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.[1] Often, an object is dualizable only when it satisfies some finiteness or compactness property.[2]
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.
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 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
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 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
\underline{Hom
Consider an object
X
(C, ⊗ ,I,\alpha,λ,\rho)
X*
X
η:I\toX ⊗ X*
\varepsilon:X* ⊗ X\toI
The object
X
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.
Rn
End(C)
C
F
G
F
G
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.
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.