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 $G$.


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