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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baez, J.: Higher Yang-Mills theory. e-print, abstract and full-text links at http://www.arxiv.org/abs/hep-th/0206130, 2002.
Barrett, J.W. and Mackaay, M.: Categorical representations of categorical groups. Privately communicated draft mansucript, 2002.
Crane, L. and Sheppard, M. D.: 2-categorical Poincare representations and state sum applications. e-preprint, abstract and full-text links at http://www.arXiv.org/abs/math.QA/0306440, 2003.
Crane, L. and Yetter, D. N.: Measurable categories and 2-groups, Appl. Categ. Struct. 13(5–6) (2005).
Ghez, P., Lima, R. and Roberts, J. E.: W * categories, Pacific J. Math. 120(1) (1985), 79–109.
Graf, S. and Mauldin, R. D.: A classification of disintegrations of measures. In Measure and Measurable Dynamics, AMS Contemp. Math, vol. 94, American Mathematical Society, 1989, pp. 147–158.
Kapranov, M. and Voevodsky, V.: 2-categories and Zamolodchikov tetrahedra equations. In Proc. Symp. Pure Math. 56, Part 2, American Mathematical Society, 1994, pp. 177–260.
Maharam, D.: Decompositions of measure algebras and spaces. Trans. Amer. Math. Soc. 69 (1950), 142–160.
Takesaki, M.: Theory of Operator Algebras, I, Springer Encyclopedia of Mathematical Sciences, vol. 124, Springer, 2002.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yetter, D.N. Measurable Categories. Appl Categor Struct 13, 469–500 (2005). https://doi.org/10.1007/s10485-005-9003-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-005-9003-6