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On the classification of 2-gerbes and 2-stacks. (English) Zbl 0818.18005

Astérisque. 225. Paris: Société mathématique de France, 160 p. FF 140.00; $ 26.00 /sc (1994).
The author investigates the sheaf theoretical and cohomological structures associated with 2-categories and, more generally, with \(n\)- categories. The basic tools for achieving this goal are the notions of stack, of gerbe, of cocycle, of torsor. The author devotes various chapters to the classification of stacks, gerbes, and torsors. He also considers various applications of the theory of nonabelian \(H^ 3\). While \(n\)-categories appear at various places it is nevertheless in the case of 2-categories that most results are developed. The bibliography is rather extensive and the author explains carefully the connections between his work and previous results by other authors.

MSC:

18G50 Nonabelian homological algebra (category-theoretic aspects)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18-02 Research exposition (monographs, survey articles) pertaining to category theory
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18D30 Fibered categories
55S45 Postnikov systems, \(k\)-invariants
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
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