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In category theory, a branch of mathematics, duality is a correspondence between properties of a category ''C'' and so-called dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category ''C''op. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about ''C'', then its dual statement is true about ''C''op. Also, if a statement is false about ''C'', then its dual has to be false about ''C''op. Given a concrete category ''C'', it is often the case that the opposite category ''C''op per se is abstract. ''C''op need not be a category that arises from mathematical practice. In this case, another category ''D'' is also termed to be in duality with ''C'' if ''D'' and ''C''op are equivalent as categories. In the case when ''C'' and its opposite ''C''op are equivalent, such a category is self-dual. ==Formal definition== We define the elementary language of category theory as the two-sorted first order language with objects and morphisms as distinct sorts, together with the relations of an object being the source or target of a morphism and a symbol for composing two morphisms. Let σ be any statement in this language. We form the dual σop as follows: # Interchange each occurrence of "source" in σ with "target". # Interchange the order of composing morphisms. That is, replace each occurrence of with Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions. ''Duality'' is the observation that σ is true for some category ''C'' if and only if σop is true for ''C''op. 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「Dual (category theory)」の詳細全文を読む スポンサード リンク
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