In Walker and Wang's article about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory $\mathcal{S_C}$ consisting of transparent objects. That quotient is promised to be modular.

I'm willing to believe this and I'd like to use it, but I can't find a mathematical reference for it, nor have I managed to work it out in all detail. *Is this a known fact? Is it treated somewhere?*

My attempt of making sense of it, is the following:

- Take a premodular category $\mathcal{C}$. That's just a ribbon category which is fusion (finitely semisimple etc.).
- Take all the transparent (trivially braiding) objects. They form a subcategory $\mathcal{S_C}$.
- Form an "exact sequence" $0 \to \mathcal{S_C} \to \mathcal{C} \to \mathcal{Q_C} \to 0$. I have no idea what exactly an exact sequence of ribbon fusion categories is or whether we even have to consider this. I'm vaguely expecting something like this:
- $\mathcal{Q_C}$ has the same objects as $\mathcal{C}$, but somehow the morphisms to and from the objects in $\mathcal{S_C}$ are identified with 0.

What I don't understand about it is:

- How does it work exactly?
- How do I prevent the monoidal unit to get killed? It's in $\mathcal{S_C}$ after all.
- Are the simple objects in the quotient also simple in $\mathcal{C}$? Do they braid the same, i.e. is the quotient obviously modular?