Editing Dual (category theory) (section)

{{Short description|Correspondence between properties of a category and its opposite}}
{{For|general notions of duality in mathematics|Duality (mathematics)}}

In [[category theory]], a branch of [[mathematics]], '''duality''' is a correspondence between the properties of a category ''C'' and the dual properties of the [[opposite category]] ''C''<sup>op</sup>. Given a statement regarding the category ''C'', by interchanging the [[Domain of a function|source]] and [[Codomain|target]] of each [[morphism]] as well as interchanging the order of [[Function composition|composing]] two morphisms, a corresponding dual statement is obtained regarding the opposite category ''C''<sup>op</sup>. 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''<sup>op</sup>. Also, if a statement is false about ''C'', then its dual has to be false about ''C''<sup>op</sup>.

Given a [[concrete category]] ''C'', it is often the case that the opposite category ''C''<sup>op</sup> per se is abstract. ''C''<sup>op</sup> 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''<sup>op</sup> are [[Equivalence of categories|equivalent as categories]].

In the case when ''C'' and its opposite ''C''<sup>op</sup> are equivalent, such a category is self-dual.<ref name="AdamekRosicky1994">{{cite book|author1=Jiří Adámek|author2=J. Rosicky|title=Locally Presentable and Accessible Categories|url=https://books.google.com/books?id=iXh6rOd7of0C&pg=PA62|year=1994|publisher=Cambridge University Press|isbn=978-0-521-42261-1|page=62}}</ref>