Localic Completion of Generalized Metric Spaces I

Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero self-distance law and the triangle inequality. We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over [0,∞], the Cauchy filters are equivalent to flat left modules. The completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to Künzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion. Non-expansive functions between gms’s lift to continuous maps between the completions. Various examples and constructions are given, including finite products. The completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable point-based reasoning for locales.