Enriched Stone-type dualities

TL;DR

This paper develops a systematic extension of Stone's duality from Boolean algebras to all compact Hausdorff spaces using enriched category theory and the unit interval as a cogenerator.

Contribution

It introduces a new duality framework for ordered and metric compact Hausdorff spaces via quantale-enriched categories, broadening classical duality results.

Findings

Duality theory for $[0,1]$-enriched categories established

Extended dualities include ordered and metric compact spaces

Framework unifies classical and enriched duality concepts

Abstract

A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces,the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval $[0,1]$ and present duality theory for ordered and metric compact Hausdorff spaces and (suitably defined) finitely cocomplete categories enriched in $[0,1]$.