Titre : | On the classification of 2-gerbes and 2-stacks | Type de document : | texte imprimé | Auteurs : | Lawrence BREEN, Auteur | Editeur : | Paris : Société Mathématique de France | Année de publication : | 1994 | Collection : | Astérisque, ISSN 0303-1179 num. 225 | Importance : | 160 p. | Langues : | Anglais (eng) | Catégories : | 18D05 18D10 18D30 18G50 55S45
| Résumé : | 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. | Note de contenu : | bibliogr. |
On the classification of 2-gerbes and 2-stacks [texte imprimé] / Lawrence BREEN, Auteur . - Société Mathématique de France, 1994 . - 160 p.. - ( Astérisque, ISSN 0303-1179; 225) . Langues : Anglais ( eng) Catégories : | 18D05 18D10 18D30 18G50 55S45
| Résumé : | 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. | Note de contenu : | bibliogr. |
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