Another view of the coarse invariant $\sigma$

TL;DR

This paper introduces a new perspective on the coarse invariant ,  by defining a metric on coarse maps and showing  as coarsely connected components, simplifying existing theorems.

Contribution

It provides a novel definition of  using a metric on coarse maps, unifying previous constructions and simplifying proofs of key properties.

Findings

 equals the set of coarsely connected components of a metric space of coarse maps.

The new formulation simplifies proofs of functoriality and coarse invariance.

The approach unifies different constructions of  in coarse geometry.

Abstract

Miller, Stibich and Moore (2010) developed a set-valued coarse invariant $\sigma\left(X,\xi\right)$ of pointed metric spaces. DeLyser, LaBuz and Tobash (2013) provided a different way to construct $\sigma\left(X,\xi\right)$ (as the set of all sequential ends). This paper provides yet another definition of $\sigma\left(X,\xi\right)$. To do this, we introduce a metric on the set $S\left(X,\xi\right)$ of coarse maps $\left(\mathbb{N},0\right)\to\left(X,\xi\right)$, and prove that $\sigma\left(X,\xi\right)$ is equal to the set of coarsely connected components of $S\left(X,\xi\right)$. As a by-product, our reformulation trivialises some known theorems on $\sigma\left(X,\xi\right)$, including the functoriality and the coarse invariance.