Cohomology of cofibred categorical groups

In this paper we study and interpret a certain non-abelian cohomology H-i(B, G), 0 less than or equal to i less than or equal to 2, of a small category B with coefficients in a B-cofibred 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 fibration over an Eilenberg-MacLane space of type K(G, i). (C) 1999 Elsevier Science B.V. All rights reserved.