Cosheaf homology

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.