Sober topological spaces valued in a quantale

TL;DR

This paper extends the concept of sobriety to topological spaces valued in a quantale, exploring their properties and relationships with quantale-valued domains using category theory.

Contribution

It introduces a new framework for sobriety in quantale-valued topological spaces and investigates their connections with quantale-valued domains.

Findings

Established an adjunction between quantale modules and quantale-valued topological spaces.

Analyzed relations between sober spaces and quantale-valued domains based on flat ideals.

Extended classical sobriety to a broader, quantale-valued setting.

Abstract

The notion of sobriety is extended to the realm of topological spaces valued in a commutative and unital quantale, via an adjunction between a category of quantale modules and the category of quantale-valued topological spaces. Relations between such sober spaces and quantale-valued domains based on flat ideals are investigated.