# Dual (category theory)

*For general notions of duality in mathematics, see duality (mathematics).*

In category theory, a branch of mathematics, **duality** is a correspondence between properties of a category *C* and so-called **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}.

## 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.

- 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

## References

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