This work arose from efforts to generalise the usual cubical boundary by using different `weights' for opposite faces, but still to obtain a chain complex, and this method was found to generalise. We describe a variant of the classical singular cubical homology theory, in which the usual boundary (n-1)-cubes of each n-cube are replaced by combinations of internal (n-1)-cubes parallel to the boundary. This defines a generalised homology theory, but the usual singular homology can be recovered by taking the quotient by the degenerate singular cubes.
Journal of Homotopy and Related Structures, Vol. 6(2011), No. 1, p. 71-101