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.
Best Answer
Here are some partial answers:
1- Usually the fusion category is the category of bifinite correspondences, i.e. Hilbert spaces with actions of $N$ and $M$ whose module dimensions are finite. Jones has a result saying that a bifinite correspondence is irreducible if and only if the algebraic module of bounded vectors is irreducible (on his website, two subfactors and the algebraic decomposition...). This means the fusion category of bifinite correspondences should be equivalent to the fusion category of algebraic bimodules (you probably need bifinite in the sense of Lueck, but this is very technical). The former category is generated by the $N-M$ correspondence $L^(M)$, and the later category is generated by $M$ as an $N-M$ bimodule (generated in the sense of taking tensor products and decomposing into irreducibles). In fact, Morrison, Peters, and Snyder use the algebraic category in their recent paper on extended Haagerup (arXiv:0909.4099v1).
2- This isn't what you're asking, but $L^2(M)$ as an $N-M$ bimodule is a Morita equivalence from $N$-Hilbert modules to $M_1$-Hilbert modules where $M_1$ is the basic construction of $N\subset M$. I just think it's an interesting point to bring up.
4- One of the best ways of constructing subfactors is via planar algebras. Given a suitable fusion category, one can construct a planar algebra. Typical examples of these nice fusion categories are the fusion categories arising from the representation theory of a finite group or a quantum group. This gives rise to a family of subfactors. In fact, since (for finite groups) there are only finitely many irreducible representations, we have that this planar algebra will be finite depth (see arXiv:0808.0764, section 4.1), and the subfactor constructed from this planar algebra will be finite depth as well. When someone says "exotic subfactor," they mean a finite index, finite depth subfactor that doesn't appear in the well known families coming from these fusion categories. To date, the best way of constructing these subfactors is to stumble upon a finite bipartite graph which doesn't appear as a fusion graph determine if it can be a principal graph for a subfactor. This has inspired a program to classify all principal graphs which can occur (see the extended Haagerup paper for a synopsis of this as well).
Tie in to 3- Two exotic subfactors, namely the Haagerup and extended Haagerup subfactor, have been constructed by finiding a subfactor planar algebra with the appropriate principal graph inside the graph planar algebra of the bipartite graph (this technique was first explored in detail in Peters' thesis). These subfactors have index equal to the square of the norm of the graph, which is the Perron-Frobenius eigenvalue. In fact, if a finite index subfactor is extremal (irreducible implies extremal), then the norm squared of the principal graph is always the Jones index. (One typically computes Jones index by computing the von Neumann dimension of the $N$-Hilbert module $L^2(M)$.)
5- I know that Kawahigashi et al. (see arXiv:0811.4128) have found a net of type $III_1$-factors corresponding to intervals on the circle. I would recommend starting there.