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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.
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.
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 productIn category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most gen in Met is given by the supreme metric mixing on the cartesian productIn mathematics, the Cartesian product (or direct product X × Y of two sets X and Y is the set of all ordered pairs whose first component is a member of X and whose second component is a member of Y''. This concept is named after Rene Descartes. X × Y { x. There is no coproduct.
We have a "forgetful" functorFor the usage in computer science, see the function object article. In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of all ( small) categories. Functors were first cons Met → SetCategory theory In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics. The category is usually denoted simply as Set . which assigns to each metric space the underlying setThis article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now, and to each short map the underlying functionIn 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 . The concept of a function is fundamental to virtu. This functor is faithful, and therefore Met is a concrete category.
Follows the Top article. See the discussion page.
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