Quantales, generalised premetrics and free locales

TL;DR

This paper extends the category of premetric spaces to a new category where objects are valued on lattices, enabling a faithful functor to topological spaces that preserves coproducts and is generated by free locales.

Contribution

It introduces a new category of metric-like objects valued on lattices, extending premetric spaces and establishing a faithful, object-surjective functor to topological spaces.

Findings

The functor from the new category to Top is faithful and surjective on objects.

Objects in the new category are generated by free locales on discrete sets.

The extension preserves coproducts, unlike the original premetric category.

Abstract

Premetrics and premetrisable spaces have been long studied and their topological interrelationships are well-understood. Consider the category ${\bf Pre}$ of premetric spaces and $\epsilon$-$\delta$ continuous functions as morphisms. The absence of the triangle inequality implies that the faithful functor ${\bf Pre} \to {\bf Top}$ - where a premetric space is sent to the topological space it generates - is not full. Moreover, the sequential nature of topological spaces generated from objects in ${\bf Pre}$ indicates that this functor is not surjective on objects either. Developed from work by Flagg and Weiss, we illustrate an extension ${\bf Pre}\hookrightarrow {\bf P} $ together with a faithful and surjective on objects left adjoint functor ${\bf P} \to {\bf Top}$ as an extension of ${\bf Pre} \to {\bf Top}$. We show this represents an optimal scenario given that ${\bf Pre} \to {\bf Top}$ preserves coproducts only. The objects in ${\bf P}$ are metric-like objects valued on value distributive lattices whose limits and colimits we show to be generated by free locales on discrete sets.