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 $\mathbf{Pre}$ of premetric spaces and $\epsilon$-$\delta$ continuous functions as morphisms. The absence of the triangle inequality implies that the faithful functor $\mathbf{Pre}\to \mathbf{Top}$ - 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 $\mathbf{Pre}$ indicates that this functor is not surjective on objects either. Developed from work by Flagg and Weiss, we illustrate an extension $\mathbf{Pre}\hookrightarrow \mathbf{P}$ together with a faithful and surjective on objects left adjoint functor $\mathbf{P}\to \mathbf{Top}$ as an extension of $\mathbf{Pre}\to \mathbf{Top}$. We show this represents an optimal scenario given that $\mathbf{Pre}\to \mathbf{Top}$ preserves coproducts only. The objects in $\mathbf{P}$ are metric-like objects valued on value distributive lattices whose limits and colimits we show to be generated by free locales on discrete sets.

General Topology, Category Theory, Premetrics, premetrizable spaces, free locale, quantales, continuity spaces, premetrics

References in corpus (1)

• Completion of continuity spaces with uniformly vanishing asymmetry

Cited by in corpus (1)

• Complete Regularity: Kopperman's duality {\it à la quantale}