A Categorical Approach to L-Convexity

TL;DR

This paper establishes a duality between enriched categories and extended L-convex sets, connecting categorical and discrete convex analysis through a new theoretical framework.

Contribution

It introduces extended L-convex sets and proves a duality theorem linking them with enriched categories, expanding the mathematical understanding of discrete convex structures.

Findings

Duality theorem between enriched categories and extended L-convex sets

Introduction of extended L-convex sets as variants of L-convex structures

Establishment of a one-to-one correspondence between the two classes

Abstract

We investigate an enriched-categorical approach to a field of discrete mathematics. The main result is a duality theorem between a class of enriched categories (called $\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-categories) and that of what we call ($\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-) extended L-convex sets. We introduce extended L-convex sets as variants of certain discrete structures called L-convex sets and L-convex polyhedra, studied in the field of discrete convex analysis. We also introduce homomorphisms between extended L-convex sets. The theorem claims that there is a one to one correspondence (up to isomorphism) between two classes. The thesis also contains an introductory chapter on enriched categories and no categorical knowledge is assumed.