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In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. ==Formal definition== Let ''C'' be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes). For all objects ''A'' and ''B'' in ''C'' we define two functors to the category of sets as follows: \pagecolor g \mapsto f\circ g for each ''g'' in Hom(''A'', ''X''). |This is a contravariant functor given by: *Hom(–,''B'') maps each object ''X'' in ''C'' to the set of morphisms, Hom(''X'', ''B'') *Hom(–,''B'') maps each morphism ''h'' : ''X'' → ''Y'' to the function *: Hom(''h'', ''B'') : Hom(''Y'', ''B'') → Hom(''X'', ''B'') given by *: for each ''g'' in Hom(''Y'', ''B''). |} The functor Hom(–,''B'') is also called the ''functor of points'' of the object ''B''. Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms. The pair of functors Hom(''A'',–) and Hom(–,''B'') are related in a natural manner. For any pair of morphisms ''f'' : ''B'' → ''B''′ and ''h'' : ''A''′ → ''A'' the following diagram commutes: Both paths send ''g'' : ''A'' → ''B'' to ''f'' ∘ ''g'' ∘ ''h''. The commutativity of the above diagram implies that Hom(–,–) is a bifunctor from ''C'' × ''C'' to Set which is contravariant in the first argument and covariant in the second. Equivalently, we may say that Hom(–,–) is a covariant bifunctor : Hom(–,–) : ''C''op × ''C'' → Set where ''C''op is the opposite category to ''C''. The notation HomC(–,–) is sometimes used for Hom(–,–) in order to emphasize the category forming the domain. 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「Hom functor」の詳細全文を読む スポンサード リンク
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