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Metric-like spaces as enriched categories: three vignettes

Simon Willerton

TL;DR

The paper develops a category-theoretic perspective on metric spaces by enriching over the extended nonnegative reals and, more generally, over the extended real line, to unify and generalize metric-like structures. It presents three vignettes—the tight span (and directed variants via the Isbell nucleus), magnitude with its Euler-characteristic interpretation and connections to biodiversity, and the Legendre–Fenchel transform approached as an R-category—showing how enriched-category methods clarify classical constructions and reveal new dualities. Key contributions include the Isbell-type adjunction and the nucleus of a profunctor, the Isbell completion as a directed tight span, the categorical framing of magnitude and its link to homology and diversity, and the R-category formulation of Legendre–Fenchel duality with Toland–Singer isometry. The work provides a unifying framework that connects metric geometry, ecological diversity measures, and convex analysis, suggesting broad applicability of enriched-categorical methods to metric-space theory and beyond.

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.

Metric-like spaces as enriched categories: three vignettes

TL;DR

The paper develops a category-theoretic perspective on metric spaces by enriching over the extended nonnegative reals and, more generally, over the extended real line, to unify and generalize metric-like structures. It presents three vignettes—the tight span (and directed variants via the Isbell nucleus), magnitude with its Euler-characteristic interpretation and connections to biodiversity, and the Legendre–Fenchel transform approached as an R-category—showing how enriched-category methods clarify classical constructions and reveal new dualities. Key contributions include the Isbell-type adjunction and the nucleus of a profunctor, the Isbell completion as a directed tight span, the categorical framing of magnitude and its link to homology and diversity, and the R-category formulation of Legendre–Fenchel duality with Toland–Singer isometry. The work provides a unifying framework that connects metric geometry, ecological diversity measures, and convex analysis, suggesting broad applicability of enriched-categorical methods to metric-space theory and beyond.

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.
Paper Structure (23 sections, 3 theorems, 36 equations, 7 figures)

This paper contains 23 sections, 3 theorems, 36 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Historical context
  3. The three vignettes
  4. Why bother with category theory?
  5. Acknowledgements
  6. Background on enriched category theory
  7. Enriched categories
  8. Metric spaces as enriched categories
  9. Examples of R+-categories
  10. Self enrichment
  11. Spaces of short maps
  12. Asymmetric Hausdorff metric on subsets of a metric space
  13. Scalar-valued functors and the Yoneda embedding
  14. The nucleus of a profunctor
  15. First vignette: the tight span
  16. ...and 8 more sections

Key Result

Proposition 1

An $\overline{\mathbb{R}}_{+}$-category, $X$, consists of a collection, $\operatorname{ob}(X)$, of objects together with the following data: Associativity and unitality are automatically satisfied due to $\overline{\mathbb{R}}_{+}$ being thin, that is, having at most one morphism between each pair of objects.

Figures (7)

  • Figure 1:
  • Figure 2:
  • Figure 3: A schematic of how Leinster generalized several notions of Euler characteristic and then specialized to finite metric spaces
  • Figure 4:
  • Figure 5: Some of the rich connections of magnitude to various areas of mathematics
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1
  • Theorem 2
  • Theorem 3