Measurable Categories

Abstract

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of bounded operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in joint work with Crane as the basis for a representation theory of (Lie) 2-groups. Several important technical results are established along the way: First it is shown that all bounded invertible additive functors (and thus a fortiori all invertible *-functors) between categories of measurable fields of Hilbert spaces are induced by invertible measurable transformations between the underlying Borel spaces. Second the distributivity of Hilbert space tensor product over direct integrals over Lusin spaces with respect to $σ$-finite measures is established. The paper concludes with a general definition of measurable bicategories.