Categorical characterisations of quasi-isometric embeddings

TL;DR

This paper provides a categorical framework for understanding quasi-isometric embeddings as regular monomorphisms in the coarsely Lipschitz category, establishing their properties and the category's structural features.

Contribution

It characterises quasi-isometric embeddings categorically and proves the coarsely Lipschitz category is coregular with an orthogonal factorisation system.

Findings

Quasi-isometric embeddings are regular monomorphisms.

The coarsely Lipschitz category is coregular.

The category admits an (Epi, RegMono) orthogonal factorisation system.

Abstract

We characterise the (closeness classes of) quasi-isometric embeddings as the regular monomorphisms in the coarsely Lipschitz category, formalising the notion that they are isomorphisms onto their image. Furthermore, we prove that the coarsely Lipschitz category is coregular, and hence admits an (Epi, RegMono)--orthogonal factorisation system. Consequently, quasi-isometric embeddings are equivalently characterised as the effective, strong, or extremal monomorphisms. Finally, we prove that the coarsely Lipschitz category is not coexact in the sense of Barr.