The category Met has metric spaces as objects and short maps as morphisms. This is a category because the composition of two short maps is again short. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a short map is a function f from a metric space X to another metric space Y such that for any we have . Here and denote metrics on and , respectively. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

The monomorphisms in Met are the injective short maps, the epimorphisms are the dense image short maps (for instance, the inclusion: , which is clearly mono, so Met is not a balanced category), and the isomorphisms are the isometries. In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...

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. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In mathematics, a singleton is a set with exactly one element. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there...

The product in Met is given by the supreme metric mixing on the cartesian product. There is no coproduct. In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ...

We have a "forgetful" functor Met → Set which assigns to each metric space the underlying set, and to each short map the underlying function. This functor is faithful, and therefore Met is a concrete category. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In category theory, a faithful functor is a functor which is injective when restricted to each set of morphisms with a given source and target. ... In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ...

Follows the Top article. See the discussion page. The category Top has topological spaces as objects and continuous maps as morphisms. ...