Formal balls of ${\sf Q}$-categories

TL;DR

This paper explores how the structure of a quantale influences the relationship between Yoneda completeness and directed completeness in ${ m Q}$-categories, especially focusing on the case of the interval [0,1] with a continuous t-norm.

Contribution

It generalizes the formal ball model to ${ m Q}$-categories and characterizes when Yoneda completeness is equivalent to directed completeness based on the properties of the t-norm.

Findings

Yoneda completeness is equivalent to directed completeness if and only if the t-norm is Archimedean.

The formal ball construction's connection to completeness depends on the structure of the quantale.

The paper extends the formal ball model to a broader class of categories with quantale structures.

Abstract

The construction of the formal ball model for metric spaces due to Edalat and Heckmann was generalized to ${\sf Q}$-categories by Kostanek and Waszkiewicz. This paper concerns the influence of the structure of the quantale ${\sf Q}$ on the connection between Yoneda completeness of ${\sf Q}$-categories and directed completeness of their sets of formal balls. In the case that ${\sf Q}$ is the interval $[0,1]$ equipped with a continuous t-norm $\&$, it is shown that in order that Yoneda completeness of each ${\sf Q}$-category be equivalent to directed completeness of its set of formal balls, a necessary and sufficient condition is that the t-norm $\&$ is Archimedean.