# Converse to Tannaka duality for rings

Let $$k$$ be a field. It's well known and easy to prove that a $$k$$-algebra $$R$$ may be recovered from the functor $$F: R-Mod \to Vect_k$$ as the endomorphisms of $$F$$. Now suppose $$\mathcal{C}$$ is a category equipped with a functor $$F: \mathcal{C} \to Vect_k$$. What are the conditions, if any, on $$\mathcal{C}$$ and $$F$$ that allow us to conclude that $$\mathcal{C} \cong End(F)- Mod$$? There is an obvious functor $$\mathcal{C} \to End(F)-Mod$$, but I'm not sure under what conditions this is essentially surjective.

• If I assume that the question means : "the specific functor $\mathcal{C}\to End(F)-Mod$ is an equivalence", then stupid necessary conditions are : $\mathcal{C}$ is a Grothendieck category, $F$ preserves all (co)limits and is faithful (the conditions on $\mathcal{C}$ don't depend on what you mean precisely) – Max Jun 20 at 14:04
• @Max since you called those conditions stupid I assume it’s obvious that they make the functor an equivalence - but I don’t see it. Could you explain? – leibnewtz Jun 20 at 21:15
• No I said stupid necessary conditions. I don't think they're enough to get the equivalence – Max Jun 20 at 21:42
• A sorry I guess I read what I wanted to read. I agree though that those are necessary – leibnewtz Jun 20 at 23:35