Dual (category theory)

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

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[edit] Examples

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

[edit] Formal definition

Let Σ be any statement of the elementary theory of an abstract category. We form the dual of Σ as follows:

  1. Replace each occurrence of "domain" in Σ with "codomain" and conversely.
  2. Replace each occurrence of g \circ f =h with f \circ g = h

Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions. For example, consider the following statements about a category \mathcal{C}:

  • f\colon A \to B
  • f is monic, i.e. for all morphisms g,h for which composition makes sense, f \circ g = f \circ h implies g = h.

The respective dual statements are

  • f\colon B \to A
  • f is epic, i.e. for all morphisms g,h for which composition makes sense, g \circ f = h \circ f implies g = h.

The duality principle asserts that if a statement is a theorem, then the dual statement is also a theorem. We take "theorem" here to mean provable from the axioms of the elementary theory of an abstract category. In practice, for a valid statement about a particular category \mathcal{C}, the dual statement is valid in the dual category \mathcal{C}^{*} ( \mathcal{C}^{op}).

[edit] Duality

The 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.

Generalising that observation, limits and colimits 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 equivalent to the opposite of the category of commutative rings. The Pontryagin duality restricts to an equivalence between the category of compact Hausdorff abelian topological groups and the opposite of the category of (discrete) abelian groups. The category of Stone spaces and continuous functions is equivalent to the opposite of the category of Boolean algebras and homomorphisms.

[edit] Dualities

A duality between categories C and D is defined as an equivalence between C and the opposite of D. The above are all examples of dualities. A self-dual category is a category equivalent to its opposite. An example of a self-dual category is the category of finite abelian groups.

[edit] Covariance and contravariance of functors

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

[edit] See also