Sheaves, principal bundles, and \v{C}ech cohomology for diffeological spaces

TL;DR

This paper develops sheaf theory and cech cohomology for diffeological spaces, showing that first-degree cohomology classes classify principal bundles, thus extending classical concepts to this broader setting.

Contribution

It introduces sheaves and cech cohomology for diffeological spaces and establishes a classification of principal bundles via cohomology.

Findings

Defined sheaves for diffeological spaces.

Constructed cech cohomology in this context.

Proved classification of principal bundles via first-degree cohomology.

Abstract

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