Principal bundles with groupoid structure: local vs. global theory and nonabelian ucech cohomology
Abstract
The aim of this paper is to review and discuss in detail local aspects of principal bundles with groupoid structure. Many results, in particular from the second and third section, are already known to some extents, but, due to the lack of a ``unified'' point of view on the subject, I decided nonetheless to (re)define all the main concepts and write all proofs; however, some results are reformulated in a more elegant way, using the division map and the generalized conjugation of a Lie groupoid. In the same framework, I discuss later generalized groupoids and Morita equivalences from a local point of view; in particular, I found a (so far as I know) new characterization of generalized morphisms coming from nonabelian ech cohomology, which allows one to view generalized morphisms as a generalization of classical descent data. I found also a factorization formula for the division map, which is the crucial point in the local formulation of Morita equivalences.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2004
 arXiv:
 arXiv:math/0404449
 Bibcode:
 2004math......4449R
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 53C10;
 58H05
 EPrint:
 97 pages