  We introduce the notion of a \emph{resolution supported on a poset}. When the
poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover
the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of
Reiner and Welker, and of Velasco, has shown that there are monomial ideals
whose minimal free resolutions are not cellular, hence cannot be supported on
any CW-poset. We show that for any monomial ideal there is a \emph{homology
CW-poset} that supports a minimal free resolution of the ideal. This allows one
to extend to every minimal resolution, essentially verbatim, techniques
initially developed to study cellular resolutions. As two demonstrations of
this process, we show that minimal resolutions of toric rings are supported on
what we call toric hcw-posets, and we give a new combinatorial proof of a
fundamental result of Miller on the relationship between Artininizations and
Alexander duality of monomial ideals.

Commutative Algebra;Algebraic Geometry;Combinatorics