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category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...
, a branch of
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the
source Source or subsource or ''variation'', may refer to: Research * Historical document * Historical source * Source (intelligence) or subsource, typically a confidential provider of non open-source intelligence * Source (journalism), a person, public ...
and
target Target may refer to: Physical items * Shooting target, used in marksmanship training and various shooting sports ** Bullseye (target), the goal one for which one aims in many of these sports ** Aiming point, in field artillery, fixed at a specifi ...
of each
morphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 Interior of the Pantheon dome, seen from beneath. The concrete for the coffered dome was laid on moulds, mounted on temporary scaffolding. Concrete is a composite material A composite material (also called a composition material or shorte ...
''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.


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 σop as follows: # Interchange each occurrence of "source" in σ with "target". # Interchange the order of composing morphisms. That is, replace each occurrence of g \circ f with 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 ''C''op.


Examples

* A morphism f\colon A \to B is a
monomorphism 220px In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of catego ...
if f \circ g = f \circ h implies g=h. Performing the dual operation, we get the statement that g \circ f = h \circ f implies g=h. For a morphism f\colon B \to A, this is precisely what it means for ''f'' to be an
epimorphism 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...
. 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 Image:Hasse diagram of powerset of 3.svg, 250px, The Hasse diagram of the power set, set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, su ...
. 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 In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th-c ...
, or of duality applied to lattices. *
Limits Limit or Limits may refer to: Arts and media * Limit (music) In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...
and limit (category theory), 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

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


References

* * * * * {{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 Category theory Duality theories, Category theory