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?

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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.

In general, taking a "center" of a higher category is looking at the endomorphisms of the identity functor. For example, if you think of a monoid as a 1-category, then the endomorphisms of the identity functor are exactly the center of the monoid. Thinking of a tensor category as a 2-category with one object this also gives the Drinfeld center.

If you think of your braided tensor category as a 3 category with one object and one morphism, then this "center" construction yields a symmetric tensor category which is exactly the subcategory of transparent objects! (This is sometimes called the "Mueger center" to distinguish it from the Drinfeld center.) So modularity is just saying that the center of the braided tensor category is trivial.

  • That's a nice way of thinking about it. Is there a good reference that explains exactly how braided categories are secretly 3-categories? I tried to work it out myself several times, but I couldn't get it. Where is the Mueger centre defined? – Manuel Bärenz Feb 16 '14 at 14:36
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    The 2-morphisms of the 3cat are the objects of the BTC and the 3-morophisms of the 3cat are the morphisms of the BTC. Vertical composition gives you the tensor product, and the Eckmann-Hilton argument gives you the braiding. The first place I know of the Mueger center being called a center is arxiv.org/abs/math/0111205 but as explained in the introduction the idea goes back further. – Noah Snyder Feb 16 '14 at 18:15
  • That's what I hear all the time, but I still can't see how the Eckmann-Hilton argument gives the braiding. That's what I tried to work out, and I never got it to work properly. Thanks for the reference, that seems like a good article! – Manuel Bärenz Feb 16 '14 at 21:51
  • You just do EH... $x \otimes y \rightarrow (x \star 1) \otimes (1 \star y) \rightarrow (x \otimes 1) \star (1 \otimes y) \rightarrow x \star y$ and then do the same thing once more to end up at $y \otimes x$. See also cheng.staff.shef.ac.uk/degeneracy/eggclock.pdf – Noah Snyder Feb 16 '14 at 22:05
  • As a warmup you might want to think through why a 2-category with one object and one 1-morphism is automatically a commutative monoid by EH. – Noah Snyder Feb 17 '14 at 3:59

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