On products in the coarse shape categories

TL;DR

This paper investigates how the coarse shape of Cartesian products of topological spaces behaves, establishing conditions under which products in the coarse shape category can be characterized and showing the compatibility of shape groups with products.

Contribution

It provides new criteria for when Cartesian products are products in the coarse shape category and explores the behavior of shape groups under product operations.

Findings

Cartesian product of two spaces admits an HPol-expansion if each factor does.

The Cartesian product of two compact Hausdorff spaces is a product in the coarse shape category.

Shape groups and coarse shape groups commute with products under certain conditions.

Abstract

The paper is devoted to the study of coarse shape of Cartesian products of topological spaces. If the Cartesian product of two spaces $X$ and $Y$ admits an HPol-expansion, which is the Cartesian product of HPol-expansions of these spaces, then $X\times Y$ is a product in the coarse shape category. As a consequence, the Cartesian product of two compact Hausdorff spaces is a product in the coarse shape category. Finally, we show that the shape groups and the coarse shape groups commute with products under some conditions.