Dual (category theory)

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 $g\; \backslash circ\; f$ with $f\; \backslash circ\; g$

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.{{sfn|Mac Lane|1978|p=33}}{{sfn|Awodey|2010|p=53-55}}

• A morphism $f\backslash colon\; A\; \backslash to\; B$ is a monomorphism if $f\; \backslash circ\; g\; =\; f\; \backslash circ\; h$ implies $g=h$. Performing the dual operation, we get the statement that $g\; \backslash circ\; f\; =\; h\; \backslash circ\; f$ implies $g=h.$ For a morphism $f\backslash colon\; B\; \backslash to\; A$, 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.

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

{{reflist}}

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• {{springer|title=Dual category|id=p/d034090}}
• {{springer|title=Duality principle|id=p/d034130}}
• {{springer|title=Duality|id=p/d034120}}
• {{Cite book|title=Categories for the Working Mathematician|last=Mac Lane|first=Saunders|date=1978|publisher=Springer New York|isbn=1441931236|edition=Second|location=New York, NY|pages=33|oclc=851741862}}
• {{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}}

{{refend}}

{{Category theory}}

Category:Category theory

Category theory