# 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
    Hom$_C$((–)$_1$ ⊗ V, (–)$_2$) → Hom$_C$((–)$_1$, 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 ${\underline {\mathrm {Hom} }}_{C}(V,\mathbb {1} _{C})$, 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 $X$ in a monoidal category $(\mathbf {C} ,\otimes ,I,\alpha ,\lambda ,\rho )$. The object $X^{*}$ is called a left dual of $X$ if there exist two morphisms
    $\eta :I\to X\otimes X^{*}$, called the coevaluation, and $\varepsilon :X^{*}\otimes X\to I$, called the evaluation,
such that the following two diagrams commute:
* and
The object $X$ is called the right dual of $X^{*}$. This definition is due to Dold & Puppe (1980).
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.

## Examples

* Consider a monoidal category (Vect$_K$, ⊗$_K$) 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.
* Consider a monoidal category (Mod$_R$, ⊗$_R$) of modules over a commutative ring R with the standard tensor product. A module M is dualizable if and only if it is a finitely generated projective module. In that case the dual object M$^∗$ is also given by the module of homomorphisms Hom$_R$(M, R).
* Consider a homotopy category of pointed spectra Ho(Sp) with the smash product as the monoidal structure. If M is a compact neighborhood retract in $\mathbb {R} ^{n}$ (for example, a compact smooth manifold), then the corresponding pointed spectrum Σ$^∞$(M$^+$) is dualizable. This is a consequence of Spanier–Whitehead duality, which implies in particular Poincaré duality for compact manifolds.
* The category $\mathrm {End} (\mathbf {C} )$ of endofunctors of a category $\mathbf {C}$ is a monoidal category under composition of functors. A functor $F$ is a left dual of a functor $G$ if and only if $F$ is left adjoint to $G$.

## 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.