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Vanessa
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Module categories over symmetric/braided monoidal categories

Given an algebraically closed field $k$ and a Noetherian commutative $k$-algebra $A$, all simple modules over $A$ are 1-dimensional

  • What is the analogous statement for symmetric monoidal $k$-linear categories?
  • What is the analogous statement for braided monoidal $k$-linear categories?

We can assume the category is Abelian and the product functor is right exact in both variables

I expect something of the sort "any simple (in some sense) module category is equivalent to $Vect$", although I don't know what's the analogue of "Noetherian" and (what's troubling me the most) I have no idea how braided and symmetric are different in this repsect

Vanessa
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