Cellular cosheaf homology are cosheaf homology

TL;DR

This paper establishes a natural correspondence between cellular cosheaf homology and cosheaf homology, enabling the definition of homology for cellular cosheaves as a dual to sheaf cohomology.

Contribution

It introduces a method to associate a cosheaf to a cellular cosheaf, allowing homology to be defined as a dual to sheaf cohomology, especially on simplicial complexes.

Findings

Borel-Moore homology of cellular cosheaves is isomorphic to the homology of an associated cosheaf.

A natural construction of a cosheaf from a cellular cosheaf is provided.

Homology of cellular cosheaves can be understood via this cosheaf association.

Abstract

A cosheaf is the dual notion of a sheaf, but we cannot define its homology as the formal dual of sheaf cohomology, in general, because of the lack of the cosheafification. A cellular cosheaf is a contravariant functor from the face poset of a CW complex to the category of abelian groups. We show that given a cellular cosheaf $F$, there is a natural way to associate a cosheaf $\widehat{F}$, for which we can define homology as the formal dual of sheaf cohomology, such that the Borel-Moore homology of $F$ is isomorphic to the homology of $\widehat{F}$ whenever the underlying CW complex of $F$ is a simplicial complex.