A Note on the Topologicity of Quantale-Valued Topological Spaces
Hongliang Lai, Walter Tholen

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 , the category - of -valued topological spaces may be introduced as a full subcategory of those -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 completely distributive, -topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to -valued relations. As a consequence, - is seen to be a topological category over , provided that is completely distributive. In this paper we give a choice-free proof that - is a topological category over under the considerably milder provision that be…
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