METRIC ENRICHMENT, FINITE GENERATION, AND THE PATH COREFLECTION

We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally ℵ1-presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry-ℵ0-generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results include the automatic completeness of a colimit of a diagram of bi-Lipschitz morphisms between complete metric spaces and a characterization of those pairs (metric space, unital C^∗-algebra) that have a tensor product in the CMet-enriched category of unital C^∗-algebras.