Approximate Injectivity

In a locally λ-presentable category, with λ a regular cardinal, classes of objects that are injective with respect to a family of morphisms whose domains and codomains are λ-presentable, are known to be characterized by their closure under products, λ-directed colimits and λ-pure subobjects. Replacing the strict commutativity of diagrams by “commutativity up to ε”, this paper provides an “approximate version” of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed ε-generalizations of the notion of λ-purity. The categorical theory is being applied to the locally א1-presentable category of Banach spaces and their linear operators of norm at most 1, culminating in a largely categorical proof for the existence of the so-called Gurarii Banach space.