Epimorphism classes and relatively maximal metrics in large-scale geometry

We consider epimorphisms and several variant notions – split, effective, regular, strong, and extremal – and determine which of these coincide in the metric coarse and coarsely Lipschitz categories. In particular, we characterise extremal epis in the coarsely Lipschitz category via a relative maximality condition on the codomain metric; this can be viewed as a morphism-relative analogue of Rosendal’s maximal metrics for topological groups.