The classifying space of a topological 2-group

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 Čech cohomology Ȟl(M,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-Bökstedt-Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and the Cech cohomology Ȟ1 (Af, G) with coefficients in a topological 2-group G, also known as "nonabelian cohomology". Then we give an elementary proof that under mild conditions on M and G there is a bijection Ȟ1(M,G) ≅ [M, B\G\] where B\G\ is the classifying space of the geometric realization of the nerve of G. 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 H(BG, Q)/(c), where c is any generator of H4(BG, Q). © Springer-Verlag Berlin Heidelberg 2009.