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| :''For general notions of duality in mathematics, see [[duality (mathematics)]].''
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| 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''<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>.
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| 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]].
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| In the case when ''C'' and its opposite ''C''<sup>op</sup> are equivalent, such a category is '''self-dual'''.
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| ==Formal definition==
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| 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.
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| Let σ be any statement in this language. We form the dual σ<sup>op</sup> as follows:
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| # Interchange each occurrence of "source" in σ with "target".
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| # Interchange the order of composing morphisms. That is, replace each occurrence of <math>g \circ f</math> with <math>f \circ g</math>
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| Informally, these conditions state that the dual of a statement is formed by reversing [[morphism|arrows]] and [[function composition|compositions]].
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| ''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>.
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| ==Examples==
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| * 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.
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| 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.
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| * 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
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| :: ''x'' ≤<sub>new</sub> ''y'' if and only if ''y'' ≤ ''x''.
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| 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.
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| * [[limit (category theory)|Limits]] and [[limit (category theory)|colimits]] are dual notions.
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| * [[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]].
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| ==See also==
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| * [[Dual object]]
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| * [[Duality (mathematics)]]
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| * [[Opposite category]]
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| ==References==
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| * {{springer|title=Dual category|id=p/d034090}}
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| * {{springer|title=Duality principle|id=p/d034130}}
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| * {{springer|title=Duality|id=p/d034120}}
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| [[Category:Category theory]]
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| [[Category:Duality theories|Category theory]]
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I'm Norwegian male :D.
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