Coarse structure

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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 [math]\displaystyle{ X }[/math] is a collection [math]\displaystyle{ \mathbf{E} }[/math] of subsets of [math]\displaystyle{ X \times X }[/math] (therefore falling under the more general categorization of binary relations on [math]\displaystyle{ X }[/math]) called controlled sets, and so that [math]\displaystyle{ \mathbf{E} }[/math] possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:

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

A set [math]\displaystyle{ X }[/math] endowed with a coarse structure [math]\displaystyle{ \mathbf{E} }[/math] is a coarse space.

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

A subset [math]\displaystyle{ B }[/math] of [math]\displaystyle{ X }[/math] is said to be a bounded set if [math]\displaystyle{ B \times B }[/math] is a controlled set.

Intuition

The controlled sets are "small" sets, or "negligible sets": a set [math]\displaystyle{ A }[/math] such that [math]\displaystyle{ A \times A }[/math] is controlled is negligible, while a function [math]\displaystyle{ f : X \to X }[/math] 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 [math]\displaystyle{ S }[/math] and a coarse structure [math]\displaystyle{ X, }[/math] we say that the maps [math]\displaystyle{ f : S \to X }[/math] and [math]\displaystyle{ g : S \to X }[/math] are close if [math]\displaystyle{ \{(f(s), g(s)) : s \in S\} }[/math] is a controlled set.

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

Examples

  • The bounded coarse structure on a metric space [math]\displaystyle{ (X, d) }[/math] is the collection [math]\displaystyle{ \mathbf{E} }[/math] of all subsets [math]\displaystyle{ E }[/math] of [math]\displaystyle{ X \times X }[/math] such that [math]\displaystyle{ \sup_{(x, y) \in E} d(x, y) }[/math] is finite. With this structure, the integer lattice [math]\displaystyle{ \Z^n }[/math] is coarsely equivalent to [math]\displaystyle{ n }[/math]-dimensional Euclidean space.
  • A space [math]\displaystyle{ X }[/math] where [math]\displaystyle{ X \times X }[/math] 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 [math]\displaystyle{ C_0 }[/math] coarse structure on a metric space [math]\displaystyle{ (X, d) }[/math] is the collection of all subsets [math]\displaystyle{ E }[/math] of [math]\displaystyle{ X \times X }[/math] such that for all [math]\displaystyle{ \varepsilon \gt 0 }[/math] there is a compact set [math]\displaystyle{ K }[/math] of [math]\displaystyle{ E }[/math] such that [math]\displaystyle{ d(x, y) \lt \varepsilon }[/math] for all [math]\displaystyle{ (x, y) \in E \setminus K \times K. }[/math] Alternatively, the collection of all subsets [math]\displaystyle{ E }[/math] of [math]\displaystyle{ X \times X }[/math] such that [math]\displaystyle{ \{(x, y) \in E : d(x, y) \geq \varepsilon\} }[/math] is compact.
  • The discrete coarse structure on a set [math]\displaystyle{ X }[/math] consists of the diagonal [math]\displaystyle{ \Delta }[/math] together with subsets [math]\displaystyle{ E }[/math] of [math]\displaystyle{ X \times X }[/math] which contain only a finite number of points [math]\displaystyle{ (x, y) }[/math] off the diagonal.
  • If [math]\displaystyle{ X }[/math] is a topological space then the indiscrete coarse structure on [math]\displaystyle{ X }[/math] consists of all proper subsets of [math]\displaystyle{ X \times X, }[/math] meaning all subsets [math]\displaystyle{ E }[/math] such that [math]\displaystyle{ E[K] }[/math] and [math]\displaystyle{ E^{-1}[K] }[/math] are relatively compact whenever [math]\displaystyle{ K }[/math] is relatively compact.

See also

References

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



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