Approximate injectivity and smallness in metric-enriched categories

TL;DR

This paper explores properties of metric-enriched categories, demonstrating that under certain conditions, approximate-injectivity classes are weakly reflective, and provides a new categorical proof of the Gurarii space's uniqueness.

Contribution

It introduces a framework for approximate injectivity in metric-enriched categories and applies it to prove the uniqueness of the Gurarii space categorically.

Findings

Approximate-injectivity classes are weakly reflective under mild conditions.

Reflection morphisms have specific properties studied in the paper.

A new categorical proof of the Gurarii space's essential uniqueness is provided.

Abstract

Properties of categories enriched over the category of metric spaces are investigated and applied to a study of constructions known from that category and the category of Banach spaces. For every class of morphisms satisfying a mild smallness condition we prove the corresponding approximate-injectivity class is weakly reflective, and we study the properties of the reflection morphisms. As an application we present a new categorical proof of the essential uniqueness of the Gurarii space.