On the classification of 2-gerbes and 2-stacks / Lawrence Breen

Bibliogr. p. 155-160

According to J. Giraud, the degree two cohomology classes of a space X with values in a non-abelian sheaf of groups describe equivalence classes of gerbes on X. We examine here the analogous concept of a 2-gerbe on a space X. It is proved that such a 2-gerbe, when suitably trivialized, is described up to equivalence by a nonabelian degree three cohomology class. An inverse construction, based on the notion of higher descent, shows that this cohomology class entirely characterizes the 2-gerbe up to equivalence. A first application of these results is a detailed description of the 2-gerbe of realizations of a lien. This embodies a vast generalization of Eilenberg and Mac Lane's well-known cohomological obstruction to the realization of an abstract kernel. Another application is the cohomological classification, it à la Postnikov, of stacks and 2-stacks with given homotopy sheaves. Finally, It is shown how this theory yields a unified approach to the problem of defining and classifying group laws (possibly constrained to satisfy appropriate commutativity conditions) on categories and 2-categories. This gives as a special case the analogous result for group laws on categories and 2-cateogries. (SMF)