Schreiber Nonabelian cocycles and their quantum symmetries

This is an old set of notes of mine on, essentially, local prequantum field theory formulated in higher geometry:

• Urs Schreiber, Nonabelian cocycles and their quantum symmetries, 2008 (pdf, pdf)

The note itself has been abandoned; the ideas have meanwhile grown into the articles that are listed at differential cohomology in a cohesive topos.

Abstract

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 $\infty$-groupoids or even to general $\infty$-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 $\sigma$-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.