Varieties of coarse spaces

TL;DR

This paper classifies the different types of coarse spaces based on their structural properties, showing that any variety containing an unbounded metric space encompasses all coarse spaces.

Contribution

It provides a complete classification of varieties of coarse spaces and characterizes when such a variety includes all coarse spaces.

Findings

Classifies all varieties of coarse spaces.

Shows that containing an unbounded metric space implies the variety includes all coarse spaces.

Establishes structural properties of coarse space varieties.

Abstract

A class $\mathfrak{M}$ of coarse spaces is called a variety if $\mathfrak{M}$ is closed under formation of subspaces, coarse images and products. We classify the varieties of coarse spaces and, in particular, show that if a variety $\mathfrak{M}$ contains an unbounded metric space then $\mathfrak{M}$ is the variety of all coarse spaces.