# 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.[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**. A category of finite-dimensional vector spaces with a 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*. A standard notion of a dual vector space *V*^{∗} has the following property. For any 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.[1] 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 , 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.[3]

## Definition

Consider an object in a monoidal category . The object is called a **left dual** of if there exist two morphsims

- , called the
**coevaluation**, and , called the**evaluation**,

such that the following two diagrams commute

and |

The object is called the **right dual** of . 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 a standard tensor product. A space*V*is dualizable if and only if it's a finite-dimensional vector space. In that 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 a standard tensor product. A module*M*is dualizable if and only if it's a finitely generated projective module. In that case the dual object*R*^{∗}is also given by the module of homomorphisms Hom_{R}(*M*,*R*). - Consider a homotopy category of pointed spectra Ho(Sp) with a smash product as a monoidal structure. If
*M*is a compact neighborhood retract in (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.[1] - The category of endofunctors of is a monoidal category under functor composition. It is true that a functor is a left dual of a functor iff , that is if is left adjoint to .[4]

## 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.

## See also

## References

- Ponto, Kate; Shulman, Michael (2014). "Traces in symmetric monoidal categories".
*Expositiones Mathematicae*.**32**(3): 248–273. arXiv:1107.6032. Bibcode:2011arXiv1107.6032P. - Becker, James C.; Gottlieb, Daniel Henry (1999). "A history of duality in algebraic topology" (PDF). In James, I.M. (ed.).
*History of topology*. North Holland. pp. 725–745. ISBN 9780444823755. - "dual object in a closed category in nLab".
*ncatlab.org*. Retrieved 11 December 2017. - See for example exercise 2.10.4 in Pavel Etingof "Tensor Categories"

- Peter Freyd and David Yetter (1989). "Braided Compact Closed Categories with Applications to Low-Dimensional Topology".
*Advances in Mathematics*.**77**(2): 156–182. doi:10.1016/0001-8708(89)90018-2. - André Joyal and Ross Street. "The Geometry of Tensor calculus II".
*Synthese Lib*.**259**: 29–68.