# Local fusion categories

A local fusion category $${\cal R}$$ is a unitary fusion category equipped with a top-faithful surjective monoidal functor to the fusion category of vector spaces: $$\beta: {\cal R} \to {\cal V}ec$$. Here, top-faithful means that the functor $$\beta$$ is injective when acting on the morphisms.

What are those local fusion categories? $${\cal R}ep(G)$$ and $${\cal V}ec_G$$ are local fusion categories, for a finite group $$G$$. Are there other examples? Is there a classification?

Let $$\mathcal{R}$$ be a fusion category and $$\beta : \mathcal{R} \to \mathrm{Vec}$$ an additive monoidal functor.
I first claim that $$\beta$$ is automatically faithful. (I know why you use "top faithful" — in higher categories, you want faithfulness just on top-morphisms — but here in 1-category land "top faithful" is just faithful.) First, note that, since $$\mathcal{R}$$ is semisimple, every additive functor out of $$\mathcal{R}$$ is exact. Second, suppose $$f : X \to Y$$ is a nonzero map in $$\mathcal{R}$$. Then by composing with the pairing $$Y \otimes Y^* \to 1$$, you get a nonzero map $$f^\# : X \otimes Y^* \to 1$$. But $$1$$ is simple, so this map is a surjection. So $$\beta(f^\#)$$ is a surjection onto $$\beta(1) = 1$$, and so $$\beta(f^\#) \neq 0$$, so $$\beta(f) \neq 0$$. For further details, see Deligne's Catégories tannakiennes.
Thus your "local fusion category" is also called "fusion category with a fibre functor". These are fully understood: such a fusion category is canonically $$\mathrm{Mod}(H)$$ for a finite-dimensional semisimple Hopf algebra $$H$$. There are many places to see the details, so I will be telegraphic. As an algebra, $$H$$ is defined as the endomorphisms of $$\beta$$-as-a-functor. Then the Hopf structure on $$H$$ comes from the monoidality of $$\beta$$.
Your two examples correspond to $$H = \mathbb{C}[G]$$, the group ring, and $$H = \mathcal{O}(G)$$, the functions on $$G$$.