Epimorphism classes and relatively maximal metrics in large-scale geometry

TL;DR

This paper investigates various types of epimorphisms in large-scale geometry, characterizing extremal epimorphisms through a relative maximality condition in the coarsely Lipschitz category, extending concepts from topological groups.

Contribution

It introduces a detailed analysis of epimorphism variants and characterizes extremal epimorphisms via a new relative maximality condition in large-scale geometric categories.

Findings

Characterization of extremal epimorphisms via relative maximality

Equivalence of different epimorphism notions in coarse categories

Extension of Rosendal's maximal metrics to coarse geometry

Abstract

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.