Four equivalent versions of nonabelian gerbes

Thomas Nikolaus and Konrad Waldorf

Vol. 264 (2013), No. 2, 355–420

DOI: 10.2140/pjm.2013.264.355

Abstract

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

Keywords

nonabelian gerbe, principal 2-bundle, 2-group, nonabelian cohomology, 2-stack

Mathematical Subject Classification 2010

Primary: 55R65

Secondary: 53C08, 55N05, 22A22

Milestones

Received: 23 April 2012

Accepted: 19 November 2012

Published: 28 July 2013