Editing Dual (category theory) (section)

==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]].