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

Abstract

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

12 pages