Beck part of this proof.

Firstly, note that by definition $\omega$ is exact and faithful, so conditions (2) and (3) in Barr-Beck are satisfied. I am not sure how to easily show condition (1), that $\omega$ has a right adjoint $R$ (adjoint functor theorems seem too general, but maybe I have missed something). Can anyone help with this?

Secondly, assume that Barr-Beck applies and that $\omega'$ induces an equivalence $\mathcal{C}\simeq\mathsf{Coalg}_{C} (\mathsf{Vecf}_k)$, where $C = \omega\circ R$ is the comonad induced on $\mathsf{Vecf}_k$ by the adjunction $\omega\dashv R$. Then $A=C(k)$ is a $k$-coalgebra, and I believe that $\mathsf{Coalg}_C (\mathsf{Vecf}_k)$ is equivalent to the category $\mathsf{Comodf}_A$ of finitely-generated comodules over the coalgebra $A$. However, I am not sure how to do this because I don't know whether $k$ can be made into a $C$-coalgebra $a_k: k\to C(k)$. If it can, I would want to define the $A$-comodule structure on a $C$-coalgebra $V$ as

$$V\xrightarrow{\sim} k\otimes V\xrightarrow{a_k\otimes \text{id}} A\otimes V,$$

but I don't know if this is correct. Can anyone confirm or correct me on this, and possibly sketch out the details of the equivalence

$$\mathsf{Coalg}_C (\mathsf{Vecf}_k) \simeq \mathsf{Comodf}_A$$

that would complete this part of the proof? Thanks for your help!

• Dear Alex, it would be wonderful if you would write a blog post about this with a complete proof! I don't think I'll ever manage to figure out the details of applying Beck's theorem here.. – Arrow Mar 22 '17 at 23:19

The Barr-Beck part of the proof is the following.

Theorem: Let $C$ be an essentially small abelian $k$-linear category and let $\omega : C \to \text{Vect}_f$ be an exact faithful functor from $C$ to the category of finite-dimensional $k$-vector spaces. Then this functor exhibits $C$ as the category of finite-dimensional comodules over a coalgebra $A$.

What we would like to do is to show 1) that $\omega$ is comonadic and 2) that the comonad is given by tensoring with a coalgebra $A$, so that coalgebras over the comonad correspond to $A$-comodules.

However, the difference between "comodule" and "finite-dimensional comodule" is extremely important here: it is in fact not possible to apply Barr-Beck directly to $\omega$ because $\omega$ in fact fails to have a right adjoint in general, as soon as $A$ is infinite-dimensional.

The fix is to pass to Ind categories on both sides, and so to consider, not $\omega$, but

$$\text{Ind}(\omega) : \text{Ind}(C) \to \text{Vect}.$$

These ind categories are locally presentable, so the locally presentable form of the adjoint functor theorem applies, and $\text{Ind}(\omega)$ has a right adjoint iff it preserves colimits. It preserves coequalizers because $\omega$ is exact, and it preserves filtered colimits by construction (I think; I haven't checked this carefully).

I am actually not sure how to complete the proof from here. I think one has to show that the right adjoint also preserves colimits, so that the corresponding comonad on $\text{Vect}$ preserves colimits. From here it is an exercise to show, e.g. using the Eilenberg-Watts theorem, that colimit-preserving comonads on $\text{Vect}$ are the same thing as coalgebras, and coalgebras over them are the same thing as comodules.

• Dear Qiaochu - sorry for getting back to this so late, but thank you for a really helpful and informative answer! I will have a think about how to show the right adjoint preserves colimits. It's good to know that applying Barr-Beck here is still not completely trivial and needs a few more ideas. – Alex Saad Mar 16 '17 at 11:01