Two-Categorical Bundles and Their Classifying Spaces

TL;DR

The paper develops a higher gauge-theory framework by defining principal $2\mathcal{C}$-bundles for topological 2-categories and proving that the geometric nerve realization $|\Delta 2\mathcal{C}|$ classifies these bundles up to concordance for good, topological $2$-categories. It connects this theory to Baas–Dundas–Rognes 2-vector bundles, Baez–Crans' variants, and cobordism/string bundles, revealing how concordance classes correspond to homotopy classes into $|Z_•|$ and tying to elliptic cohomology via relevant spectra. The authors establish a robust toolkit: a topological Kan condition, goodness criteria, concordance theory for simplicial spaces, and a small-object argument to replace $Z_•$ with a Kan-fibrant model, culminating in a general classification theorem. This provides a flexible, higher-categorical classification mechanism with broad implications for 2-categorical topology and associated cohomology theories. Overall, the work systematizes classifying spaces for higher bundles and lays groundwork for further exploration of 2-categorical topology and 2-vector bundle cohomology.

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.

Theorems & Definitions (81)

• Theorem 1.1
• Definition 2.1
• Proposition 2.2
• Remark 2.3
• Definition 2.4
• Proposition 2.5
• Definition 2.6
• Proposition 2.7
• Proposition 2.8
• Definition 2.9