# Masaryk University

Publication details

# Approximate injectivity

Authors THOLEN Walter 2018 Article in Periodical Applied Categorical Structures http://dx.doi.org/10.1007/s10485-017-9510-2 enriched category; locally presentable category; pure morphism; injective object; approximate injectivity class; Urysohn space; Gurarii space In a locally $\lambda$-presentable category, with $\lambda$ a regular cardinal, classes of objects that are injective with respect to a family of morphisms whose domains and codomains are $\lambda$-presentable, are known to be characterized by their closure under products, $\lambda$-directed colimits and $\lambda$-pure subobjects. Replacing the strict commutativity of diagrams by commutativity up to $\eps$", this paper provides an approximate version" of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed $\eps$-generalizations of the notion of $\lambda$-purity. The categorical theory is being applied to the locally $\aleph_1$-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.