TITLE: How do I describe a fusion category given a subfactor?
QUESTION [14 upvotes]: 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.
(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.

REPLY [2 votes]: There are lots of great answers above, I'm just going to try for as succinct of answers as I can to all the questions.

In the II_1 case, the category is the subcategory of the category of bimodules over N tensor generated by M.  (Similarly you get another category by reversing the roles of N and M).  For III_1 factors you can use endomorphisms instead of bimodules.  This tensor category is fusion iff the subfactor is finite depth.  You can consider either algebraic bimodules or L^2 bimodules and you end up with the same category by a preprint of Vaughan's.
Morita equivalent with the equivalence given by an appropriate bimodule category of N-M bimodules.
The Jones index measures the FP dimension not of the whole category, but of your particular favorite object M.
There are no deeply satisfying constructions yet.  All the constructions are by hands and somewhat messy.  The descriptions in Emily Peters thesis and related papers at least give a nice description of the category by generators and relations, but the proof that they exist is still very computational.  Everyone would love if someone managed to give a really conceptual construction of one of these beasts.