Tight spans, Isbell completions and semi-tropical modules

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.