Some aspects of cosheaves on diffeological spaces

Abstract

We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincaré groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of the D-topological structure. We also study quasicosheaves, defined by pre-cosheaves which respect the colimit over covering generating families, and prove that cosheaves are quasi-cosheaves. Finally, a so-called quasi-Čech homology with values in pre-cosheaves is established for diffeological spaces.

Diffeological Čech cohomology

“…Diffeological Čech cohomology is formulated in terms of covering generating families to detect compatible data on them in the zero degree (Proposition 4.6), and gives rise to an exact ∂ -functor of the section functor for sheaves on diffeological spaces (Proposition 4.7). It is also dual to quasi-Čech homology defined in [2].…”