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

lmcs:2640 - Logical Methods in Computer Science, August 15, 2017, Volume 13, Issue 3 - https://doi.org/10.23638/LMCS-13(3:12)2017
A Note on the Topologicity of Quantale-Valued Topological SpacesArticle

Authors: Hongliang Lai ; Walter Tholen

    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.


    Volume: Volume 13, Issue 3
    Published on: August 15, 2017
    Accepted on: July 8, 2017
    Submitted on: August 15, 2017
    Keywords: Computer Science - Logic in Computer Science
    Funding:
      Source : OpenAIRE Graph
    • Funder: Natural Sciences and Engineering Research Council of Canada

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