Interpretation of equivariant homology groups of a cosheaf

Determine a rigorous mathematical interpretation for equivariant homology groups of a cellular cosheaf equipped with a finite group action (a G-cosheaf). Clarify how such equivariant homology should be defined and related to the cosheaf chain complexes, and ascertain its connections to established notions in equivariant homology (e.g., Bredon-type constructions) and to structural quantities in symmetric graphic statics.

Background

Throughout the paper, homology of cosheaf chain complexes is computed with an accompanying finite group action, and irreducible components are studied via representation theory. However, an explicit notion of "equivariant homology of a cosheaf"—analogous to classical equivariant homology theories—is not defined or interpreted.

The authors state that it is unclear how to interpret equivariant homology groups of a cosheaf, suggesting the need to develop a formal definition and understanding, potentially linking to frameworks such as Bredon homology or to the behavior of quotient frameworks in rigidity theory.