Metric enrichment, finite generation, and the path comonad

Abstract

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 intrinsic complete metric spaces is locally ℵ 1 -presentable, closed monoidal, and comonadic over CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry-ℵ 0 -generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results include the automatic completeness of a colimit of bi-Lipschitz morphisms of 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.