  We use the natural homeomorphism between a regular CW-complex X and its face
poset P_X to establish a canonical isomorphism between the cellular chain
complex of X and the result of applying the poset construction of
arXiv:0806.4532v1 to P_X. For a monomial ideal whose free resolution is
supported on a regular CW-complex, this isomorphism allows the free resolution
of the ideal to be realized as a CW-poset resolution. Conversely, any CW-poset
resolution of a monomial ideal gives rise to a resolution supported on a
regular CW-complex.
