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In mathematics, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. == Formal definition == Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: the collection of objects is ob(''S''), the collection of morphisms is hom(''S''), and the identities and composition are as in ''C''. There is an obvious faithful functor ''I'' : ''S'' → ''C'', called the inclusion functor which takes objects and morphisms to themselves. Let ''S'' be a subcategory of a category ''C''. We say that ''S'' is a full subcategory of ''C'' if for each pair of objects ''X'' and ''Y'' of ''S'' : A full subcategory is one that includes ''all'' morphisms between objects of ''S''. For any collection of objects ''A'' in ''C'', there is a unique full subcategory of ''C'' whose objects are those in ''A''. 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「Subcategory」の詳細全文を読む スポンサード リンク
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