Diffeological Čech 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 Čech cohomology, which is an exact ∂ -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 Čech cohomology to the de Rham cohomology. Moreover, we characterize the isomorphism classes of diffeological fiber, principal, and vector bundles as (non-abelian) diffeological Čech cohomology in degree 1.