Editing Category of metric spaces

{{Short description|Category whose objects are metric spaces and whose morphisms are metric maps}}
In [[category theory]], '''Met''' is a [[Category (mathematics)|category]] that has [[metric space]]s as its [[object (category theory)|objects]] and [[metric map]]s ([[Continuous_function#Continuous_functions_between_metric_spaces|continuous]] [[Function (mathematics)|functions]] between metric spaces that do not increase any pairwise distance) as its [[morphism]]s. This is a category because the [[Function composition|composition]] of two metric maps is again a metric map. It was first considered by {{harvtxt|Isbell|1964}}.

==Arrows==
The [[monomorphism]]s in '''Met''' are the [[injective]] metric maps. The [[epimorphism]]s are the metric maps for which the [[Domain of a function|domain]] of the map has a [[Dense set|dense]] [[Image (mathematics)|image]] in the [[Range of a function|range]]. The [[isomorphism]]s are the [[Isometry|isometries]], i.e. metric maps which are injective, [[surjective]], 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]].

==Objects==
The [[Empty set|empty]] metric space is the [[initial object]] of '''Met'''; any [[singleton (mathematics)|singleton]] metric space is a [[terminal object]]. Because the initial object and the terminal objects differ, there are no [[zero object]]s in '''Met'''.

The [[injective object]]s in '''Met''' are called [[injective metric space]]s. Injective metric spaces were introduced and studied first by {{harvtxt|Aronszajn|Panitchpakdi|1956}}, prior to the study of '''Met''' as a category; they may also be defined intrinsically in terms of a [[Helly family|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 [[Embedding|embedded]], called its metric envelope or [[tight span]].

==Products and functors==
The [[Product (category theory)|product]] of a finite [[Set (mathematics)|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 (category theory)|coproduct]] in '''Met'''.

The [[forgetful functor]] '''Met''' → '''[[Category of sets|Set]]''' assigns to each metric space the underlying [[Set (mathematics)|set]] of its points, and assigns to each metric map the underlying set-theoretic function. This functor is [[Faithful functor|faithful]], and therefore '''Met''' is a [[concrete category]].

==Related categories==
'''Met''' is not the only category whose objects are metric spaces; others include the category of [[uniform continuity|uniformly continuous functions]], the category of [[Lipschitz continuity|Lipschitz functions]] and the category of [[quasi-Lipschitz mapping]]s. The metric maps are both uniformly continuous and Lipschitz, with Lipschitz constant at most one.

== See also ==

* {{annotated link|Category of groups}}
* {{annotated link|Category of sets}}
* {{annotated link|Category of topological spaces}}
* {{annotated link|Category of topological spaces with base point}}
* {{annotated link|Category of topological vector spaces}}

== References ==

*{{citation
 | author1-link = Nachman Aronszajn | last1 = Aronszajn | first1 = N. | last2 = Panitchpakdi | first2 = P.
 | title = Extensions of uniformly continuous transformations and hyperconvex metric spaces
 | journal = Pacific Journal of Mathematics
 | volume = 6
 | year = 1956
 | issue = 3 | pages = 405–439
 | url = http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103043960 | doi=10.2140/pjm.1956.6.405| doi-access = free}}.
*{{citation
 | last1 = Deza | first1 = Michel Marie | author1-link = Michel Deza
 | last2 = Deza | first2 = Elena | author2-link = Elena Deza
 | contribution = Category of metric spaces
 | page = 38
 | publisher = Springer-Verlag
 | title = Encyclopedia of Distances
 | url = https://books.google.com/books?id=LXEezzccwcoC&pg=PA38
 | year = 2009| isbn = 9783642002342 }}.
*{{citation
 | last = Isbell | first = J. R. | authorlink = John R. Isbell
 | title = Six theorems about injective metric spaces
 | journal = [[Comment. Math. Helv.]]
 | volume = 39
 | issue = 1
 | year = 1964
 | pages = 65–76
 | url = http://www.digizeitschriften.de/resolveppn/GDZPPN002058340 | doi = 10.1007/BF02566944| s2cid = 121857986 }}.

{{Metric spaces}}

[[Category:Categories in category theory|Metric spaces]]
[[Category:Metric geometry]]