  We study the polytopes of affine maps between two polytopes -- the
hom-polytopes. The hom-polytope functor has a left adjoint -- tensor product
polytopes. The analogy with the category of vector spaces is limited, as we
illustrate by a series of explicit examples exhibiting various extremal
properties. The main challenge for hom-polytopes is to determine their
vertices. A polytopal analogue of the rank-nullity theorem amounts to
understanding how the vertex maps behave relative to their surjective and
injective factors. This leads to interesting classes of surjective maps. In the
last two sections we focus on two opposite extremal cases -- when the source
and target polytopes are both polygons and are either generic or regular.
