Principal bundles, groupoids, and connections

Abstract

We clarify in which precise sense the theory of principal bundles and the theory of groupoids are equivalent; and how this equivalence of theories, in the differentiable case, reflects itself in the theory of connections. The method used is that of synthetic differential geometry. Introduction. In this note, we make explicit a sense in which the theory of principal fibre bundles is equivalent to the theory of groupoids; and in particular, how the differential geometric notion of connection appears in this equivalence. For the latter, we shall utilize the method of Synthetic Differential Geometry, which has for its base the notion of “first neighbourhood of the diagonal” of a manifold. Basically, the notions of “principal bundle” and “groupoid”, their essential equivalence, and the notion of connection in this context, were described by Ehresmann, [4], [5] etc. It is classical to formulate the notion of electromagnetic field in terms of a connection in a principal G-bundle, with G the (abelian) group U(1). Recent particle physics is extending this “gauge theory” viewpoint to non-commutative G, and also to higher “connective structures”, see e.g. [3] and references there. To cope with the mathematical complications arising in this extension process, Breen and Messing [2], [3] found it helpful to utilize some of the formal or “synthetic” method elaborated in the last decades (as in e.g. [7]). The present note hopefully also provides a contribution to such “synthetic gauge theory”, by combining it more firmly with the theory of groupoids. The main vehicle for the relationship between principal bundles and groupoids is a functor which to a principal bundle P associates a transitive groupoid PP; this “Ehresmann functor” (terminology of Pradines [20]) is however, as pointed out in loc. cit., not an equivalence of categories in the sense of category theory, since it is not 2000 Mathematics Subject Classification: 51K10, 53C05, 58H05, 20L05.