Sheaves, principal bundles, and Čech cohomology for diffeological spaces

Abstract

The purpose of this note is to define sheaves for diffeological spaces and give a construction of their Čech cohomology. As an application, we prove that the first degree Čech cohomology classes for the sheaf of smooth functions to an abelian diffeological group $G$ classify the diffeological principal $G$-bundles.

28 pages; v2: besides minor corrections and additions, we also added a subsection comparing this cohomology with that of Iglesias-Zemmour