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Two-Categorical Bundles and their Classifying Spaces

Published online by Cambridge University Press:  23 February 2012

Nils A. Baas ,
Marcel Bökstedt  and
Tore August Kro
Nils A. Baas
Affiliation:
Department of Mathematical Sciences, NTNU, NO-7034 Trondheim, Norwaybaas@math.ntnu.no
Marcel Bökstedt
Affiliation:
Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmarkmarcel@imf.au.dk
Tore August Kro
Affiliation:
Østfold University College, NO-1757 Halden, Norwaytore.a.kro@hiof.no
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Abstract

For a 2-category 2C we associate a notion of a principal 2C-bundle. For the 2-category of 2-vector spaces, in the sense of M.M. Kapranov and V.A. Voevodsky, this gives 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 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. Another 2-category of 2-vector spaces has been proposed by J.C. Baez and A.S. Crans. A calculation using our main theorem shows that in this case the theory of principal 2-bundles splits, up to concordance, as two copies of ordinary vector bundle theory. 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.

Keywords

2-bundlesclassifying spacesgeometric nerve
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 2 , October 2012 , pp. 299 - 369
Copyright
Copyright © ISOPP 2012

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