Alireza Ahmadi - Department of Math. Yazd University Yazd, Iran
Akbar Dehghan Nezhad - School of Mathematics, Iran University of Science and Technology, Narmak,Tehran, ۱۶۸۴۶--۱۳۱۱۴, Iran

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 Poincar\'{e} 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 quasi-cosheaves, defined by pre-cosheaves which respect the colimit over covering generating families, and prove that cosheaves are quasi-cosheaves. Finally, a so-called quasi-\v{C}ech homology with values in pre-cosheaves is established for diffeological spaces.

Cosheaves, quasi-cosheaves, site of plots, covering generating families, quasi-v{C}ech homology, diffeological spaces