Commit 51cb2d56 authored by Jannes Bantje's avatar Jannes Bantje

finish coarse structures appendix (for now)

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......@@ -583,6 +583,63 @@ The closeness relation obtained from a metric space can now be abstracted:
The \enquote{most controlled} subset of $X \times X$ is the diagonal $\Delta \subset X \times X$.
The notion of controlled sets conversely determines the coarse structure, since one can show, that two maps $f,g \colon S \to X$ are closes if and only if the image of $(f,g) \colon S \to X \times X$ is controlled (\cite[Prop.~6.1.4]{higson_roe}).
The notion of controlled subsets allows us to generalise additional concepts from metric spaces:
\begin{definition}
A subset $B \subseteq X$ is called \Index{bounded}, if the one-element family $\set*{B}$ is uniformly bounded, i.e. $B \times B$ is controlled.
\end{definition}
\begin{lemma}{\cite[Lem.~6.1.6]{higson_roe}}
A non-empty subset $B \subseteq X$ of a coarse space $X$ is bounded if and only if the inclusion $B \hookrightarrow X$ is close to a constant map.
\end{lemma}
In our examples focused on manifolds, there will be coarse \emph{and} topological information, which comes as a locally compact topology.
We combine this as follows:
\begin{definition}
Let $X$ be locally compact space.
A coarse structure on $X$ is called \Index[coarse structure!proper]{proper} if
\begin{enumerate}[(i)]
\item $X$ has a uniformly bounded open cover and
\item every bounded subset of $X$ has compact closure
\end{enumerate}
\end{definition}
These properties assure, that bounded subsets coincide with the ones having compact closure.
This notion is obviously consistent with our earlier use of \enquote{proper}, since the coarse structure associated to any metric space is proper, if the metric space is proper in the sence that closed balls are compact.
To get a category we still have to define, what coarse maps are in this general setting:
\begin{definition}
Let $X$ and $Y$ be coarse spaces.
$f \colon X \to Y$ is called a coarse map, if
\begin{enumerate}[(i)]
\item for all close maps $p,q$ into $X$, the composites $f \circ p$ and $f \circ q$ are close maps into $Y$ and
\item $f$ is proper in the sense, that preimages of bounded sets are bounded.
\end{enumerate}
$X$ and $Y$ are \Index{coarsely equivalent}, if there are coarse maps $f \colon X \to Y$ and $g \colon Y \to X$ such that both composition are close the the respective identity.
\end{definition}
To build up intuition we discuss a few examples:
\begin{example}
\begin{itemize}
\item Consider the natural metric coarse structures on $\mathbb{R}^2$ and $\mathbb{R}$.
The coordinate projections are not coarse, since preimages of bounded sets are clearly not bounded.
The norm function however is.
\item The inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$ is a coarse equivalence, where the inverse is given by $x \mapsto \lfloor x\rfloor$.
\item Let $M$ be a complete, simply connected Riemannian manifold of non-positive curvature and $p \in M$.
The tangential space $\Tang_p M$ is Euclidean and the exponential map
\[
\exp \colon \Tang_p M \To{} M
\]
is a diffeomorphism by the Cartan--Hadamard theorem.
The inverse $\log \colon M \to \Tang_p M$ decreases distances and it follows, that $\log$ is a coarse map.
\end{itemize}
\end{example}
% chapter coarse_structures (end)
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