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Concept
Dual object
Summary
In , a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for in arbitrary . It is only a partial generalization, based upon the categorical properties of duality for finite-dimensional vector spaces. An object admitting a dual is called a dualizable object. In this formalism, infinite-dimensional vector spaces are not dualizable, since the dual vector space V∗ doesn't satisfy the axioms. Often, an object is dualizable only when it satisfies some finiteness or compactness property.
A in which each object has a dual is called autonomous or rigid. The category of finite-dimensional vector spaces with the standard tensor product is rigid, while the is not.
Let V be a finite-dimensional vector space over some field K. The standard notion of a dual vector space V∗ has the following property: for any K-vector spaces U and W there is an adjunction HomK(U ⊗ V,W) = HomK(U, V∗ ⊗ W), and this characterizes V∗ up to a unique isomorphism. This expression makes sense in any category with an appropriate replacement for the tensor product of vector spaces. For any (C, ⊗) one may attempt to define a dual of an object V to be an object V∗ ∈ C with a natural isomorphism of bifunctors
HomC((–)1 ⊗ V, (–)2) → HomC((–)1, V∗ ⊗ (–)2)
For a well-behaved notion of duality, this map should be not only natural in the sense of category theory, but also respect the monoidal structure in some way. An actual definition of a dual object is thus more complicated.
In a C, i.e. a monoidal category with an internal Hom functor, an alternative approach is to simulate the standard definition of a dual vector space as a space of functionals. For an object V ∈ C define V∗ to be , where 1C is the monoidal identity. In some cases, this object will be a dual object to V in a sense above, but in general it leads to a different theory.
Consider an object in a . The object is called a left dual of if there exist two morphisms
called the coevaluation, and , called the evaluation,
such that the following two diagrams commute:
The object is called the right dual of .
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