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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.

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  • $\begingroup$ 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) $\endgroup$ – Max Jun 20 at 14:04
  • $\begingroup$ @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? $\endgroup$ – leibnewtz Jun 20 at 21:15
  • $\begingroup$ No I said stupid necessary conditions. I don't think they're enough to get the equivalence $\endgroup$ – Max Jun 20 at 21:42
  • $\begingroup$ A sorry I guess I read what I wanted to read. I agree though that those are necessary $\endgroup$ – leibnewtz Jun 20 at 23:35

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