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

Given a concrete category C, it is often the case that the opposite category Cop per se is abstract. Cop 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 Cop are equivalent as categories.

In the case when C and its opposite Cop are equivalent, such a category is self-dual.

Formal definition
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 σop as follows: Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions.
 * 1) Interchange each occurrence of "source" in σ with "target".
 * 2) Interchange the order of composing morphisms. That is, replace each occurrence of $$g \circ f$$ with $$f \circ g$$

Duality is the observation that σ is true for some category C if and only if σop is true for Cop.

Examples

 * A morphism $$f\colon A \to B$$ is a monomorphism if $$f \circ g = f \circ h$$ implies $$g=h$$. Performing the dual operation, we get the statement that $$g \circ f = h \circ f$$ implies $$g=h.$$ For a morphism $$f\colon B \to A$$, this is precisely what it means for f to be an epimorphism. In short, the property of being a monomorphism is dual to the property of being an epimorphism.

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 Cop is an epimorphism.


 * An example comes 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 partial order relation ≤new by


 * x ≤new y if and only if y ≤ x.

This example on orders is a special case, since partial orders correspond to a certain kind of category in which Hom(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, or of duality applied to lattices.


 * Limits and colimits are dual notions.
 * Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called Eckmann–Hilton duality.