Convergence and quantale-enriched categories

TL;DR

This paper explores the relationship between quantale-enriched categories with compact Hausdorff topology and ultrafilter-quantale-enriched categories, demonstrating that such topologies ensure certain completeness properties in the categories.

Contribution

It extends Nachbin's topology and order theory to quantale-enriched categories, establishing new links between topology and categorical completeness.

Findings

Compact Hausdorff topology guarantees Cauchy completeness.

Presence of topology implies codirected completeness.

Comparison between different enriched categorical structures.

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 $\mathcal{V}$-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.