Dual (category theory)

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.