# 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}. 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 with

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 is a monomorphism if implies . Performing the dual operation, we get the statement that implies For a morphism , 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.

## References

- Jiří Adámek; J. Rosicky (1994).
*Locally Presentable and Accessible Categories*. Cambridge University Press. p. 62. ISBN 978-0-521-42261-1. - Mac Lane 1978, p. 33.
- Awodey 2010, p. 53-55.

- Hazewinkel, Michiel, ed. (2001) [1994], "Dual category",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Hazewinkel, Michiel, ed. (2001) [1994], "Duality principle",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Hazewinkel, Michiel, ed. (2001) [1994], "Duality",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Mac Lane, Saunders (1978).
*Categories for the Working Mathematician*(Second ed.). New York, NY: Springer New York. p. 33. ISBN 1441931236. OCLC 851741862. - Awodey, Steve (2010).
*Category theory*(2nd ed.). Oxford: Oxford University Press. pp. 53–55. ISBN 0199237182. OCLC 740446073.