Vertex maps between Δ, ▭, and (sic)

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 and are easily understood. In this work we describe the vertex sets of , , and . The emergent pattern in our arguments is reminiscent of diagram chasing in homological algebra.