  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.
