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In mathematics, a functor is a type of mapping between categories which is applied in category theory. Functors can be thought of as homomorphisms between categories. In the category of small categories, functors can be thought of more generally as morphisms. Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are generally applicable in areas within mathematics that category theory can make an abstraction of. The word ''functor'' was borrowed by mathematicians from the philosopher Rudolf Carnap, who used the term in a linguistic context:〔Carnap, The Logical Syntax of Language, p. 13–14, 1937, Routledge & Kegan Paul〕 see function word. ==Definition== Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that〔Jacobson (2009), p. 19, def. 1.2.〕 * associates to each object an object in ''D'', * associates to each morphism in ''C'' a morphism in ''D'' such that the following two conditions hold: * * for every object * * for all morphisms and That is, functors must preserve identity morphisms and composition of morphisms. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「functor」の詳細全文を読む スポンサード リンク
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