Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 12 1 2020 01 01 Some aspects of cosheaves on diffeological spaces 123 147 87119 10.29252/cgasa.12.1.123 EN Alireza Alireza Ahmadi Department of Math. Yazd University Yazd, Iran Akbar Dehghan Nezhad School of Mathematics, Iran University of Science and Technology, Narmak,Tehran, 16846--13114, Iran Journal Article 2018 10 17 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. https://cgasa.sbu.ac.ir/article_87119_6b625005860bfe6a4bd5da17a099b89b.pdf