# Category of metric spaces

Category of metric spaces

The category Met, first considered by Isbell (1964), has metric spaces as objects and metric maps or short maps as morphisms. This is a category because the composition of two metric maps is again metric.

The monomorphisms in Met are the injective metric maps, the epimorphisms are the dense image metric maps (for instance, the inclusion: $mathbb\left\{Q\right\}submathbb\left\{R\right\}$, which is clearly mono, so Met is not a balanced category), and the isomorphisms are the isometries.

The empty set (considered as a metric space) is the initial object of Met; any singleton metric space is a terminal object. There are thus no zero objects in Met.

The product in Met is given by the supreme metric mixing on the cartesian product. There is no coproduct.

We have a "forgetful" functor MetSet which assigns to each metric space the underlying set, and to each metric map the underlying function. This functor is faithful, and therefore Met is a concrete category.

The injective objects in Met are called injective metric spaces. They were introduced and studied first by Aronszajn and Panitchpakdi (1956), who named them "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. It is related to the notion of the universal (or maximal) metric space aimed at its subspace.

References

*cite journal
author = Aronszajn, N.; Panitchpakdi, P.
title = Extensions of uniformly continuous transformations and hyperconvex metric spaces
journal = Pacific Journal of Mathematics
volume = 6
year = 1956
pages = 405–439
url = http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103043960

*cite journal
author = Isbell, J. R.
title = Six theorems about injective metric spaces
journal = Comment. Math. Helv.
volume = 39
year = 1964
pages = 65–76
url = http://www.digizeitschriften.de/resolveppn/GDZPPN002058340 | doi = 10.1007/BF02566944

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