Tight spans, Isbell completions and semi-tropical modules (Q2847375)

In this paper it is shown that the Isbell completion of categories, hence the directed tight span of a metric space, has two different semi-tropological module structures. This led to understanding a link between two unrelated constructions of Isbell via categorical completeness to connections with tropical algebra. Moreover, the author proves the Isbell completion is cocomplete by using the fact that the space of presheaves is cocomplete. Additionally it is shown that for a generalized metric space X the isometry from X into the Isbell completion I(X) is both categorically continuous and cocontinuous. Hence I(X) has two semi-tropical module structures making it into a metric semi-tropical module and a co-metric semi-tropical module.