# Category theory

**Category theory** is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

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A category is formed by two sorts of objects, the objects of the category, and the morphisms, which relate two objects called the *source* and the *target* of the morphism. One often says that a morphism is an *arrow* that *maps* its source to its target. Morphisms can be *composed* if the target of the first morphism equals the source of the second one, and morphism composition has similar properties as function composition (associativity and existence of identity morphisms). Morphisms are often some sort of function, but this is not always the case. For example, a monoid may be viewed as a category with a single object, whose morphisms are the elements of the monoid.

The second fundamental concept of category is the concept of a functor, which plays the role of a morphism between two categories and it maps objects of to objects of and morphisms of to morphisms of in such a way that sources are mapped to sources and targets are mapped to targets (or, in the case of a contravariant functor, sources are mapped to targets and *vice-versa*). A third fundamental concept is a natural transformation that may be viewed as a morphism of functors.