For two general polytopal complexes the set of face-wise affine maps between them is shown to be a polytopal complex in an algorithmic way. The resulting algorithm for computing the affine hom-complex is analyzed in detail. There is also a natural tensor product of polytopal complexes, which is the left adjoint functor for Hom. This extends the corresponding facts from single polytopes, systematic study of which was initiated in , . Explicit examples of computations of the resulting structures are included. In the special case of simplicial complexes, the affine hom-complex is a functorial subcomplex of Kozlov’s combinatorial hom-complex , which generalizes Lovász’ well-known construction  for graphs.
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Malkhaz Bakuradze, Alexander Gamkrelidze, Joseph Gubeladze, Affine hom-complexes. Port. Math. 73 (2016), no. 3, pp. 183–205DOI 10.4171/PM/1984