Dual (category theory)
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^{op}. 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 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 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.^{[1]}
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^{op} as follows:
- Interchange each occurrence of "source" in σ with "target".
- Interchange the order of composing morphisms. That is, replace each occurrence of with
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 σ^{op} is true for C^{op}.
Examples
- A morphism is a monomorphism if implies . Performing the dual operation, we get the statement that implies For a morphism , 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^{op} 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) 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
References
- ↑ Jiří Adámek; J. Rosicky (1994). Locally Presentable and Accessible Categories. Cambridge University Press. p. 62. ISBN 978-0-521-42261-1.
- Hazewinkel, Michiel, ed. (2001), "Dual category", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Hazewinkel, Michiel, ed. (2001), "Duality principle", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Hazewinkel, Michiel, ed. (2001), "Duality", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4