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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. 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. 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. Also, if a statement is false about C, then its dual has to be false about C.

Given a concrete category C, 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.

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