Dual (category theory)

Dual (category theory)

In category theory, an abstract branch of mathematics, the dual of a category is the category formed by reversing all the morphisms of . That is, we take to be the category with objects that are those of , but with the morphisms from X to Y in being the morphisms from Y to X in . Hence, the dual of a dual of a category is itself.

It is also often 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 groups and that of (discrete) abelian groups. The category of Stone spaces and continuous functions is the opposite of the category of Boolean algebras and homomorphisms.

One other way in which the concept is used is to remove the distinction between covariant and contravariant functors: a contravariant functor to is equally a functor to the opposite of .