Dual object

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-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 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 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 Hom"K"("U" ⊗ "V","W") = Hom"K"("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 monoidal category ("C", ⊗) one may attempt to define a dual of an object "V" to 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 category "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 formula_1, where 1"C" 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 formula_2 in a monoidal category formula_3. The object formula_4 is called a left dual of formula_2 if there exist two morphisms
such that the following two diagrams commute:
The object formula_2 is called the right dual of formula_4. 
This definition is due to .
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 (respectively right) dual is sometimes called a left (respectively right) autonomous category. Algebraic geometers call it a left (respectively 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.
Traces.
Any endomorphism "f" of a dualizable object admits a trace, which is a certain endomorphism of the monoidal unit of "C". This notion includes, as very special cases, the trace in linear algebra and the Euler characteristic of a chain complex.