Convergence and quantale-enriched categories

Abstract

Generalising Nachbin's theory of ``topology and order'', in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $V$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category.

Keywords

• Ordered compact Hausdorff space
• metric space
• approach space
• sober space
• Cauchy completness
• quantale-enriched category