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}. 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 ''C''^{op}. Also, if a statement is false about ''C'', then its dual has to be false about ''C''^{op}.
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 $g\; \backslash circ\; f$ with $f\; \backslash circ\; g$
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 $f\backslash colon\; A\; \backslash to\; B$ is a monomorphism if $f\; \backslash circ\; g\; =\; f\; \backslash circ\; h$ implies $g=h$. Performing the dual operation, we get the statement that $g\; \backslash circ\; f\; =\; h\; \backslash circ\; f$ implies $g=h.$ For a morphism $f\backslash colon\; B\; \backslash to\; A$, 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 ''C''^{op} 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.

See also

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

References

* * * * * {{Cite book|title=Category theory|last=Awodey|first=Steve|date=2010|publisher=Oxford University Press|isbn=978-0199237180|edition=2nd|location=Oxford|pages=53–55|oclc=740446073 Category:Category theory Category theory

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

Examples

* A morphism $f\backslash colon\; A\; \backslash to\; B$ is a monomorphism if $f\; \backslash circ\; g\; =\; f\; \backslash circ\; h$ implies $g=h$. Performing the dual operation, we get the statement that $g\; \backslash circ\; f\; =\; h\; \backslash circ\; f$ implies $g=h.$ For a morphism $f\backslash colon\; B\; \backslash to\; A$, 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 ''C''

See also

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

References

* * * * * {{Cite book|title=Category theory|last=Awodey|first=Steve|date=2010|publisher=Oxford University Press|isbn=978-0199237180|edition=2nd|location=Oxford|pages=53–55|oclc=740446073 Category:Category theory Category theory