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Two-categorical bundles and their classifying spaces

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Two-categorical bundles and their classifying spaces. / Baas, Nils A.; Bökstedt, M.; Kro, T.A.

I: Journal of K-Theory, Bind 10, Nr. 2, 01.10.2012, s. 299-369.

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avisTidsskriftartikelForskningpeer review

Harvard

Baas, NA, Bökstedt, M & Kro, TA 2012, 'Two-categorical bundles and their classifying spaces', Journal of K-Theory, bind 10, nr. 2, s. 299-369. https://doi.org/10.1017/is012001012jkt181

APA

Baas, N. A., Bökstedt, M., & Kro, T. A. (2012). Two-categorical bundles and their classifying spaces. Journal of K-Theory, 10(2), 299-369. https://doi.org/10.1017/is012001012jkt181

CBE

MLA

Vancouver

Baas NA, Bökstedt M, Kro TA. Two-categorical bundles and their classifying spaces. Journal of K-Theory. 2012 okt. 1;10(2):299-369. doi: 10.1017/is012001012jkt181

Author

Baas, Nils A. ; Bökstedt, M. ; Kro, T.A. / Two-categorical bundles and their classifying spaces. I: Journal of K-Theory. 2012 ; Bind 10, Nr. 2. s. 299-369.

Bibtex

@article{ac98245da86741779a00aa812cae8977,
title = "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.",
keywords = "2-bundles, classifying spaces, geometric nerve",
author = "Baas, {Nils A.} and M. B{\"o}kstedt and T.A. Kro",
note = "Copyright 2012 Elsevier B.V., All rights reserved.",
year = "2012",
month = oct,
day = "1",
doi = "10.1017/is012001012jkt181",
language = "English",
volume = "10",
pages = "299--369",
journal = "Journal of K-Theory",
issn = "1865-2433",
publisher = "Cambridge University Press",
number = "2",

}

RIS

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

VL - 10

SP - 299

EP - 369

JO - Journal of K-Theory

JF - Journal of K-Theory

SN - 1865-2433

IS - 2

ER -