In Ehresmann's footsteps: from group geometries to groupoid geometries
The geometric understanding of Cartan connections led Charles Ehresmann from the Erlangen program of (abstract) transformation groups to the enlarged program of Lie groupoid actions, via the basic concept of structural groupoid acting through the fibres of a (smooth) principal fibre bundle or of its associated bundles, and the basic examples stemming from the manifold of jets (fibred by its source or target projections). We show that the remarkable relation arising between the actions of the structural group and the structural groupoid (which are mutually determined by one another and commuting) may be viewed as a very special (unsymmetrical!) instance of a general fully symmetric notion of "conjugation between principal actions" and between "associated actions", encapsulated in a nice "butterfly diagram". In this prospect, the role of the local triviality looks more incidental, and may be withdrawn, allowing to encompass and bring together much more general situations. We describe various examples illustrating the ubiquity of this concept in Differential Geometry, and the way it unifies miscellaneous aspects of fibre bundles and foliations. We also suggest some tracks (to be developed more extensively elsewhere) for a more efficient implementation of the basic principle presently known as "internalization", pioneered by Ehresmann in his very general theory of "structured" categories and functors, towards the more special but very rich and far-reaching study of the above-mentioned Lie groupoid actions. Still now, due to misleading and conflicting terminologies, the latter concept seems too often neglected (and sometimes misunderstood) by too many geometers, and has long been generally ignored or despised by most "pure categorists", though it will be presented here as one of the gems of Ehresmann's legacy.