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 $G$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.