Approximate injectivity

TL;DR

This paper extends classical categorical injectivity concepts to metric-enriched categories, providing an approximate characterization and applying it to Banach spaces to prove the existence of the Gurarii space.

Contribution

It introduces an approximate injectivity framework in metric-enriched categories and applies it to Banach spaces, proving the existence of the Gurarii space categorically.

Findings

Characterization of approximate injectivity via closure properties

Development of $oldsymbol{ ext{ extit{ extepsilon}}}$-purity in metric categories

Categorical proof of the Gurarii Banach space existence

Abstract

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 $\varepsilon$", this paper provides an "approximate version" of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed $\varepsilon$-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.