Category of metric spaces

In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of two metric maps is again a metric map. It was first considered by Isbell (1964).

The monomorphisms in Met are the injective metric maps, maps that do not map two points into a single point. The epimorphisms are the metric maps in which the domain of the map has a dense image in the range. The isomorphisms are the isometries, metric maps that are one-to-one, onto, and distance-preserving.

As an example, the inclusion of the rational numbers into the real numbers 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 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 injective objects in Met are called injective metric spaces. Injective metric spaces were introduced and studied first by Aronszajn & Panitchpakdi (1956), 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.

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