# Localization for coarse homology theories

## TL;DR

The paper introduces Bredon-style equivariant coarse homology theories, demonstrating their localization properties and approximation capabilities, with applications to algebraic and topological equivariant coarse K-homology and a coarse Segal localization theorem.

## Contribution

It defines Bredon-style equivariant coarse homology theories and proves their localization properties, providing a new framework and approximation method for general equivariant coarse homology theories.

## Key findings

- Bredon-style theories satisfy localization theorems.
- General equivariant coarse homology can be approximated by Bredon-style theories.
- Established a coarse analog of Segal's localization theorem.

## Abstract

We introduce the notion of a Bredon-style equivariant coarse homology theory. We show that such a Bredon-style equivariant coarse homology theory satisfies localization theorems and that a general equivariant coarse homology theory can be approximated by a Bredon-style version. We discuss the special case of algebraic and topological equivariant coarse $K$-homology and obtain the coarse analog of Segal's localization theorem.