The Classifying Space of a Topological 2-Group

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 Ȟ(M, G) or the set of homotopy classes [M, BG], where BG is the classifying space of G. Here we review work by Bartels, Jurčo, 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 Čech cohomology Ȟ(M,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 Ȟ(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 H(BG,Q).