Fibrations of groupoids

Abstract

The notion of a covering morphism of groupoids has been developed by P. J. Higgins [4, 51 and shown to be a convenient tool in algebra, even for purely group theoretic results. That covering morphisms of groupoids model conveniently the covering maps of spaces is shown in [l]. If we weaken the conditions for a covering morphism we obtain what we shall call a fibration of groupoids, and our purpose is to explore this notion. The main results are that, even if we start in the category of groups, then certain constructions lead naturally to fibrations of groupoids; that for fibrations of groupoids we can obtain a family of exact sequences of a type familiar to homotopy theorists; and that these exact sequences include not only the bottom part of the usual exact sequence of a fibration of spaces, but also the well known six term exact sequences in the non-Abelian cohomolog!; of groups [6]. A further advantage of our procedure is that the same setup leads naturally to a definition of non-Abelian cohomology in dimensions 0 and 1 of a groupoid with coefficients in a groupoid. This cohomology (which will be dealt with elsewhere) generalises a non-Abelian cohomology of n group with coefficients in a groupoid which has been developed by A. Frohlich (unpublished) with a view to applications in Galois cohomology. Another question not touched on here is possible application of these methods to the non-bbelian Hz. There is some overlap of this paper with techniques used by J. Gray in 1131. However, the aims of that paper are quite different from ours, and so the theory is developed here from the beginning.