  We study the vertices of the polytopes of all affine maps (a.k.a.
hom-polytopes) between higher dimensional simplices, cubes, and crosspolytopes.
Systematic study of general hom-polytopes was initiated in [3]. The study of
such vertices is the classical aspect of a conjectural homological theory of
convex polytopes. One quickly runs into open problems even for simple source
and target polytopes. The vertices of Hom(simplex_m,-) and Hom(-,cube_n) are
easily understood. In this work we describe the vertex sets of
Hom(box_m,simplex_n), Hom(diamond_m,simplex_n), and Hom(diamond_m,diamond_n).
The emergent pattern in our arguments is reminiscent of diagram chasing in
homological algebra.
