# Dual object

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In [category theory](/source/category_theory), a branch of [mathematics](/source/mathematics), a '''dual object''' is an analogue of a [dual vector space](/source/dual_vector_space) from [linear algebra](/source/linear_algebra) for [objects](/source/Object_(category_theory)) in arbitrary [monoidal categories](/source/Monoidal_category). It is only a partial generalization, based upon the categorical properties of [duality](/source/Duality_(mathematics)) for [finite-dimensional](/source/Dimension_(vector_space)) [vector space](/source/vector_space)s. 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''<sup>∗</sup> doesn't satisfy the axioms.<ref name="traces">{{cite journal| first1 = Kate | last1 = Ponto | first2 = Michael | last2 = Shulman|author2-link = Michael Shulman (mathematician)|title = Traces in symmetric monoidal categories | journal = [Expositiones Mathematicae](/source/Expositiones_Mathematicae) | volume = 32 | issue = 3 | year = 2014 | pages = 248–273 | arxiv = 1107.6032| bibcode = 2011arXiv1107.6032P | doi=10.1016/j.exmath.2013.12.003 | doi-access=free }}</ref> Often, an object is dualizable only when it satisfies some finiteness or [compactness](/source/Compact_space) property.<ref>{{cite book| last1 = Becker | first1 = James C. | last2 = Gottlieb | first2 = Daniel Henry | editor-last=James | editor-first = I.M. | title = History of topology | publisher= North Holland | date = 1999 | pages = 725–745 | chapter=A history of duality in algebraic topology | isbn=978-0-444-82375-5 | chapter-url=http://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf}} </ref>

A [category](/source/Category_(mathematics)) in which each object has a dual is called '''autonomous''' or '''rigid'''. The category of finite-dimensional vector spaces with the standard [tensor product](/source/tensor_product) is rigid, while the [category of all vector spaces](/source/category_of_vector_spaces) is not. 

==Motivation==
Let ''V'' be a finite-dimensional vector space over some [field](/source/field_(mathematics)) ''K''. The standard notion of a [dual vector space](/source/dual_vector_space) ''V''<sup>∗</sup> has the following property: for any ''K''-vector spaces ''U'' and ''W'' there is an [adjunction](/source/adjoint_functors) Hom<sub>''K''</sub>(''U'' ⊗ ''V'',''W'') = Hom<sub>''K''</sub>(''U'', ''V''<sup>∗</sup> ⊗ ''W''), and this characterizes ''V''<sup>∗</sup> up to a unique [isomorphism](/source/isomorphism). This expression makes sense in any category with an appropriate replacement for the [tensor product](/source/tensor_product) of vector spaces. For any [monoidal category](/source/monoidal_category) (''C'', ⊗) one may attempt to define a dual of an object ''V'' to be an object ''V''<sup>∗</sup> ∈ ''C'' with a [natural isomorphism](/source/natural_isomorphism) of [bifunctor](/source/bifunctor)s
:Hom<sub>''C''</sub>((–)<sub>1</sub> ⊗ ''V'', (–)<sub>2</sub>) → Hom<sub>''C''</sub>((–)<sub>1</sub>, ''V''<sup>∗</sup> ⊗ (–)<sub>2</sub>)
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.<ref name="traces" /> An actual definition of a dual object is thus more complicated.

In a [closed monoidal category](/source/closed_monoidal_category) ''C'', i.e. a monoidal category with an [internal Hom](/source/internal_Hom) functor, an alternative approach is to simulate the standard definition of a dual vector space as a space of [functionals](/source/functional_(mathematics)). For an object ''V'' ∈ ''C'' define ''V''<sup>∗</sup> to be <math>\underline{\mathrm{Hom}}_C(V, \mathbb{1}_C)</math>, where 1<sub>''C''</sub> 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.<ref>{{nlab|id=dual+object+in+a+closed+category|title=dual object in a closed category}}</ref>

==Definition==

Consider an object <math>X</math> in a [monoidal category](/source/monoidal_category) <math>(\mathbf{C},\otimes, I, \alpha, \lambda, \rho)</math>. The object <math>X^*</math> is called a '''left dual''' of <math>X</math> if there exist two morphisms
:<math>\eta:I\to X\otimes X^*</math>,  called the '''coevaluation''', and <math>\varepsilon:X^*\otimes X\to I</math>, called the '''evaluation''', 
such that the following two diagrams commute:
{|
| 350px
| width="100pt" style="text-align: center;" | and 
| 350px
|}

The object <math>X</math> is called the '''right dual''' of <math>X^*</math>. 
This definition is due to {{harvtxt|Dold|Puppe|1980}}.

Left duals are canonically isomorphic when they exist, as are right duals.  When ''C'' is [braided](/source/braided_monoidal_category) (or [symmetric](/source/symmetric_monoidal_category)), every left dual is also a right dual, and vice versa. 

If we consider a monoidal category as a [bicategory](/source/bicategory) with one object, a dual pair is exactly an [adjoint pair](/source/adjoint_pair).

