Dual (category theory)

Correspondence between properties of a category and its opposite

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

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 Cop. 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 Cop. (Cop 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 Cop. Also, if a statement is false about C, then its dual has to be false about Cop. (Compactly saying, S for C is true if and only if its dual for Cop is true.)

Given a concrete category C, it is often the case that the opposite category Cop per se is abstract. Cop 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 Cop are equivalent as categories.

In the case when C and its opposite Cop are equivalent, such a category is self-dual.[1]

## Formal definition

[edit]

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 σop as follows:

1. Interchange each occurrence of "source" in σ with "target".
2. Interchange the order of composing morphisms. That is, replace each occurrence of  g ∘ f {\displaystyle g\circ f} with  f ∘ g {\displaystyle 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 σop is true for Cop.[2][3]

## Examples

[edit]

* A morphism  f : A -> B {\displaystyle f\colon A\to B} is a monomorphism if  f ∘ g = f ∘ h {\displaystyle f\circ g=f\circ h} implies  g = h {\displaystyle g=h} . Performing the dual operation, we get the statement that  g ∘ f = h ∘ f {\displaystyle g\circ f=h\circ f} implies  g = h . {\displaystyle g=h.} This reversed morphism  f : B -> A {\displaystyle 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 Cop (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

[edit]

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

## References

[edit]

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

* "Dual category", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
* "Duality principle", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
* "Duality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
* 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 978-0199237180. OCLC 740446073.

* v
* t
* e

Category theory

Key concepts

| Key concepts|

* Category
* Abelian
* Additive
* Concrete
* Pre-abelian
* Preadditive
* Bicategory
* Adjoint functors
* CCC
* Commutative diagram
* End
* Exponential
* Functor
* Kan extension
* Morphism
* Natural transformation
* Universal property
* Yoneda lemma

|

Universal constructions|

| Limits|

* Terminal objects
* Products
* Equalizers
* Kernels
* Pullbacks
* Inverse limit

|

Colimits|

* Initial objects
* Coproducts
* Coequalizers
* Cokernels and quotients
* Pushout
* Direct limit

Algebraic categories|

* Sets
* Relations
* Magmas
* Groups
* Abelian groups
* Rings (Fields)
* Modules (Vector spaces)

Constructions on categories|

* Free category
* Functor category
* Kleisli category
* Opposite category
* Quotient category
* Product category
* Comma category
* Subcategory

Higher category theory

| Key concepts|

* Categorification
* Enriched category
* Higher-dimensional algebra
* Homotopy hypothesis
* Model category
* Simplex category
* String diagram
* Topos

|

n-categories|

| Weak n-categories|

* Bicategory (pseudofunctor)
* Tricategory
* Tetracategory
* Kan complex
* ∞-groupoid
* ∞-topos

|

Strict n-categories|

* 2-category (2-functor)
* 3-category

Categorified concepts|

* 2-group
* 2-ring
* En-ring
* (Traced)(Symmetric) monoidal category
* n-group
* n-monoid

* Category
* Outline
* Glossary

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template