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Two-categorical bundles and their classifying spaces

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  • Nils A. Baas, Department of Mathematical Sciences, NTNU, Trondheim, Norway
  • M. Bökstedt
  • T.A. Kro, Østfold University College, Norway
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
Volume10
Issue2
Pages (from-to)299-369
Number of pages71
ISSN1865-2433
DOIs
Publication statusPublished - 1 Oct 2012

    Research areas

  • 2-bundles, classifying spaces, geometric nerve

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