Memoirs of the American Mathematical Society 2012; 120 pp; softcover Volume: 219 ISBN10: 0821872842 ISBN13: 9780821872840 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/219/1032
 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 finitedimensional \(2\)vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinitedimensional \(2\)vector spaces called "measurable categories" (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinitedimensional 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 subintertwinersfeatures not seen in ordinary group representation theory and study irreducible and indecomposable representations and intertwiners. They also study "irretractable" representationsanother 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
