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the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take C^op to be the category with objects that are those of C, but with the morphisms from X to Y in C^op being the morphisms from Y to X in C. Hence, the dual of a dual of a category is itself. The dual category is also called the opposite category.

Examples come from reversing the direction of inequalities in a partial order. So if X is a set and ≤ a partial order relation , we can define a new ≤_new by the definition

x ≤_new y if and only if y ≤ x.

For example, there are opposite pairs child/parent, or descendant/ancestor.

This is a special case, since partial orders correspond to a certain kind of category in which Mor(A,B) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). For example, if we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws.

Generalising that observation, inverse limits and direct limits are interchanged when one passes to the opposite category. This is immediately useful, when one can identify the opposite category in concrete terms. For example the category of affine schemes is the opposite of the category of commutative rings. The Pontryagin duality restricts to the duality between the category of compact Hausdorff abelian topological groupIn mathematics, a topological group ''G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. Here, G × G is viewed as a topological space by using the produs and that of (discrete) abelian groups. The category of Stone spaces and continuous functions is the opposite of the category of Boolean algebraIn mathematics and computer science, Boolean algebras or Boolean lattices are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and complemes and homomorphisms.

One other way in which the concept is used is to remove the distinction between covariantIn category theory, see covariant functor. In tensor analysis, a covariant coordinate system is reciprocal to a corresponding contravariant coordinate system. Roughly speaking, a covariant tensor is a vector field that defines the topology of a space; it and contravariantContravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projection) used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system bein functorFor the usage in computer science, see the function object article. In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of all ( small) categories. Functors were first conss: a contravariant functor to C is equally a functor to the opposite of C.

See also: Duality (mathematics)In mathematics, duality has numerous meanings. They are inter-connected, mostly; without there being a single, master duality. The duality of linear algebra is one of the fundamental sources. De Morgan dual in logic dual (category theory) dual polyhedron