Combinatorial notions relating to principal fibre bundles

Abstract

This article aims at clarifying the relationship between principal fibre bundles and groupoids, and along with it, the relationship between connections in the bundle or groupoid, and the associated connection forms. These notions are essentially due to E. Cartan [2] and to Ehresmann [4], [5], who in them saw some of the fundamental aspects of differential geometry, cf. also [IO]. To this end, we introduce the notion of pregroupoid over a base ‘space’ B (‘space’ may mean either ‘topological space’, ‘smooth manifold’, or ‘object in a topos (;5’ ‘, and accordingly for ‘map’ (or ‘operation’ or ‘law’)). We use the word ‘set’ synonymously with ‘space’, and give some standard comments for this abuse below. Formally, a pregroupoid over B is a set E-B over B equipped with a partially defined ternary operation A, satisfying certain equations. In essence, a pregroupoid over B is the same as a principal fibre bundle, or torsor, over B, but whereas for a torsor, a group has to be given in advance, a pregroupoid canonically creates its own group. Also, by a dual construction, a pregroupoid creates a groupoid over B. Identifying the pregroupoid with a principal bundle H, this groupoid is Ehresmann’s HH- ‘, [4]. In the context of differential geometry, a typical example of a pregroupoid is the bundle E of orthonormal frames on a Riemannian manifold B; for x,y,z such frames, with x and z being frames at the same point of B, L(x, y,z) is the frame (at the same point as y) which has the same coordinates in terms of y as z does in terms of x. To describe the relationship between connections and connection forms, we need to assume that the base ‘space’ B comes equipped with a reflexive symmetric ‘neighbour’ relation. Except for the two trivial extreme cases, topological spaces do not carry any natural relation of this kind, nor do smooth manifolds. However, for the latter, the method of synthetic differential geometry (cf. e.g. [7]) becomes available: essentially, it consists in embedding the category Mf of smooth manifolds into a suitable ‘well-adapted’ topos (5’. When viewed in 6, any smooth manifold does