The n-Category Café

Here’s what it’s about:

A ‘2-group’ is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on ‘2-vector spaces’, which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called ‘measurable categories’ (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie 2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and sub-intertwiners — features not seen in ordinary group representation theory. We classify irreducible and indecomposable representations and intertwiners. We also classify ‘irretractable’ representations — another feature not seen in ordinary group representation theory. Finally, we argue that measurable categories equipped with some extra structure deserve to be considered ‘separable 2-Hilbert spaces’, and compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras.

I’ll try to explain this stuff soon in This Week’s Finds. Right now I’ll just throw it out there as a Christmas present to the universe. If you catch typos or other errors, please let me know.

It’s long, but that’s because it’s self-contained. We start by explaining what a 2-category is, what a 2-group is, what a Kapranov–Voevodsky 2-vector space is, what a measurable category is, and so on… and there are big fat appendices explaining all the measure theory. So, anyone who has taken a class in real analysis and a class in group theory can read this — if trapped with nothing else to do for a sufficiently long time.

Speaking of which: the weather in most of the USA has been dreadful over the last few weeks: blizzard, ice storms, even clouds here in Riverside. Lots of flights have been delayed or canceled; lots of roads have been shut down. So, if you’ve had trouble traveling, you have my sympathies. And if you’re about to fly somewhere, print out a copy of this paper and take it with you! By the time you reach your destination, you may be an expert on 2-group representation theory.

Merry Christmas! Happy Hanukkah! Cool Kwanzaa! Wild Winter Solstice! And a fantastic Festivus!