# Categories of coarse groups: quasi-homomorphisms and functorial coarse structures

## TL;DR

This paper explores the large-scale geometry of groups using coarse structures, introducing quasi-homomorphisms and functorial structures to enhance categorical understanding.

## Contribution

It introduces a new class of group coarse structures via cardinal invariants and employs quasi-homomorphisms for a categorical framework.

## Key findings

- Development of group coarse structures using cardinal invariants
- Introduction of quasi-homomorphisms as large-scale homomorphisms
- Examples of functorial coarse structures

## Abstract

Coarse geometry is the study of large-scale properties of spaces. In this paper we study group coarse structures (i.e., coarse structures on groups that agree with the algebraic structures), by using group ideals. We introduce a large class of examples of group coarse structures induced by cardinal invariants. In order to enhance the categorical treatment of the subject, we use quasi-homomorphisms, as a large-scale counterpart of homomorphisms. In particular, the localisation of a category plays a fundamental role. We then define the notion of functorial coarse structures and we give various examples of those structures.