Applied Categorical Structures

, Volume 13, Issue 5, pp 469–500

Measurable Categories

Authors

    • Department of MathematicsKansas State University
Article

DOI: 10.1007/s10485-005-9003-6

Cite this article as:
Yetter, D.N. Appl Categor Struct (2005) 13: 469. doi:10.1007/s10485-005-9003-6

Abstract

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of 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 [3] and [4] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are established along the way: first it is shown that all invertible additive bounded 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 and second we establish the distributivity of Hilbert space tensor product over direct integrals over Lusin spaces with respect to σ-finite measures. The paper concludes with a general definition of measurable bicategories.

Mathematics Subject Classifications (2000)

Primary 18D05Secondary 18D10, 28A20

Key words

monoidal categorymonoidal bicategoryBorel spacedirect integralmeasurable cetegory
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Copyright information

© Springer Science+Business Media, Inc. 2005