A pullback diagram in the coarse category

TL;DR

This paper introduces a new product construction in the coarse category of metric spaces, enabling the development of a homotopy theory and proving the existence of finite colimits.

Contribution

It defines an asymptotic product as a pullback in the coarse category, expanding the tools for coarse geometric and homotopy theoretical analysis.

Findings

Asymptotic product is well-defined for visual or geodesic spaces.

Finite colimits exist in the coarse category.

A natural homotopy theory for coarse metric spaces is established.

Abstract

This paper studies the asymptotic product of two metric spaces. It is well defined if one of the spaces is visual or if both spaces are geodesic. In this case the asymptotic product is the pullback of a limit diagram in the coarse category. Using this product construction we can define a homotopy theory on coarse metric spaces in a natural way. We prove that all finite colimits exist in the coarse category.