Two-Categorical Bundles and Their Classifying Spaces
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.
Cite
@article{arxiv.math/0612549,
title = {Two-Categorical Bundles and Their Classifying Spaces},
author = {Nils. A. Baas and Marcel Bokstedt and Tore August Kro},
journal= {arXiv preprint arXiv:math/0612549},
year = {2008}
}
Comments
LaTex, 64 pages, revised version