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 ''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. (''C''^op is composed by reversing every morphism of ''C''.) Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement ''S'' 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. (Compactly saying, ''S'' for ''C'' is true if and only if its dual for ''C''^op is true.)

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.

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 σ^op as follows:
# Interchange each occurrence of "source" in σ with "target".
# Interchange the order of composing morphisms. That is, replace each occurrence of $g \circ f$ with $f \circ g$
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.

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 $g=h.$ This reversed morphism $f\colon B \to A$ is by definition precisely 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 ''C''^op (composed by reversing all morphisms in ''C'') 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'') (a set of all morphisms from ''A'' to ''B'' of a category) 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.

See also

* Adjoint functor
* Dual object
* Duality (mathematics)
* Opposite category
* Pulation square

References

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{{Category theory

Category theory
Category theory