Two-Categorical Bundles and their Classifying Spaces. Baas, N. A., Bökstedt, M., & Kro, T. A. Journal of K-Theory, 10(2):299–369, October, 2012. ![Two-Categorical Bundles and their Classifying Spaces [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper doi abstract bibtex 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.
@article{baas_two-categorical_2012,
title = {Two-{Categorical} {Bundles} and their {Classifying} {Spaces}},
volume = {10},
issn = {1865-2433, 1865-5394},
url = {https://www.cambridge.org/core/journals/journal-of-k-theory/article/twocategorical-bundles-and-their-classifying-spaces/2CA1CF6541AF7FF61932A872BF136FE1},
doi = {10.1017/is012001012jkt181},
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.},
language = {en},
number = {2},
urldate = {2018-06-07},
journal = {Journal of K-Theory},
author = {Baas, Nils A. and Bökstedt, Marcel and Kro, Tore August},
month = oct,
year = {2012},
keywords = {2-bundles, classifying spaces, geometric nerve},
pages = {299--369},
}
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