Editing Category theory (section)

{{Short description|General theory of mathematical structures}}
{{More sources needed|date=November 2024}}

[[File:Commutative diagram for morphism.svg|right|thumb|200px|Schematic representation of a category with objects ''X'', ''Y'', ''Z'' and morphisms ''f'', ''g'', {{nowrap|1=''g'' ∘ ''f''}}. (The category's three identity morphisms 1<sub>''X''</sub>, 1<sub>''Y''</sub> and 1<sub>''Z''</sub>, if explicitly represented, would appear as three arrows, from the letters ''X'', ''Y'', and ''Z'' to themselves, respectively.)]]

'''Category theory''' is a general theory of [[mathematical structure]]s and their relations. It was introduced by [[Samuel Eilenberg]] and [[Saunders Mac Lane]] in the middle of the 20th century in their foundational work on [[algebraic topology]].<ref>{{Citation |last=Marquis |first=Jean-Pierre |title=Category Theory |date=2023 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/fall2023/entries/category-theory/ |access-date=2024-04-23 |edition=Fall 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref> Category theory is used in almost all areas of mathematics. In particular, many constructions of new [[mathematical object]]s from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include [[quotient space (disambiguation)|quotient space]]s, [[direct product]]s, completion, and [[duality (mathematics)|duality]].

Many areas of [[computer science]] also rely on category theory, such as [[functional programming]] and [[Semantics (computer science)|semantics]].

A [[category (mathematics)|category]] is formed by two sorts of [[mathematical object|objects]]: the [[object (category theory)|object]]s of the category, and the [[morphism]]s, which relate two objects called the ''source'' and the ''target'' of the morphism. Metaphorically, 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. Morphism composition has similar properties as [[function composition]] ([[associativity]] and existence of an [[identity element|identity morphism]] for each object). Morphisms are often some sort of [[function (mathematics)|function]]s, 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 theory is the concept of a [[functor]], which plays the role of a morphism between two categories <math>\mathcal{C}_1</math> and <math>\mathcal{C}_2</math>: it maps objects of <math>\mathcal{C}_1</math> to objects of <math>\mathcal{C}_2</math> and morphisms of <math>\mathcal{C}_1</math> to morphisms of <math>\mathcal{C}_2</math> 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.