Abstract

The aim of these notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are for the most part based on the author’s texts [1]-[4]. Our main goal is to describe the construction which associates to a gerbe or a 2-gerbe the corresponding non-abelian cohomology class. We begin by reviewing the well-known theory for principal bundles and show how to extend this to biprincipal bundles (a.k.a bitorsors). After reviewing the definition of stacks and gerbes, we construct the cohomology class associated to a gerbe. While the construction presented is equivalent to that in [4], it is clarified here by making use of diagram (5.1.9), a definite improvement over the corresponding diagram [4] (2.4.7), and of (5.2.7). After a short discussion regarding the role of gerbes in algebraic topology, we pass from 1 − to 2−gerbes. The construction of the associated cohomology classes follows the same lines as for 1-gerbes, but with the additional degree of complication entailed by passing from 1- to 2-categories, so that it now involves diagrams reminiscent of those in [5]. Our emphasis will be on explaining how the fairly elaborate equations which define cocycles and coboundaries may be reduced to terms which can be described in the tradititional formalism of non-abelian cohomology. Since the concepts discussed here are very general, we have at times not made explicit the mathematical

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