Diffeological \v{C}ech cohomology

TL;DR

This paper introduces a new cohomology theory for diffeological spaces, called diffeological Čech cohomology, which connects to de Rham cohomology and classifies bundles, providing tools for analyzing complex geometric structures.

Contribution

It develops diffeological Čech cohomology as an exact functor, establishes a Mayer-Vietoris sequence, and links it to de Rham cohomology and bundle classification.

Findings

Established a generalized Mayer-Vietoris sequence.

Connected diffeological Čech cohomology to de Rham cohomology.

Characterized bundle isomorphism classes via degree 1 cohomology.

Abstract

Motivated by problems in which data are given over covering generating families, we suggest a new cohomology theory for diffeological spaces, called diffeological \v{C}ech cohomology, which is an exact $ \partial $-functor of the section functor for sheaves on diffeological spaces. As applications, under the situations of a setup, i) the generalized Mayer-Vietoris sequence for a diffeological space is established; ii) a version of the de Rham theorem is obtained, which connects diffeological \v{C}ech cohomology to the de Rham cohomology. Moreover, we characterize the isomorphism classes of diffeological fiber, principal, and vector bundles as (non-abelian) diffeological \v{C}ech cohomology in degree 1.