A semisimple braided category with duals is called modular when a certain matrix $S$ is invertible. The components $S\_{AB}$ are indexed by (isomorphism classes of) simple objects of the category and one computes $S\_{AB}$ by colouring the Hopf link with (representants of) $A$ and $B$ and evaluates the resulting diagram. One can show that a category is modular iff there are no "transparent" objects (objects that braid trivially with every other object) besides the monoidal unit.


Is being modular a specific property, say in the category of braided categories? Is being a modular category equivalent to being the limit of some diagram or satisfying some diagram?



Is "being a modular category" a universal or categorical/algebraic property?
The Drinfeld center $Z(C)$ of a braided category "contains" $C$ and $\bar C$ (the category with opposite braiding) and therefore also $C\boxtimes \bar C$ and you can show that the following is equivalent


1) $C$ is modular


2) $Z(C)$ is equivalent with $C\boxtimes \bar C$.


In other words, for a braided category $C$ there is a natural notion of a center $Z(C)$ and a natural embedding $C\boxtimes \bar C$ in $Z(C)$ which is an equivalence if and only if $C$ is modular.


