# Module categories over symmetric/braided monoidal categories

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

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."
• How does it look like in the 4-category of braided categories, monoidal bimodule categories, bimodule categories, functors and natural transformations? Is the 1-morphism $\text{Rep}H$ Morita equivalent to $\text{Vect}$? Oct 21, 2015 at 8:08
• Noah you completely misunderstood my intention. Probably I need to reformulate. I am not asking about algebra objects in symmetric monoidal categories. I'm trying to categorify the statement i.e. consider module categories over symmetric monoidal categories. A module category $\mathcal{M}$ over a symmetric monoidal category $\matcal{A}$ is a category equipped with a linear (or even right exact) bifunctor $\otimes: \mathcal{A} \times \mathcal{M} \rightarrow \mathcal{M}$ satisfying the appropriate conditions Feb 18, 2012 at 17:08