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's 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. 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's 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. A category of finite-dimensional vector spaces with a 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. A standard notion of a dual vector space V∗ has the following property. For any 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 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 H o m _ C ( V , 1 C ) {displaystyle {underline {mathrm {Hom} }}_{C}(V,mathbb {1} _{C})} , 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 X {displaystyle X} in a monoidal category ( C , ⊗ , I , α , λ , ρ ) {displaystyle (mathbf {C} ,otimes ,I,alpha ,lambda , ho )} . The object X ∗ {displaystyle X^{*}} is called a left dual of X {displaystyle X} if there exist two morphsims