Dual (category theory)

For general notions of duality in mathematics, see 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^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.

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:

1. Interchange each occurrence of "source" in σ with "target".
2. 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 σ^op is true for C^op.

Examples

• A morphism $f\colon A\to B$ is a monomorphism 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. 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

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

References

• "Dual category", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Duality principle", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Duality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. p. 33. ISBN 1441931236. OCLC 851741862.
• Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. pp. 53–55. ISBN 978-0199237180. OCLC 740446073.