I felt like following up on Kate's question. There were some good motivational answers there.
Given a pair of factors M < N, there is a standard way to construct a 2-category whose objects are M and N, whose morphism categories are the categories of bimodules, and whose composition is described by some kind of Connes product. If I restrict to the endomorphism category of M, I get a monoidal category structure, but I don't know how to say anything about it. Here's a barrage of questions:
- When people talk about fusion categories coming from subfactors, are they referring to the endomorphism category of one of the factors?
- How are the endomorphism categories of M and N related? Are they equivalent? Are they Koszul dual?
- Does the Jones index say something concrete about the category, like Frobenius-Perron dimension? (How does one compute Jones index, anyway?)
- How do people go about constructing exotic subfactors? Do they just arise in nature? I'm totally okay with pointers to references here.
I should get a braided tensor structure from a net of factors on a circle. Is this the center of the fusion category, and is it in the literature?
Edit: Based on the (fantastically illuminating) responses, it seems that my bonus question doesn't make sense, because the M-M bimodule fusion category depends on the choice of N in an essential way. Maybe the phrase "conformal defect" should be used somewhere. If I come up with a suitable replacement, I'll present it as a separate question.