Monotone-light factorizations in coarse geometry

TL;DR

This paper develops large scale analogues of topological monotone and light maps, establishing a factorization system in coarse geometry and exploring their properties and connections to classical notions.

Contribution

It introduces coarsely monotone and coarsely light maps, forming a new factorization system in the coarse category, and links coarse and classical concepts via the Higson corona.

Findings

Coarsely monotone and light maps form a factorization system in the coarse category.

Coarsely monotone maps are stable under certain pullbacks in the coarse category.

Coarsely light maps preserve properties like finite asymptotic dimension and exactness.

Abstract

We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar way to classically monotone maps, and prove that coarsely monotone maps are stable under those pullbacks which exist in the coarse category. For the case of maps between proper metric spaces, we exhibit some connections between the coarse and classical notions of monotone and light using the Higson corona. Finally, we look at some coarse properties which are preserved by coarsely light maps such as finite asymptotic dimension and exactness, and make some remarks on the situation for groups and group homomorphisms.