–
Room P3.10, Mathematics Building
John Baez, Department of Mathematics, University of California at Riverside
Topological 2-Groups and Their Classifying Spaces
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 and a space , principal -bundles over are classified by either the Cech cohomology or the set of homotopy classes , where is the classifying space of . Here we review work by Bartels, Jurco, Baas-Bökstedt-Kro, and others generalizing this result to topological 2-groups. We explain various viewpoints on topological 2-groups and the Cech cohomology with coefficients in a topological 2-group , also known as "nonabelian cohomology". Then we sketch a proof that under mild conditions on and there is a bijection between and , where is the classifying space of the geometric realization of the nerve of . Applying this result to the ''string 2-group" of a simply-connected compact simple Lie group , we obtain a theory of characteristic classes for principal -2-bundles.