Two-categorical bundles and their classifying spaces

TY - JOUR

T1 - Two-categorical bundles and their classifying spaces

AU - Baas, Nils A.

AU - Bökstedt, M.

AU - Kro, T.A.

N1 - Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012/10/1

Y1 - 2012/10/1

N2 - 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.

AB - 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.

KW - 2-bundles

KW - classifying spaces

KW - geometric nerve

UR - http://www.scopus.com/inward/record.url?scp=84869167062&partnerID=8YFLogxK

U2 - 10.1017/is012001012jkt181

DO - 10.1017/is012001012jkt181

M3 - Journal article

AN - SCOPUS:84869167062

SN - 1865-2433

VL - 10

SP - 299

EP - 369

JO - Journal of K-Theory

JF - Journal of K-Theory

IS - 2

ER -