A Note on the Topologicity of Quantale-Valued Topological Spaces

For a quantale $\mathsf{V}$, the category $\mathsf{V}$-$\mathbf{T}\mathbf{o}\mathbf{p}$ of $\mathsf{V}$-valued topological spaces may be introduced as a full subcategory of those $\mathsf{V}$-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 $\mathsf{V}$ completely distributive, $\mathsf{V}$-topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to $\mathsf{V}$-valued relations. As a consequence, $\mathsf{V}$-$\mathbf{T}\mathbf{o}\mathbf{p}$ is seen to be a topological category over $\mathbf{S}\mathbf{e}\mathbf{t}$, provided that $\mathsf{V}$ is completely distributive. In this paper we give a choice-free proof that $\mathsf{V}$-$\mathbf{T}\mathbf{o}\mathbf{p}$ is a topological category over $\mathbf{S}\mathbf{e}\mathbf{t}$ under the considerably milder provision that $\mathsf{V}$ be a spatial coframe. When $\mathsf{V}$ is a continuous lattice, that provision yields complete distributivity of $\mathsf{V}$ in the constructive sense, hence also in the ordinary sense whenever the Axiom of Choice is granted.