In Walker and Wang's [article](http://arxiv.org/abs/1104.2632) 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?



Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories?
Short answer: Look for papers on "deequivariantization". (I think the original references are by Müger and Brugières, but I am not sure whether they used the term "deequivariantization".)


Longer answer: 


The procedure mentioned in the paper is actually a sort of predual to the deequivariantization procedure found in the literature, in the sense that $Rep(C/S) \cong Deeq(Rep(C))$. So it is very closely related to deequivariantization, but not exactly the same thing.


You attempt was on the right track. One starts with $C$, then adds isomorphisms between objects of $S$ and the trivial object. This has to be done carefully, of course. One uses the fact that $S \cong Rep(G)$ for some finite group $G$, and $Rep(G)$ has a fiber functor.


A fancier way of thinking of it goes as follows. Since $S$ is symmetric monoidal we can think of it as an $n$-category for any $n$, and in particular we can think of it as a 4-category. The 3-category $C$ then becomes a module 3-category for the 4-category $S$. We can construct another module 3-category $F$ for $S$ by using the isomorphism $S\cong Rep(G)$ and the fiber functor for $Rep(G)$. Now we can now define
$$
 C/S = C \otimes\_S F
$$


$F$ is actually a $S$-$G\_4$ bimodule 3-category, where $G\_4$ denotes the finite group $G$ thought of as a 4-category. It follows that $C/S$ has a $G$ action. (A $G$ action on a 3-category is the same thing as a $G\_4$-module structure on that 3-category.)


I have a preprint which fills in the details of the above arguments, but it is not yet ready for the arXiv. If you go to [this page](http://canyon23.net/math/talks/) and look at the slides from talks at Vienna and Princeton (Feb 2014), you can find some of the details.


