Sequential ends and nonstandard infinite boundaries of coarse spaces

TL;DR

This paper investigates the properties of certain functors related to coarse spaces, proving injectivity of a natural transformation in specific cases using nonstandard analysis, thus addressing open problems from prior work.

Contribution

It demonstrates that a key natural transformation is injective for all metrisable spaces within a certain subcategory, advancing understanding of coarse space boundaries.

Findings

Proves injectivity of a natural transformation for metrisable spaces.

Shows the composition of two transformations explains the surjectivity and injectivity properties.

Partially answers open problems from previous research.

Abstract

This paper is an addendum to the author's previous paper [#Im20a]. Miller et al. [#MSM10] introduced a functor $\sigma\colon\mathbf{pCoarse}\to\mathbf{Sets}$, where $\mathbf{pCoarse}$ is the category of pointed coarse spaces and coarse maps. DeLyser et al. [#DLT13] introduced a functor $\varepsilon\colon\mathbf{pCoarse}\to\mathbf{Sets}$, and proved that $\varepsilon$ coincides with $\sigma$ on $\mathbf{pMetr}$ (the full subcategory of metrisable spaces). Using techniques of nonstandard analysis, the author in [#Ima20a] provided a functor $\iota\colon\mathscr{C}\subseteq\mathbf{pCoarse}\to\mathbf{Sets}$, where $\mathscr{C}$ is an arbitrary small full subcategory, and a natural transformation $\omega\colon\sigma\restriction\mathscr{C}\Rightarrow\iota$. The surjectivity of $\omega$ has been proved for all proper geodesic metrisable spaces, while the injectivity has remained open. In this note, we first pointed out that $\omega$ is the composition of two natural transformations $\varphi\restriction\mathscr{C}\colon\sigma\restriction\mathscr{C}\Rightarrow\varepsilon\restriction\mathscr{C}$ and $\omega'\colon\varepsilon\restriction\mathscr{C}\Rightarrow\iota$, and then show that $\omega'$ is injective for all spaces in $\mathscr{C}$. As a corollary, $\omega$ is injective for all metrisable spaces in $\mathscr{C}$. This partially answers some of the problems posed in [#Ima20a].