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In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. ==Definition== If ''F'' and ''G'' are functors between the categories ''C'' and ''D'', then a natural transformation ''η'' from ''F'' to ''G'' is a family of morphisms that satisfy two requirements. # The natural transformation must associate to every object ''X'' in ''C'' a morphism between objects of ''D''. The morphism ''η''''X'' is called the component of ''η'' at ''X''. # Components must be such that for every morphism we have: ::: The last equation can conveniently be expressed by the commutative diagram : 175px If both ''F'' and ''G'' are contravariant, the horizontal arrows in this diagram are reversed. If ''η'' is a natural transformation from ''F'' to ''G'', we also write or . This is also expressed by saying the family of morphisms is natural in ''X''. If, for every object ''X'' in ''C'', the morphism ''η''''X'' is an isomorphism in ''D'', then ''η'' is said to be a (or sometimes natural equivalence or isomorphism of functors). Two functors ''F'' and ''G'' are called ''naturally isomorphic'' or simply ''isomorphic'' if there exists a natural isomorphism from ''F'' to ''G''. An infranatural transformation ''η'' from ''F'' to ''G'' is simply a family of morphisms . Thus a natural transformation is an infranatural transformation for which for every morphism . The naturalizer of ''η'', nat(''η''), is the largest subcategory of ''C'' containing all the objects of ''C'' on which ''η'' restricts to a natural transformation. 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「Natural transformation」の詳細全文を読む スポンサード リンク
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