The Classifying Space of a Topological 2-Group

TL;DR

This paper extends the classification of principal bundles to principal 2-bundles associated with topological 2-groups, providing a bijection between nonabelian Cech cohomology and homotopy classes of maps into classifying spaces, with applications to string 2-groups.

Contribution

It offers an elementary proof of the classification of principal 2-bundles via nonabelian cohomology and classifying spaces, generalizing classical results to topological 2-groups.

Findings

Established a bijection between Cech cohomology with coefficients in a topological 2-group and homotopy classes of maps into the classifying space.

Applied the classification to the string 2-group, deriving rational characteristic classes from cohomology of BG.

Extended classical bundle classification results to the setting of topological 2-groups and 2-categories.

Abstract

Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bokstedt-Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and Cech cohomology with coefficients in a topological 2-group C, also known as 'nonabelian cohomology'. Then we give an elementary proof that under mild conditions on M and C there is a bijection between the first Cech cohomology of M with coefficients in C and [M,B|C|] where B|C| is the classifying space of the geometric realization of the nerve of C. Applying this result to the 'string 2-group' String(G) of a simply-connected compact simple Lie group G, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of the rational cohomology of BG modulo the ideal generated by c, where c is any nonzero element in the 4th cohomology of BG.