Generalized bornological coarse spaces and coarse motivic spectra

TL;DR

This paper extends the concept of bornological coarse spaces by removing the boundedness condition, creating a more general framework, and shows that the associated motivic coarse spectra remain equivalent under this generalization.

Contribution

It introduces a complete and co-complete generalization of bornological coarse spaces and proves the invariance of motivic coarse spectra under this extension.

Findings

The generalized category is complete and co-complete.

The inclusion functor induces an equivalence of motivic coarse spectra.

Equivalence of coarse homology theories on both categories.

Abstract

We generalize the notion of a bornology by omitting the condition that a one-point-subset is bounded and obtain a complete and co-complete generalization of the category of bornological coarse spaces. Then we imitate the construction of motivic coarse spectra in this new setting and show that the inclusion functor from the category of bornological coarse spaces to its generalization induces an equivalence of motivic coarse spectra. In particular, for any stable co-complete $\infty$-category $C$, it induces an equivalence between the category of $C$-valued coarse homology theories on bornological coarse spaces and the category of $C$-valued coarse homology theories on generalized bornological coarse spaces.