Enriched purity and presentability in Banach spaces

Abstract

The category of Banach spaces and linear maps of norm is locally -presentable but not locally finitely presentable. We prove, however, that is locally finitely presentable in the enriched sense over complete metric spaces. Moreover, in this sense, pure morphisms are just ideals of Banach spaces. We characterize classes of Banach spaces approximately injective with respect to sets of morphisms having finite-dimensional domains and separable codomains.