Enriched purity and presentability in Banach spaces

TL;DR

This paper explores the enriched categorical structure of Banach spaces, revealing that while $Ban$ isn't locally finitely presentable in the classical sense, it is in the enriched sense, with pure morphisms corresponding to ideals.

Contribution

It establishes that $Ban$ is locally finitely presentable in the enriched sense over complete metric spaces and characterizes pure morphisms as ideals of Banach spaces.

Findings

Banach spaces form an enriched locally finitely presentable category.

Pure morphisms in this setting are exactly ideals of Banach spaces.

Characterization of approximately injective Banach spaces to certain morphisms.

Abstract

The category $Ban$ of Banach spaces and linear maps of norm $\leq 1$ is locally $\aleph_1$-presentable but not locally finitely presentable. We prove, however, that $Ban$ 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 to sets of morphisms having finite-dimensional domains and separable codomains.