INSTITUTE OF MATHEMATICS · POLISH ACADEMY OF SCIENCES
BANACH CENTER
PUBLICATIONS
ISSN: 0137-6934(p) 1730-6299(e)
In Ehresmann's footsteps: from group geometries to groupoid geometries
Jean Pradines1 Banach Center Publ. 76 (2007), 87-157
Abstract: 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and may be withdrawn, allowing to encompass and bring together much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.
