COMPLETIONS OF NON-SYMMETRIC METRIC SPACES VIA ENRICHED CATEGORIES

It is known from [13] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0, infinity]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits commuting in the base category [0, infinity] with the conical terminal object and cotensors. Those can be interpreted in metric terms as very general filters, which we call filters of type 1. This correspondence extends the one between minimal Cauchy filters and weights which are adjoint as modules. Translating elements of enriched category theory into the metric context, one obtains a notion of convergence for filters of type 1 with a related completeness notion for spaces, for which there exists a universal completion. Another smaller class of flat presheaves is also considered both in the context of both metric spaces and preorders. (The latter being enrichments over the monoidal closed category 2.) The corresponding completion for preorders is the so-called dcpo completion.