A $C_0$ coarse structure for families of pseudometrics and the Higson-Roe functor

TL;DR

This paper introduces a $C_0$ coarse structure for families of pseudometrics, offering a new topological perspective on the relationship between coarse spaces and compactifications, and establishing new functors and categorical equivalences.

Contribution

It defines a $C_0$ coarse structure for pseudometric families, linking coarse geometry with topology and compactifications, and develops new functors and categorical relations.

Findings

Established a $C_0$ coarse structure for pseudometrics.

Connected coarse structures with compactifications via new functors.

Proved equivalences of categories relating coarse and topological structures.

Abstract

This paper deepens into the relations between coarse spaces and compactifications, by defining a $C_0$ coarse structure attached to a family of pseudometrics. This definition allow us to give a more topological point of view on the relations between coarse structures and compactifications ---like the Higson-Roe compactification, corona and functor and the topological coarse structure attached to a compactification---, define new functors and giving new relations between them, in particular, some equivalences of categories.