Lie Groupoids as generalized atlases
Differential Geometry
2007-11-15 v1 Category Theory
Abstract
Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the "virtual structure" of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This "structure" keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings and surmersions, yielding a very polymorphous unifying theory. We suggest further developments.
Cite
@article{arxiv.0711.2077,
title = {Lie Groupoids as generalized atlases},
author = {Jean Pradines},
journal= {arXiv preprint arXiv:0711.2077},
year = {2007}
}
Comments
34 pages, lecture delivered at the 5th Conference on Geometry ans Topology of Manifolds, Krynica (Poland), April 2003