==Examples==
* Consider a monoidal category (Vect<sub>''K''</sub>, ⊗<sub>''K''</sub>) 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''<sup>∗</sup> coincides with the standard notion of a [dual vector space](/source/dual_vector_space).
* Consider a monoidal category (Mod<sub>''R''</sub>, ⊗<sub>''R''</sub>) of [modules](/source/module_(mathematics)) over a [commutative ring](/source/commutative_ring) ''R'' with the standard [tensor product](/source/Tensor_product_of_modules). A module ''M'' is dualizable if and only if it is a [finitely generated](/source/finitely_generated_module) [projective module](/source/projective_module). In that case the dual object ''M''<sup>∗</sup> is also given by the module of [homomorphisms](/source/Module_homomorphism) Hom<sub>''R''</sub>(''M'', ''R'').<ref>{{harvnb|Dold|Puppe|1980|p=88}}</ref>
* Consider a [homotopy category](/source/homotopy_category) of [pointed](/source/Pointed_space) [spectra](/source/Spectrum_(topology)) Ho(Sp) with the [smash product](/source/smash_product) as the monoidal structure. If ''M'' is a [compact](/source/compact_space) [neighborhood retract](/source/deformation_retract) in <math>\mathbb{R}^n</math> (for example, a compact smooth [manifold](/source/manifold)), then the corresponding pointed spectrum Σ<sup>∞</sup>(''M''<sup>+</sup>) is dualizable. This is a consequence of [Spanier–Whitehead duality](/source/Spanier%E2%80%93Whitehead_duality), which implies in particular [Poincaré duality](/source/Poincar%C3%A9_duality) for compact manifolds.<ref name="traces"/>
* The category <math>\mathrm{End}(\mathbf{C})</math> of [endofunctor](/source/endofunctor)s of a category <math>\mathbf{C}</math> is a monoidal category under composition of [functor](/source/functor)s. A functor <math>F</math> is a left dual of a functor <math>G</math> if and only if <math>F</math> is left adjoint to <math>G</math>.<ref>See for example {{cite book |last1=Nikshych|first1=D.|author2-link=Pavel Etingof |last2=Etingof|first2=P.I.|last3=Gelaki|first3=S.|last4=Ostrik|first4=V. |chapter=Exercise 2.10.4 |title=Tensor Categories |publisher=American Mathematical Society |series=Mathematical Surveys and Monographs |volume=205 |date=2016 |isbn=978-1-4704-3441-0 |pages=41 |url={{GBurl|Z6XLDAAAQBAJ|pg=PR7}}}}</ref>

== 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](/source/Algebraic_geometry) call it a '''left''' (respectively  right) '''[rigid category](/source/rigid_category)'''.  A monoidal category where every object has both a left and a right dual is called an '''[autonomous category](/source/autonomous_category)'''.  An autonomous category that is also [symmetric](/source/symmetric_monoidal_category) is called a '''[compact closed category](/source/compact_closed_category)'''.

==Traces==
Any endomorphism ''f'' of a dualizable object admits a [trace](/source/categorical_trace), which is a certain endomorphism of the monoidal unit of ''C''. This notion includes, as very special cases, the [trace in linear algebra](/source/trace_(linear_algebra)) and the [Euler characteristic](/source/Euler_characteristic) of a [chain complex](/source/chain_complex).

==See also==
* [Dualizing object](/source/Dualizing_object)

== References ==
{{reflist}}
{{refbegin}}
* {{Citation|last1=Dold|first1=Albrecht|author1link = Albrecht Dold|author2link = Dieter Puppe|last2=Puppe|first2=Dieter|chapter=Duality, trace, and transfer|title=Proceedings of the International Conference on Geometric Topology (Warsaw, 1978)|pages=81–102 |publisher=PWN-Polish Scientific Publishers |year=1980|mr=656721 |isbn=9788301017873 |oclc=681088710}}
* {{cite journal
| author1-link = Peter Freyd |first1=Peter |last1=Freyd |first2=David |last2=Yetter
| title = Braided Compact Closed Categories with Applications to Low-Dimensional Topology
| journal = [Advances in Mathematics](/source/Advances_in_Mathematics)
| volume = 77
| pages = 156–182
| year = 1989
| doi = 10.1016/0001-8708(89)90018-2
| issue = 2
| doi-access = free
}}
* {{cite journal
| author1-link =  André Joyal |first1=André |last1=Joyal |author2-link=Ross Street |first2=Ross |last2=Street
| title = The Geometry of Tensor calculus II
| journal = Synthese Library
| volume = 259
| pages = 29–68 |url=http://www.math.mq.edu.au/~street/GTCII.pdf |citeseerx=10.1.1.532.1533
}}
{{refend}}

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Category:Monoidal categories

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Adapted from the Wikipedia article [Dual object](https://en.wikipedia.org/wiki/Dual_object) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Dual_object?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.