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 they continue this work.

They begin with a detailed study of measurable categories. Then they give a geometrical description of the measurable representations, intertwiners and \(2\)-intertwiners for any skeletal measurable \(2\)-group. They study tensor products and direct sums for representations, and various concepts of subrepresentation. They describe direct sums of intertwiners, and sub-intertwiners--features not seen in ordinary group representation theory and study irreducible and indecomposable representations and intertwiners. They also study "irretractable" representations--another feature not seen in ordinary group representation theory. Finally, they 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.

Table of Contents

• Introduction
• Representations of \(2\)-groups
• Measurable categories
• Representations on measurable categories
• Conclusion
• Appendix A. Tools from measure theory
• Bibliography