Metric-like spaces as enriched categories: three vignettes

TL;DR

This paper explores the connection between metric spaces and enriched categories, illustrating how category theory provides a unifying framework and offering three mathematical examples demonstrating its usefulness.

Contribution

It introduces a category-theoretic perspective on metric spaces, showing how they can be viewed as enriched categories, and discusses three mathematical applications.

Findings

Unified framework for metric spaces and categories

Application to tight span, magnitude, Legendre-Fenchel transform

Enhanced understanding of metric space structures

Abstract

This is a write-up of a talk given at the CATMI meeting in Bergen in July 2023, and is an introduction to a category-theoretic perspective on metric spaces. A metric space is a set of points such that between each pair of points there is a number -- the distance -- such that the triangle inequality is satisfied; a small category is a set of objects such that between each pair of objects there is a set -- the hom-set -- such that elements of the hom-sets can be composed. The analogy between the structures that can be made in to a common generalization of the two structures, so that both are examples of enriched categories. This gives a bridge between category theory and metric space theory. I will describe this and three examples from around mathematics where this perspective has been useful or interesting. The examples are related to the tight span, the magnitude and the Legendre-Fenchel transform.