Baas, Nils A.^{2}; Bökstedt, M.^{4}; Kro, T.A.^{3}

Affiliations:

^{1} Department of Mathematics, Science and Technology, Aarhus University^{2} Department of Mathematical Sciences, NTNU, Trondheim^{3} Østfold University College, Halden^{4} Department of Mathematics, Science and Technology, Aarhus University

DOI:

10.1017/is012001012jkt181

Abstract:

For a 2-category 2C we associate a notion of a principal 2C-bundle. In case of the 2-category of 2-vector spaces in the sense of M.M. Kapranov and V.A. Voevodsky this gives the the 2-vector bundles of N.A. Baas, B.I. Dundas and J. Rognes. Our main result says that the geometric nerve of a good 2-category is a classifying space for the associated principal 2-bundles. In the process of proving this we develop a lot of powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. A calculation based on the main theorem shows that the principal 2-bundles associated to the 2-category of 2-vector spaces in the sense of J.C. Baez and A.S. Crans split, up to concordance, as two copies of ordinary vector bundles. When 2C is a cobordism type 2-category we get a new notion of cobordism-bundles which turns out to be classified by the Madsen-Weiss spaces.

Type:

Journal article

Language:

English

Published in:

Journal of K-theory, 2012, Vol 10, Issue 2, p. 299-369