Coarse structure

In the mathematical fields of geometry and topology, a coarse structure on a set X is a collection of subsets of the cartesian product X × X with certain properties which allow the large-scale structure of metric spaces and topological spaces to be defined.

The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. Coarse geometry and coarse topology provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.

Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.

Definition

A coarse structure on a set $X$ is a collection $𝐄$ of subsets of $X \times X$ (therefore falling under the more general categorization of binary relations on $X$ ) called controlled sets, and so that $𝐄$ possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:

Identity/diagonal:
The diagonal $Δ = {(x, x) : x \in X}$ is a member of $𝐄$ —the identity relation.
Closed under taking subsets:
If $E \in 𝐄$ and $F \subseteq E,$ then $F \in 𝐄 .$
Closed under taking inverses:
If $E \in 𝐄$ then the inverse (or transpose) $E^{- 1} = {(y, x) : (x, y) \in E}$ is a member of $𝐄$ —the inverse relation.
Closed under taking unions:
If $E, F \in 𝐄$ then their union $E \cup F$ is a member of $𝐄 .$
Closed under composition:
If $E, F \in 𝐄$ then their product $E \circ F = {(x, y) : there exists z \in X such that (x, z) \in E and (z, y) \in F}$ is a member of $𝐄$ —the composition of relations.

A set $X$ endowed with a coarse structure $𝐄$ is a coarse space.

Let $E \in 𝐄$ a controlled set. For a subset $K$ of $X,$ the set $E [K]$ is defined as ${x \in X : (x, k) \in E for some k \in K} .$ We define the section of $E$ by $x$ to be the set $E [{x}],$ also denoted $E_{x} .$ The symbol $E^{y}$ denotes the set $E^{- 1} [{y}] .$ These are forms of projections.

A subset $B$ of $X$ is said to be a bounded set if $B \times B$ is a controlled set.

Intuition

The controlled sets are "small" sets, or "negligible sets": a set $A$ such that $A \times A$ is controlled is negligible, while a function $f : X \to X$ such that its graph is controlled is "close" to the identity. In the bounded coarse structure, these sets are the bounded sets, and the functions are the ones that are a finite distance from the identity in the uniform metric.

Coarse maps

Given a set $S$ and a coarse structure $X,$ we say that the maps $f : S \to X$ and $g : S \to X$ are close if ${(f (s), g (s)) : s \in S}$ is a controlled set.

For coarse structures $X$ and $Y,$ we say that $f : X \to Y$ is a coarse map if for each bounded set $B$ of $Y$ the set $f^{- 1} (B)$ is bounded in $X$ and for each controlled set $E$ of $X$ the set $(f \times f) (E)$ is controlled in $Y .$ $X$ and $Y$ are said to be coarsely equivalent if there exists coarse maps $f : X \to Y$ and $g : Y \to X$ such that $f \circ g$ is close to ${id}_{Y}$ and $g \circ f$ is close to ${id}_{X} .$

Examples

The bounded coarse structure on a metric space $(X, d)$ is the collection $𝐄$ of all subsets $E$ of $X \times X$ such that $sup_{(x, y) \in E} d (x, y)$ is finite. With this structure, the integer lattice $ℤ^{n}$ is coarsely equivalent to $n$ -dimensional Euclidean space. This has remarkable implications in solid state physics, in particular in the field of topological insulators.
A space $X$ where $X \times X$ is controlled is called a bounded space. Such a space is coarsely equivalent to a point. A metric space with the bounded coarse structure is bounded (as a coarse space) if and only if it is bounded (as a metric space).
The trivial coarse structure only consists of the diagonal and its subsets. In this structure, a map is a coarse equivalence if and only if it is a bijection (of sets).
The $C_{0}$ coarse structure on a metric space $(X, d)$ is the collection of all subsets $E$ of $X \times X$ such that for all $ε > 0$ there is a compact set $K$ of $E$ such that $d (x, y) < ε$ for all $(x, y) \in E ∖ K \times K .$ Alternatively, the collection of all subsets $E$ of $X \times X$ such that $\overline{{(x, y) \in E : d (x, y) \geq ε}}$ is compact.
The discrete coarse structure on a set $X$ consists of the diagonal $Δ$ together with subsets $E$ of $X \times X$ which contain only a finite number of points $(x, y)$ off the diagonal.
If $X$ is a topological space then the indiscrete coarse structure on $X$ consists of all proper subsets of $X \times X,$ meaning all subsets $E$ such that $E [K]$ and $E^{- 1} [K]$ are relatively compact whenever $K$ is relatively compact.

Bounded sets

Let $ℬ = {B \subseteq X : B \times B \in 𝐄}$ be the collection of all bounded sets of a coarse space $X$ . Say that a coarse structure $𝐄$ on $X$ is coarsely connected if ${(x, y)} \in 𝐄$ for all $(x, y) \in X \times X$ . Then $ℬ$ is a bornology on $X$ if and only if $𝐄$ is coarsely connected. For example, if $X$ has at least two points and $𝐄$ is the trivial coarse structure, then $ℬ$ is not a bornology. Bounded, discrete and indiscrete coarse structures are coarsely connected.

References

1 2 Hoffland, Christian Stuart. Course structures and Higson compactification. OCLC 76953246.

John Roe, Lectures in Coarse Geometry, University Lecture Series Vol. 31, American Mathematical Society: Providence, Rhode Island, 2003. Corrections to Lectures in Coarse Geometry
Roe, John (June–July 2006). "What is...a Coarse Space?" (PDF). Notices of the American Mathematical Society. 53 (6): 669. Retrieved 2008-01-16.

[Course_structures_and_Higson_compactification-1] 1 2 Hoffland, Christian Stuart. Course structures and Higson compactification. OCLC 76953246.

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