A Note on the Topologicity of Quantale-Valued Topological Spaces

TL;DR

This paper proves that the category of quantale-valued topological spaces is topological over Set when the quantale is a spatial coframe, broadening previous results that required complete distributivity.

Contribution

It provides a choice-free proof that quantale-valued topological spaces form a topological category over Set under milder conditions than previously known.

Findings

${ m V}$-Top is a topological category over Set when ${ m V}$ is a spatial coframe.

The proof does not require the Axiom of Choice.

Results extend the characterization of ${ m V}$-topological spaces via ultrafilter monad.

Abstract

For a quantale ${\sf{V}}$, the category $\sf V$-${\bf Top}$ of ${\sf{V}}$-valued topological spaces may be introduced as a full subcategory of those ${\sf{V}}$-valued closure spaces whose closure operation preserves finite joins. In generalization of Barr's characterization of topological spaces as the lax algebras of a lax extension of the ultrafilter monad from maps to relations of sets, for ${\sf{V}}$ completely distributive, ${\sf{V}}$-topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to ${\sf{V}}$-valued relations. As a consequence, ${\sf{V}}$-$\bf Top$ is seen to be a topological category over $\bf Set$, provided that ${\sf{V}}$ is completely distributive. In this paper we give a choice-free proof that ${\sf{V}}$-$\bf Top$ is a topological category over $\bf Set$ under the considerably milder provision that ${\sf{V}}$ be a spatial coframe. When ${\sf{V}}$ is a continuous lattice, that provision yields complete distributivity of ${\sf{V}}$ in the constructive sense, hence also in the ordinary sense whenever the Axiom of Choice is granted.