# 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.

## Key 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.