Extension Theory for Local Groupoids

We intend to provide an algebraic framework in which some of the algebraic theory of connections become identical to some of the (Eilenberg Mac Lane) theory of extensions of non-abelian groups. In particular, the Bianchi identity for the curvature of a connection is related to one of the crucial equations in extension theory. The Bianchi identity as a purely combinatorial fact was dealt with in [12], where the relationship to differential geometry was substantiated; this relationship will not be an issue here, and we do not presuppose [12]. The algebraic notion underlying our project is that of local groupoid; this is a, rather evident, widening of Van Est’s and Swierczkowski’s notion of local group [16], which is the context in which the latter studied extension theory. But by considering local groups only, one fails to bring the reflexive symmetric graphs into the scope, and they are the local groupoids that carry the connection theory. Extension theory for (global) groupoids was studied by Brown and Higgins [2], but again, the graphs are not included under groupoids either. Finally, Kirill Mackenzie, in [15] and elsewhere studied extension theory for Lie algebroids, which is explicitly presented as including (differential geometric) connection theory, and is modelled on the classical extension theory. In some sense, Lie algebroids are infinitesimal (a special case of local) groupoids, but I haven’t yet been able to get Mackenzie’s results (notably [15] Theorem IV.3.20) on Lie Algebroids out as part of the theory to be presented here; its relationship to his theory is therefore at present only an analogy. I want to thank him for his interest and some correspondence which prompted the present research. I also want to thank Professor Van Est, who in 1991 sent me an inspiring letter and a 1976 manuscript [5], advo-