Four equivalent versions of nonabelian gerbes

Abstract

We recall and partially improve four versions of smooth, nonabelian gerbes: Cech 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.