# How nontrivial can "central extensions of ribbon fusion categories" be?

In a sense, this is a follow up on this question, but one PhD programme later.

Let $\mathcal{C}$ be ribbon fusion. By $\mathcal{C}'$, we denote the symmetric centre, i.e. the full subcategory of objects that trivially braid with every other object in $\mathcal{C}$. Let us require that $\mathcal{C}$ is modularisable, i.e. $\mathcal{C}' \simeq \operatorname{Rep}G$ for a finite group $G$. We then have the following "exact sequence of ribbon fusion categories": $$\mathcal{C}' \hookrightarrow \mathcal{C} \twoheadrightarrow \widetilde{\mathcal{C}}$$ Here, $\widetilde{\mathcal{C}}$ is the modularisation of $\mathcal{C}$. I'm calling this an exact sequence since $\mathcal{C}'$ is being sent to the copy of vector spaces sitting inside the modularisation as the category spanned by the tensor unit, and in a certain sense, $\mathcal{C}'$ is the biggest such subcategory of $\mathcal{C}$.

This looks a lot like a central extension. So another way of describing a (modularisable) ribbon fusion category would then be as a central extension of a modular category by the representations of a finite group (i.e. a Tannakian symmetric fusion category).

It's known that central extensions of groups are parametrised by second group cohomology. Is there a similar characterisation of ribbon fusion categories in terms of "central extensions"? I.e. can every modularisable ribbon fusion category be described by a modular category, a symmetric fusion category and some extra data?

Deequivariantization (of which modularization is a special case) is inverse to equivariantization. That is, you can recover $\mathcal{C}$ as the category of G-equivariant objects in $\tilde{\mathcal{C}}$ for some action of G on $\tilde{\mathcal{C}}$. So to classify such extensions you want to classify G-actions. (Note that this can be a bit tricky because G can act nontrivially on the category even if it acts trivially on iso classes of objects.)
• Awesome! Is there work on classifying $G$-actions on a given modular category? (A quick search didn't bring up anything, but I might searching for the wrong keywords.) May 5 '16 at 17:09