双対 (圏論)
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 σ as follows:
σ において各「始域」と「終域」と入れ替える.
射を合成する順序を入れ替える.つまり,各 g ∘ f {\displaystyle g\circ f} を f ∘ g {\displaystyle f\circ g} に置き換える.
インフォーマルには,これらの条件はステートメントの双対は矢と合成を逆にすることによって作られるといっている.
Duality is the observation that σ is true for some category C if and only if σ is true for C.