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The category Ord has preordered sets as objects and monotonic functions as morphisms. This is a category because the composition of two monotonic functions preserves monotonicity.
The monomorphisms in Ord are the injective monotonic functions.
The empty set (considered as a preordered set) is the initial object of Ord; any singleton preordered set is a terminal object. There are thus no zero objects in Ord.
The product in Ord is given by the product order on the cartesian product.
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 Ord → 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 preordered set 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 monotonic function 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 faithfulIn 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 other words, a functor F : C → D is faithful if the maps : are injective for every pair of objects X an, and therefore Ord is a concrete categoryIn 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 function.
Follows the TopCategory theory The category Top has topological spaces as objects and continuous maps as morphisms. This is a category because the composition of two continuous maps is again continuous. The study of Top and of properties of topological spaces using the article. See the discussion page.
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