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A categorical perspective on extended metric-topological spaces

Enrico Pasqualetto, Timo Schultz, Janne Taipalus

TL;DR

The work develops a categorical framework for extended metric-topological spaces, proving that the central category $\mathbf{ExtMetTop}$ is bicomplete and hence closed under all small limits and colimits. It introduces the e.m.t.-fication procedure via the functor $\mathsf{emt}$, which makes $\mathbf{ExtMetTop}$ a reflective subcategory of a pre-extended setting, enabling explicit (co)limit descriptions. A compactification theory is built through the functor $\gamma$, yielding compact e.m.t. spaces that satisfy a universal property with respect to maps from non-compact targets, and connecting to Stone–Čech constructions. The paper also situates ${\bf ExtMetTop}$ within a hierarchy of coreflective subcategories, including $\mathbf{Tych}$ and $\mathbf{ExtMet}$, via discretisation and topology-enrichment functors, and develops lambda-truncation, metric completion, and geodesification to facilitate geometric analysis on these spaces. Altogether, the results provide a robust universal-construction toolbox for metric-topological analysis in infinite-dimensional and non-smooth settings, including compactification, completion, and geodesic reduction, with potential applications to metric-measure problems and optimal transport in extended spaces.

Abstract

Motivated by the analysis and geometry of metric-measure structures in infinite dimensions, we study the category of extended metric-topological spaces, along with many of its distinguished subcategories (such as the one of compact spaces). One of the main achievements is the proof of the bicompleteness (i.e. of the existence of all small limits and colimits) of the aforementioned categories.

A categorical perspective on extended metric-topological spaces

TL;DR

The work develops a categorical framework for extended metric-topological spaces, proving that the central category is bicomplete and hence closed under all small limits and colimits. It introduces the e.m.t.-fication procedure via the functor , which makes a reflective subcategory of a pre-extended setting, enabling explicit (co)limit descriptions. A compactification theory is built through the functor , yielding compact e.m.t. spaces that satisfy a universal property with respect to maps from non-compact targets, and connecting to Stone–Čech constructions. The paper also situates within a hierarchy of coreflective subcategories, including and , via discretisation and topology-enrichment functors, and develops lambda-truncation, metric completion, and geodesification to facilitate geometric analysis on these spaces. Altogether, the results provide a robust universal-construction toolbox for metric-topological analysis in infinite-dimensional and non-smooth settings, including compactification, completion, and geodesic reduction, with potential applications to metric-measure problems and optimal transport in extended spaces.

Abstract

Motivated by the analysis and geometry of metric-measure structures in infinite dimensions, we study the category of extended metric-topological spaces, along with many of its distinguished subcategories (such as the one of compact spaces). One of the main achievements is the proof of the bicompleteness (i.e. of the existence of all small limits and colimits) of the aforementioned categories.
Paper Structure (15 sections, 26 theorems, 82 equations)

This paper contains 15 sections, 26 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological notions
  4. Metric notions
  5. Limits and colimits in Set, Top and ExtPMet
  6. Extended metric-topological spaces
  7. The category ExtMetTop
  8. Catalogue of categories
  9. The e.m.t.-fication functor
  10. Bicompleteness of ExtMetTop
  11. The compactification functor
  12. Tych and ExtMet are coreflective subcategories of ExtMetTop
  13. Truncation, metric-completion and geodesification functors
  14. Compendium of category theory
  15. Equivalent definitions of extended metric-topological space

Key Result

Theorem 2.1

Let $({\rm X},\tau)$ be a topological space. Then there exist a compact Hausdorff space $(\beta{\rm X},\beta\tau)$ and a continuous map $i\colon{\rm X}\to\beta{\rm X}$ such that the following universal property holds: if $({\rm Y},\tau_{\rm Y})$ is a compact Hausdorff space and $\varphi\colon{\rm X}

Theorems & Definitions (75)

  • Theorem 2.1: Stone--Čech compactification
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5: Completion of an extended metric space
  • proof
  • Remark 2.6
  • Definition 2.7: Pre-extended pseudometric-topological space
  • Remark 2.8
  • Definition 2.9: Continuous-short map
  • ...and 65 more