Cosheaf homology
Andrei V. Prasolov
TL;DR
The paper develops a comprehensive cosheaf homology theory within a pro-categorical and site-theoretic framework. It builds left satellites, spectral sequences, and hypercovering methods to relate cosheaf homology to Čech and shape pro-homology, proving key isomorphisms in Hausdorff paracompact spaces and revealing equivalences with singular homology for Alexandroff spaces. This approach addresses limitations of classical Set- and Ab-valued cosheaves by employing pro-objects and quasi-projective resolutions, yielding robust computations and new structural insights. Overall, the work unifies cosheaf homology with shape theory, hypercoverings, and Alexandroff-space topology, offering versatile tools for homology calculations in generalized settings.
Abstract
In this paper the cosheaf homology is investigated from different viewpoints: the behavior under site morphisms, connections with Cech homology via spectral sequences, and description of cosheaf homology using hypercoverings. It is proved that in the case of Hausdorff paracompact spaces, the cosheaf homology in general is isomorphic to the Cech homology, and for a constant cosheaf is isomorphic to the shape pro-homology. In the case of Alexandroff spaces, including finite and locally finite spaces, the cosheaf homology is isomorphic to the singular homology.