Four Equivalent Versions of Non-Abelian Gerbes

TL;DR

This paper demonstrates the equivalence of four different frameworks for smooth non-abelian gerbes, connecting recent developments and establishing new relations among them.

Contribution

It proves the equivalence of four versions of non-abelian gerbes and establishes new links between non-abelian cohomology, bundle gerbes, and principal 2-bundles.

Findings

Proves bijection between continuous and smooth non-abelian cohomology.

Establishes explicit equivalence between bundle gerbes and principal 2-bundles as 2-stacks.

Partially improves four existing frameworks for non-abelian gerbes.

Abstract

We recall and partially improve four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results we prove are a bijection between continuous and smooth non-abelian cohomology, and an explicit equivalence between bundle gerbes and principal 2-bundles as 2-stacks.