The note itself has been abandoned; the ideas have meanwhile grown into the articles that are listed at differential cohomology in a cohesive topos.
Nonabelian cohomology may be regarded as a generalization of group cohomology to the case where both the group itself as well as the coefficient object are allowed to be generalized to -groupoids or even to general -categories. Cocycles in nonabelian cohomology in particular represent higher principal bundles (gerbes) – possibly equivariant, possibly with connection – as well as the corresponding associated higher vector bundles.
We propose a systematic formalization of the -model quantum field theory associated with a given nonabelian cocycle, regarded as a background field, expanding on constructions studied in Freed, Willerton, Bartlett.
In a series of examples we show how this formalization reproduces familiar structures, for instance in Dijkgraaf-Witten theory and in the Yetter model.