  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.
