On the classification of $2$-gerbes and $2$-stacks

On the classification of $2$-gerbes and $2$-stacks

Lawrence BREEN
On the classification of $2$-gerbes and $2$-stacks
  • Année : 1994
  • Tome : 225
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
  • Class. Math. : 18G50, 18D05, 18D10, 18D30, 55S45
  • Nb. de pages : 160
  • ISSN : 0303-1179
  • DOI : 10.24033/ast.271

According to J. Giraud, the degree two cohomology es of a space $X$ with values in a non-abelian sheaf of groups describe equivalence es 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 . An inverse construction, based on the notion of higher descent, shows that this cohomology 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 ification, 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 ifying 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.

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