Two-categorical bundles and their classifying spaces

Nils A. Baas, M. Bökstedt, T.A. Kro

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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.
Original languageEnglish
JournalJournal of K-Theory
Pages (from-to)299-369
Number of pages71
Publication statusPublished - 1 Oct 2012


  • 2-bundles
  • classifying spaces
  • geometric nerve


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