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Some aspects of cosheaves on diffeological spaces. (English) Zbl 1448.18016
Summary: 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 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-Čech homology with values in pre-cosheaves is established for diffeological spaces.
MSC:
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
57P99 Generalized manifolds
58A40 Differential spaces
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