Approximate Injectivity and Smallness in Metric-Enriched Categories

Properties of categories enriched over the category of metric spaces areinvestigated and applied to a study of constructions known from that categoryand the category of Banach spaces. For every class of morphisms satisfying amild smallness condition we prove the corresponding approximate-injectivityclass is weakly reflective, and we study the properties of the reflectionmorphisms. As an application we present a new categorical proof of theessential uniqueness of the Gurarii space.