# Dual (category theory)

__: Correspondence between properties of a category and its opposite__

**Short description**

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.^{[1]}

## 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 [math]\displaystyle{ g \circ f }[/math] with [math]\displaystyle{ f \circ g }[/math]

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}.^{[2]}^{[3]}

## Examples

- A morphism [math]\displaystyle{ f\colon A \to B }[/math] is a monomorphism if [math]\displaystyle{ f \circ g = f \circ h }[/math] implies [math]\displaystyle{ g=h }[/math]. Performing the dual operation, we get the statement that [math]\displaystyle{ g \circ f = h \circ f }[/math] implies [math]\displaystyle{ g=h. }[/math] For a morphism [math]\displaystyle{ f\colon B \to A }[/math], 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

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

## References

- ↑ Jiří Adámek; J. Rosicky (1994).
*Locally Presentable and Accessible Categories*. Cambridge University Press. p. 62. ISBN 978-0-521-42261-1. https://books.google.com/books?id=iXh6rOd7of0C&pg=PA62. - ↑ Mac Lane 1978, p. 33.
- ↑ Awodey 2010, p. 53-55.

- Hazewinkel, Michiel, ed. (2001), "Dual category",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/d034090 - Hazewinkel, Michiel, ed. (2001), "Duality principle",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/d034130 - Hazewinkel, Michiel, ed. (2001), "Duality",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/d034120 - Mac Lane, Saunders (1978).
*Categories for the Working Mathematician*(Second ed.). New York, NY: Springer New York. pp. 33. ISBN 1441931236. OCLC 851741862. - Awodey, Steve (2010).
*Category theory*(2nd ed.). Oxford: Oxford University Press. pp. 53–55. ISBN 978-0199237180. OCLC 740446073.

Original source: https://en.wikipedia.org/wiki/Dual (category theory).
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