Hom-polytopes

TL;DR

This paper explores the structure of hom-polytopes, affine maps between polytopes, examining their vertices, extremal properties, and special cases involving polygons, with insights into their categorical and geometric characteristics.

Contribution

It introduces the concept of hom-polytopes and their tensor product, analyzes their vertices, and studies extremal cases for polygons, expanding understanding of affine map polytopes.

Findings

Characterization of vertices of hom-polytopes

Identification of extremal properties in specific cases

Analysis of affine maps between polygons

Abstract

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.