Homotopy theory with bornological coarse spaces

TL;DR

This paper develops an axiomatic framework for coarse homology theories on bornological coarse spaces, introduces a category of motivic coarse spectra, and explores classification and examples, including coarse K-homology.

Contribution

It provides a new axiomatic characterization, constructs motivic coarse spectra, and advances the understanding of coarse homology theories and their classifications.

Findings

Transformations inducing equivalences on discrete spaces extend to finite asymptotic dimension spaces.

Constructs a category of motivic coarse spectra.

Discusses coarse K-homology as a key example.

Abstract

We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the classification of coarse homology theories and the construction of examples. We show that if a transformation between coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. The example of coarse K-homology will be discussed in detail.