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Concept
Category of metric spaces
Résumé
In , Met is a that has metric spaces as its 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 .
The monomorphisms in Met are the injective metric maps. The epimorphisms are the metric maps for which the domain of the map has a dense in the range. The isomorphisms are the isometries, i.e. metric maps which are injective, surjective, 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 .
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 , 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.
The 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 , but it is finitely complete. There is no in Met.
The forgetful functor Met → assigns to each metric space the underlying set of its points, and assigns to each metric map the underlying set-theoretic function.
Source officielle
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