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

Auteur principal : Breen, Lawrence , 1944-, AuteurType de document : MonographieCollection : Astérisque, 225Langue : anglais.Pays : France.Éditeur : Paris : Société Mathématique de France, 1994Description : 1 vol. (160 p.) ; 24 cmISSN : 0303-1179.Bibliographie : Bibliogr. p. 155-160.Sujet MSC : 18G50, Homological algebra in category theory, derived categories and functors, Nonabelian homological algebra
18N10, Higher categories and homotopical algebra, 2-categories, bicategories, double categories
18D30, Category theory; homological algebra - Categorical structures, Fibered categories
18M05, Monoidal categories and operads, Monoidal categories, symmetric monoidal categories
55S45, Operations and obstructions in algebraic topology, Postnikov systems, k-invariants
En-ligne : Résumé
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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)

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