In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of duality for finite-dimensionalvector 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 category 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 category of all vector spaces is not.
Motivation
Let V be a finite-dimensional vector space over some fieldK. The standard notion of a dual vector spaceV∗ has the following property: for any K-vector spaces U and W there is an adjunction HomK = HomK, 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 monoidal category one may attempt to define a dual of an object Vto be an object V∗ ∈ C with a natural isomorphism of bifunctors 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 closed monoidal categoryC, 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.
Definition
Consider an object in a monoidal category. The object is called a left dual of if there exist two morphisms such that the following two diagrams commute: The object is called the right dual of. This definition is due to. Left duals are canonically isomorphic when they exist, as are right duals. When C is braided, 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.
Examples
Consider a monoidal category of vector spaces over a field K with the standard tensor product. A space V is dualizable if and only if it is finite-dimensional, and in this case the dual object V∗ coincides with the standard notion of a dual vector space.
A monoidal category where every object has a left dual is sometimes called a leftautonomous category. Algebraic geometers call it a leftrigid 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.
