# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1705.08671