Quantales, generalised premetrics and free locales

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.