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In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors. Functor categories are of interest for two main reasons: * many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; * every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting. == Definition == Suppose ''C'' is a small category (i.e. the objects and morphisms form a set rather than a proper class) and ''D'' is an arbitrary category. The category of functors from ''C'' to ''D'', written as Fun(''C'', ''D''), Funct(''C'',''D'') or ''D''''C'', has as objects the covariant functors from ''C'' to ''D'', and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if μ(''X'') : ''F''(''X'') → ''G''(''X'') is a natural transformation from the functor ''F'' : ''C'' → ''D'' to the functor ''G'' : ''C'' → ''D'', and η(''X'') : ''G''(''X'') → ''H''(''X'') is a natural transformation from the functor ''G'' to the functor ''H'', then the collection η(''X'')μ(''X'') : ''F''(''X'') → ''H''(''X'') defines a natural transformation from ''F'' to ''H''. With this composition of natural transformations (known as vertical composition, see natural transformation), ''D''''C'' satisfies the axioms of a category. In a completely analogous way, one can also consider the category of all ''contravariant'' functors from ''C'' to ''D''; we write this as Funct(''C''op,''D''). If ''C'' and ''D'' are both preadditive categories (i.e. their morphism sets are abelian groups and the composition of morphisms is bilinear), then we can consider the category of all additive functors from ''C'' to ''D'', denoted by Add(''C'',''D''). 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 category」の詳細全文を読む スポンサード リンク
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