# 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: $mathbb\{Q\}submathbb\{R\}$, 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 **Met** → **Set** 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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