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:

1. When people talk about fusion categories coming from subfactors, are they referring to the endomorphism category of one of the factors?
2. How are the endomorphism categories of M and N related? Are they equivalent? Are they Koszul dual?
3. Does the Jones index say something concrete about the category, like Frobenius-Perron dimension? (How does one compute Jones index, anyway?)
4. How do people go about constructing exotic subfactors? Do they just arise in nature? I'm totally okay with pointers to references here.
5. (bonus) 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.

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:

1. When people talk about fusion categories coming from subfactors, are they referring to the endomorphism category of one of the factors?
2. How are the endomorphism categories of M and N related? Are they equivalent? Are they Koszul dual?
3. Does the Jones index say something concrete about the category, like Frobenius-Perron dimension? (How does one compute Jones index, anyway?)
4. How do people go about constructing exotic subfactors? Do they just arise in nature? I'm totally okay with pointers to references here.
5. (bonus) 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?