Betweenness relations in a categorical setting

TL;DR

This paper explores betweenness relations through category theory, demonstrating the existence of a homogeneous relation and properties like reflectivity, with applications to lattice theory and completions.

Contribution

It introduces a categorical framework for betweenness relations, showing the existence of a homogeneous relation and the reflectivity of antisymmetric cases, linking to lattice completions.

Findings

Existence of a countable homogeneous betweenness relation.

Reflectivity of the subcategory of antisymmetric betweenness relations.

Connections to distributive complete lattices and Dedekind-MacNeille completion.

Abstract

We apply a categorical lens to the study of betweenness relations by capturing them within a topological category, fibred in lattices, and study several subcategories of it. In particular, we show that its full subcategory of finite objects forms a Fraiss\'{e} class implying the existence of a countable homogenous betweenness relation. We furthermore show that the subcategory of antisymmetric betweenness relations is reflective. As an application we recover the reflectivity of distributive complete lattices within complete lattices, and we end with some observations on the Dedekind-MacNeille completion.