From e784f7f6bcfbdec98096579f78f584bbfe2f1297 Mon Sep 17 00:00:00 2001 From: Jannes Bantje Date: Wed, 24 Jun 2020 16:18:49 +0200 Subject: [PATCH] add definition of coarse structure --- contents/appendix.tex | 33 +++++++++++++++++++++++++++++++++ contents/coarse.tex | 3 ++- 2 files changed, 35 insertions(+), 1 deletion(-) diff --git a/contents/appendix.tex b/contents/appendix.tex index ddef200..5197007 100644 --- a/contents/appendix.tex +++ b/contents/appendix.tex @@ -542,4 +542,37 @@ In fact being contractible is not just a property of the total space of universa % section thom_isomorphism (end) % chapter appendix_bundle_theory (end) +\chapter{Coarse Structures} % (fold) +\label{cha:coarse_structures} + +Given a metric space $X$ with metric $d$ the topologist naturally considers the topology induced by $d$. +This only captures the \enquote{small scale structure} of the metric space $X$, since the metric +$+ d'(x,y) \coloneqq \min \set[\big]{d(x,y),1} +$ +induces the same topology. +In this appendix we want to study the dual procedure of capturing the \enquote{large scale structure} of a metric space. + +In \cref{sec:general_theory_modules} we already mentioned the concept of two maps $f,g \colon S \to X$ (for some set $S$) being \emph{close} (see \cref{def:coarse_cat}). +Closeness is clearly an equivalence relation and if the diameter of $X$ is finite, any two such maps are close. +We can modify $d$ in another way dual to $d'$ by removing all the small scale information: +For $x \neq y$ set +$+ d''(x,y) \coloneqq \max \set[\big]{d(x,y),1} +$ +The closeness relation is preserved by this operation. +The closeness relation obtained from a metric space can now be abstracted: + +\begin{definition}{\cite[Def.~6.1.2]{higson_roe}} + Let $X$ be a set. + A \Index{coarse structure} on $X$ is a collection of equivalence relations, called \enquote{being close}, on each set of maps $S \to X$ such that the following properties hold: + \begin{enumerate}[(i)] + \item If $f,g \colon S \to X$ are close and $q \colon S' \to S$ is any map, then $f \circ q$ and $g \circ q$ are close. + \item If $S = S' \cup S''$ and $f,g \colon S \to X$ are maps, whose restrictions to $S'$ and $S''$ are close, then $f$ and $g$ are close. + \item Any two constant maps are close to each other. + \end{enumerate} +\end{definition} + +% chapter coarse_structures (end) + diff --git a/contents/coarse.tex b/contents/coarse.tex index 8477547..0a5f950 100644 --- a/contents/coarse.tex +++ b/contents/coarse.tex @@ -234,7 +234,7 @@ We now look at the more specialised setting of the coarse category: \begin{definition} For us, metrics are allowed to take the value infinity. - A metric space $X$ is \Index[proper metric space]{proper}, if all closed bounded sets are compact. + A metric space $X$ is called \Index[proper metric space]{proper}, if all closed bounded sets are compact. \end{definition} Given any map $f \colon X \to Y$ between proper metric spaces, the \Index{expansion function} of $f$, denoted $\omega_f \colon [0,\infty) \to \benbrace*{0,\infty}$ is defined by @@ -257,6 +257,7 @@ Given any map $f \colon X \to Y$ between proper metric spaces, the \Index{expans \begin{remark} \begin{itemize} + \item In \cref{cha:coarse_structures} we give more background on coarse structures. \item \Textcite{rudi} calls the first property \enquote{bornologous}, referring to the language of bornological spaces, in which bounded sets are axiomatised in a manner similar to open sets in a topological space. \item The finiteness of the expansion can also be formulated in a manner similar to the $\varepsilon$-$\delta$-definition of continuity, where the roles of $\varepsilon$ and $\delta$ are swapped, a \enquote{$\delta$-$\varepsilon$-definition} so to speak. \item Assuming that all spaces are proper has the following implication: Properness of a map $f \colon X \to Y$ in the sense above is equivalent to requiring, that the preimage of bounded sets is a finite union of bounded sets. -- GitLab