Affine hom-complexes

TL;DR

This paper develops an algorithmic framework for affine hom-complexes between polytopal complexes, introduces a tensor product operation, and explores their properties, extending classical results from polytopes and graphs.

Contribution

It introduces a systematic method to construct affine hom-complexes for polytopal complexes and establishes a tensor product as a left adjoint, extending known polytope and graph theories.

Findings

Affine hom-complexes form polytopal complexes with an explicit algorithm.

A tensor product of polytopal complexes is defined as a left adjoint to Hom.

In the simplicial case, the affine hom-complex relates to Kozlov's and Lovasz's complexes.

Abstract

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 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 [6,12]. 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 [14], which generalizes Lovasz' well-known construction [15] for graphs.