ANOTHER COBOUNDARY OPERATOR FOR DIFFERENTIAL FORMS WITH VALUES IN THE LIE ALGEBRA OF A GROUP  A CHAPTER IN SYNTHETIC DIFFERENTIAL GEOMETRY OF GROUPOIDS

Hirokazu Nishimura (Japan)

Received January 5, 2007

Abstract

Kock [5] has considered differential forms with values in a group in a context where neighborhood relations are available. By doing so, he has made it clear where the socalled Maurer-Cartan formula should come from. In this paper, while we retain the classical definition of differential form with values in the Lie algebra of a group, we propose another definition of coboundary operator for the de Rham complex in a highly general microlinear context, in which neighborhood relations are no longer in view. Using this new definition of the coboundary operator, it is to be shown that the main result of Kock’s paper mentioned above still prevails in our general microlinear context. Our considerations will be carried out within the framework of groupoids.

Keywords and phrases: differential form, synthetic differential geometry, groupoid, coboundary operator, Maurer-Cartan equation.