Localic completion of generalized metric spaces I

By S. Vickers. Theory and Applications of Categories, 2005

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Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 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 equiv- alent to the Yoneda completion in its netwise form due to Kunzi 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.

Subfields: Computational Theory and Mathematics, Computer Networks and Communications, Geometry and Topology

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