Category of metric spaces
Encyclopedia
In category-theoretic mathematics
, Met is a category that has metric space
s as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphism
s. This is a category because the composition
of two metric maps is again a metric map. It was first considered by .
The monomorphism
s in Met are the injective metric maps, maps that do not map two points into a single point. The epimorphism
s are the metric maps in which the domain of the map has a dense image in the range. The isomorphism
s are the isometries
, metric maps that are one-to-one, onto, and distance-preserving. As an example, the inclusion of the rational number
s into the real number
s is a monomorphism and an epimorphism, but it is clearly not an isomorphism; this example shows that Met is not a balanced category.
The empty set
(considered as a metric space) is the initial object
of Met; any singleton metric space is a terminal object. Because the initial object and the terminal objects differ, there are no zero objects in Met.
The product
of a finite set of metric spaces in Met is a metric space that has the cartesian product
of the spaces as its points; the distance in the product space is given by the supremum of the distances in the base spaces. That is, it is the product metric with the sup norm. However, the product of an infinite set of metric spaces may not exist, because the distances in the base spaces may not have a supremum. That is, Met is not a complete category
, but it is finitely complete. There is no coproduct in Met.
The "forgetful" functor
Met → Set
assigns to each metric space the underlying set of its points, and assigns to each metric map the underlying set-theoretic function
. This functor is faithful, and therefore Met is a concrete category
.
The injective object
s in Met are called injective metric space
s. Injective metric spaces were introduced and studied first by , prior to the study of Met as a category; they may also be defined intrinsically in terms of a Helly property
of their metric balls, and because of this alternative definition Aronszajn and Panitchpakdi named these spaces hyperconvex spaces. Any metric space has a smallest injective metric space into which it can be isometrically embedded, called its metric envelope or tight span
.
Met is not the only category whose objects are metric spaces; others include the category of uniformly continuous functions
and the category of Lipschitz functions
. The metric maps are both uniformly continuous and Lipschitz, with Lipschitz constant at most one.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, Met is a category that has metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s. This is a category because the composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of two metric maps is again a metric map. It was first considered by .
The monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
s in Met are the injective metric maps, maps that do not map two points into a single point. The epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
s are the metric maps in which the domain of the map has a dense image in the range. The isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s are the isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
, metric maps that are one-to-one, onto, and distance-preserving. As an example, the inclusion of the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s into the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s is a monomorphism and an epimorphism, but it is clearly not an isomorphism; this example shows that Met is not a balanced category.
The empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
(considered as a metric space) is the initial object
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
of Met; any singleton metric space is a terminal object. Because the initial object and the terminal objects differ, there are no zero objects in Met.
The product
Product (category theory)
In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
of a finite set of metric spaces in Met is a metric space that has the cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the spaces as its points; the distance in the product space is given by the supremum of the distances in the base spaces. That is, it is the product metric with the sup norm. However, the product of an infinite set of metric spaces may not exist, because the distances in the base spaces may not have a supremum. That is, Met is not a complete category
Complete category
In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C where J is small has a limit in C. Dually, a cocomplete category is one in which all small colimits exist...
, but it is finitely complete. There is no coproduct in Met.
The "forgetful" functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
Met → Set
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
assigns to each metric space the underlying set of its points, and assigns to each metric map the underlying set-theoretic function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
. This functor is faithful, and therefore Met is a concrete category
Concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...
.
The injective object
Injective object
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories...
s in Met are called injective metric space
Injective metric space
In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces...
s. Injective metric spaces were introduced and studied first by , prior to the study of Met as a category; they may also be defined intrinsically in terms of a Helly property
Helly family
In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an empty intersection has k or fewer sets in it. In other words, any subfamily such that every k-fold intersection is non-empty has non-empty total intersection.The k-Helly property is the property...
of their metric balls, and because of this alternative definition Aronszajn and Panitchpakdi named these spaces hyperconvex spaces. Any metric space has a smallest injective metric space into which it can be isometrically embedded, called its metric envelope or tight span
Tight span
In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes...
.
Met is not the only category whose objects are metric spaces; others include the category of uniformly continuous functions
Uniform continuity
In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f and f be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between x and y cannot...
and the category of Lipschitz functions
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
. The metric maps are both uniformly continuous and Lipschitz, with Lipschitz constant at most one.