On di-injective \(T_{0}\)-quasi-metric spaces (Q2401575)

In this paper, the author shows that given a \(T_0\)-quasi-metric space, then its Isbell-convex hull is injective in the category whose objects are \(T_0\)-quasi-metric spaces with arrows the nonexpansive maps. Several characterisations of the Isbell-convex hull of a \(T_0\)-quasi-metric space are also provided as well as the space of all Katětov function pairs on a \(T_0\)-quasi-metric space (see [\textit{H.-P. A. Künzi} and \textit{M. Sanchis}, Topology Appl. 159, No. 3, 711--720 (2012; Zbl 1242.54012) and Math. Struct. Comput. Sci. 25, No. 8, 1685--1691 (2015; Zbl 1369.54020)]). It is worth noting that \textit{J. R. Isbell} has proved in [Comment. Math. Helv. 39, 65--76 (1964; Zbl 0151.30205)] that every metric space has a unique hyperconvex hull. In addition, if the space is compact then so is its hyperconvex hull. In [Topology Appl. 159, No. 9, 2463--2475 (2012; Zbl 1245.54023)], \textit{E. Kemajou} et al. showed a result analogous to Isbell's theorem: Every \(T_0\)-quasi-metric space has a \(q\)-hyperconvex hull, which is joincompact if the space is joincompact.