Here is a reference -first result in google:
The author (who now works in a bank) needs his construction of sheaf homology to prove a criterion for the vanishing of certain integrals of differential forms, though the paper treats general topological spaces.
The overarching goal seems interesting, as the names of Langlands, Igusa, Denef are mentioned, but is beyond me, unhelpfully.
Anyway that looks like very nice mathematics. There are actually 2 constructions of sheaf homology. Both rely on triangulations, the first proceeds defining modules of sections on simplices and restriction maps to subsimplices and extending by linearity to complexes. The second, used to prove Poincaré duality/isomorphism with (usual) sheaf cohomology, proceeds defining sheaves "on simplicial complexes" and taking the homology of the chain complex of global sections of these.
The 2 constructions are isomorphic, and I guess they could be isomorphic to a more direct construction which only uses open covers.
What is the paper you are referring to?