Dual (category theory)

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 categoryC. 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. (C is composed by reversing every morphism of C.) Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement S is true about C, then its dual statement is true about C. Also, if a statement is false about C, then its dual has to be false about C. (Compactly saying, S for C is true if and only if its dual for C is true.)

Given a concrete categoryC, it is often the case that the opposite category C per se is abstract. C 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 are equivalent as categories.

In the case when C and its opposite C are equivalent, such a category is self-dual.