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Dual object

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Dual object

In category theory, a branch of mathematics, it is possible to define a concept of dual object generalizing the concept of dual space in linear algebra.

Contents

A category in which each object has a dual is called autonomous or rigid.

Definition

Consider an object X in a monoidal category ( C , ⊗ , I , α , λ , ρ ) . The object X ∗ is called a left dual of X if there exist two morphsims

η : I → X ⊗ X ∗ , called the coevaluation, and ε : X ∗ ⊗ X → I , called the evaluation,

such that the following two diagrams commute

The object X is called the right dual of X ∗ . Left duals are canonically isomorphic when they exist, as are right duals. When C is braided (or symmetric), every left dual is also a right dual, and vice versa.

If we consider a monoidal category as a bicategory with one object, a dual pair is exactly an adjoint pair.

Categories with duals

A monoidal category where every object has a left (resp. right) dual is sometimes called a left (resp. right) autonomous category. Algebraic geometers call it a left (resp. right) rigid category. A monoidal category where every object has both a left and a right dual is called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.

References

Dual object Wikipedia
(Text) CC BY-SA


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