Towards probabilistic partial metric spaces: Diagonals between distance distributions

TL;DR

This paper explores the categorical foundations of probabilistic partial metric spaces by analyzing diagonals between distance distributions within the quantale framework, revealing key structural properties.

Contribution

It characterizes diagonals between distance distributions in the quantaloid, providing a foundational categorical framework for probabilistic partial metric spaces.

Findings

Diagonals between distance distributions are precisely characterized.

One-step functions uniquely determine the diagonals set.

The quantale of distance distributions is non-divisible with respect to continuous t-norms.

Abstract

The quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories. Motivated by the categorical interpretation of partial metric spaces, we are led to investigate the quantaloid of diagonals between distance distributions, which is expected to establish the categorical foundation of probabilistic partial metric spaces. Observing that the quantale of distance distributions w.r.t. an arbitrary continuous t-norm is non-divisible, we precisely characterize diagonals between distance distributions, and prove that one-step functions are the only distance distributions on which the set of diagonals coincides with the generated down set.