Opposite (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 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.

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^op as follows:

Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions.

Duality is the observation that σ is true for some category C if and only if σ^op is true for C^op.

Applying duality, this means that a morphism in some category C is a monomorphism if and only if the reverse morphism in the opposite category C^op is an epimorphism.

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