Artin-Wederburn says that if you have a semisimple algebra then it is a product of matrix algebras over division rings.

Suppose that $C$ is a fusion category over the complex numbers (if you want to assume pivotal or similar things, that's fine, but don't assume symmetric or braided). Suppose that $X$ is an algebra object in $C$. That is $X$ is an object in $C$ together with a multiplication map $X \otimes X \rightarrow X$ and a unit etc. We call $X$ semisimple if the category of $X$-module objects in $C$ is semisimple. Is there some good analogue of Artin-Wedderburn?