| Popis: |
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 Ĥ1(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 Ĥ1(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 Ĥ1(M, G) ≡ [M,B|G|]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 groupG, 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 4(BG,Q). |
