Journal of Pure and Applied Algebra 143 (1999) 107–154 www.elsevier.com/locate/jpaa Cohomology of cobred categorical groups A.M. Cegarra a;∗ , L. Fern andez b a Departamento de  Algebra, Facultad de Ciencias, Universidad de Granada 18071, Granada, Spain b Departamento de Matem atica Aplicada, Facultad de Ciencias, Universidad de Granada 18071, Granada, Spain Dedicated to Michael Barr on his 60th birthday Abstract In this paper we study and interpret a certain non-abelian cohomology H i (B; G), 0 ≤ i ≤ 2, of a small category B with coecients in a B-cobred categorical group G. The work is developed in a purely abstract setting, but several examples that indicate a very close connection with algebraic and topological problems are discussed explicitly. For instance, we obtain two new interpretations for the Brauer group of a Galois extension of commutative rings: an algebraic one in terms of equivalence classes of torsors over the Galois group and a topological one in terms of homotopy classes of cross-sections for a bration over an Eilenberg–MacLane space of type K (G; 1). c  1999 Elsevier Science B.V. All rights reserved. A categorical group is a monoidal groupoid in which each object has a quasi-inverse with respect to the tensor product. Since a groupoid with only one object is just a group, a categorical group with only one object is a group together with a second group struc- ture, given by the tensor product, and both with the same unit. The interchange law implies that both multiplications are equal and the group is commutative. Therefore, categorical groups with only one object are the same as abelian groups. This paper is concerned with a certain kind of non-abelian cohomology H i (B; G), 0 ≤ i ≤ 2, dened for a small category B, with coecients taken as bundles of categorical groups, in- stead of bundles of abelian groups, i.e., B-modules, as is usual. We consider coecients in B-categorical groups since they arise in numerous algebraic and topological prob- lems, such as the classication of arbitrary extensions of groups, the classication of Azumaya algebras over a commutative ring, or the classication of the cross-sections ∗ Corresponding author. Partially supported by DGICYT PB94-0823. 0022-4049/99/$ – see front matter c  1999 Elsevier Science B.V. All rights reserved. PII: S0022-4049(98)00109-1