Editing Dual (category theory)

{{Short description|Correspondence between properties of a category and its opposite}}
{{For|general notions of duality in mathematics|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]] ''C''<sup>op</sup>. Given a statement regarding the category ''C'', by interchanging the [[Domain of a function|source]] and [[Codomain|target]] of each [[morphism]] as well as interchanging the order of [[Function composition|composing]] two morphisms, a corresponding dual statement is obtained regarding the opposite category ''C''<sup>op</sup>. 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''<sup>op</sup>. Also, if a statement is false about ''C'', then its dual has to be false about ''C''<sup>op</sup>.

Given a [[concrete category]] ''C'', it is often the case that the opposite category ''C''<sup>op</sup> per se is abstract. ''C''<sup>op</sup> 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''<sup>op</sup> are [[Equivalence of categories|equivalent as categories]].

In the case when ''C'' and its opposite ''C''<sup>op</sup> are equivalent, such a category is self-dual.<ref name="AdamekRosicky1994">{{cite book|author1=Jiří Adámek|author2=J. Rosicky|title=Locally Presentable and Accessible Categories|url=https://books.google.com/books?id=iXh6rOd7of0C&pg=PA62|year=1994|publisher=Cambridge University Press|isbn=978-0-521-42261-1|page=62}}</ref>

==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<sup>op</sup> as follows: 
# Interchange each occurrence of "source" in σ with "target".
# Interchange the order of composing morphisms. That is, replace each occurrence of <math>g \circ f</math> with <math>f \circ g</math>
Informally, these conditions state that the dual of a statement is formed by reversing [[morphism|arrows]] and [[function composition|compositions]].

''Duality'' is the observation that σ is true for some category ''C'' if and only if σ<sup>op</sup> is true for ''C''<sup>op</sup>.{{sfn|Mac Lane|1978|p=33}}{{sfn|Awodey|2010|p=53-55}}

==Examples==

* A morphism <math>f\colon A \to B</math> is a [[monomorphism]] if <math>f \circ g = f \circ h</math> implies <math>g=h</math>. Performing the dual operation, we get the statement that <math>g \circ f = h \circ f</math> implies <math>g=h.</math> For a morphism <math>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''<sup>op</sup> is an epimorphism.

* An example comes from reversing the direction of inequalities in a [[partial order]]. So if ''X'' is a [[Set (mathematics)|set]] and ≤ a partial order relation, we can define a new partial order relation ≤<sub>new</sub> by

:: ''x'' ≤<sub>new</sub> ''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 theory|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 (order theory)|duality]] applied to lattices.

* [[limit (category theory)|Limits]] and [[limit (category theory)|colimits]] are dual notions.
* [[Fibration]]s and [[cofibration]]s 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==
{{reflist}}
{{refbegin}}
* {{springer|title=Dual category|id=p/d034090}}
* {{springer|title=Duality principle|id=p/d034130}}
* {{springer|title=Duality|id=p/d034120}}
* {{Cite book|title=Categories for the Working Mathematician|last=Mac Lane|first=Saunders|date=1978|publisher=Springer New York|isbn=1441931236|edition=Second|location=New York, NY|pages=33|oclc=851741862}}
* {{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}}
{{refend}}

{{Category theory}}

[[Category:Category theory]]
[[Category:Duality theories|Category theory]]