Tight spans, Isbell completions and semi-tropical modules

TL;DR

This paper explores the categorical Isbell completion for generalized metric spaces, demonstrating its equivalence to the directed tight span and revealing its semi-tropical module structures.

Contribution

It establishes the Isbell completion as an analogue of the tight span and connects categorical completion with semi-tropical modules, providing new insights into metric space completions.

Findings

Isbell completion coincides with the directed tight span

The Isbell completion admits two semi-tropical module structures

Categorical completion relates to semi-tropical modules

Abstract

In this paper we consider the categorical Isbell completion construction for generalized metric spaces in the sense of Lawvere. We show that this is an analogue of the tight span construction of classical metric spaces, and that the Isbell completion coincides with the directed tight span of Hirai and Koichi. The notions of categorical completion and cocompletion are related to the existence of semi-tropical module structure, and it is shown that the Isbell completion (hence the directed tight span) has two different semi-tropical module structures.