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 categor y B with coefficients in a B-cofibred categorical group G. The work is deve loped in a purely abstract setting, but several examples that indicate a ve ry close connection with algebraic and topological problems are discussed e xplicitly For instance, we obtain two new interpretations for the Brauer gr oup of a Galois extension of commutative rings: an algebraic one in terms o f equivalence classes of torsors over the Galois group and a topological on e in terms of homotopy classes of cross-sections for a fibration over an Ei lenberg-MacLane space of type K(G, i). (C) 1999 Elsevier Science B.V. All r ights reserved.