TITLE: Module categories over symmetric/braided monoidal categories
QUESTION [6 upvotes]: Given an algebraically closed field $k$ and a finitely generated 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 have no idea how braided and symmetric are different in this repsect

REPLY [8 votes]: Consider C = Rep(G) for a finite group G.  This is a symmetric tensor category.  Any subgroup $H \subset G$ yields a module category Rep(H) with the action given by restricting and then tensoring.  This module category is simple in any appropriate sense, and in no interesting sense is it "1-dimensional."