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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 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]}

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

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

such that the following two diagrams commute:

and |

The object is called the **right dual** of .
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.

- 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 (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 a category is a monoidal category under composition of functors. A functor is a left dual of a functor if and only if is left adjoint to .
^{[4]}

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.

- ^
^{a}^{b}^{c}Ponto, Kate; Shulman, Michael (2014). "Traces in symmetric monoidal categories".*Expositiones Mathematicae*.**32**(3): 248–273. arXiv:1107.6032. Bibcode:2011arXiv1107.6032P. doi:10.1016/j.exmath.2013.12.003. **^**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 978-0-444-82375-5.**^**dual object in a closed category at the*n*Lab**^**See for example Nikshych, D.; Etingof, P.I.; Gelaki, S.; Ostrik, V. (2016). "Exercise 2.10.4".*Tensor Categories*. Mathematical Surveys and Monographs. Vol. 205. American Mathematical Society. p. 41. ISBN 978-1-4704-3441-0.

- Dold, Albrecht; Puppe, Dieter (1980), "Duality, trace, and transfer",
*Proceedings of the International Conference on Geometric Topology (Warsaw, 1978)*, PWN-Polish Scientific Publishers, pp. 81–102, ISBN 9788301017873, MR 0656721, OCLC 681088710 - Freyd, Peter; Yetter, David (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. - Joyal, André; Street, Ross. "The Geometry of Tensor calculus II" (PDF).
*Synthese Library*.**259**: 29–68. CiteSeerX 10.1.1.532.1533.