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.<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 | 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 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 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.<ref name="traces" /> 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 <math>\underline{\mathrm{Hom}}_C(V, \mathbb{1}_C)</math>, 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.<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 <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 (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'').<ref>{{harvnb|Dold|Puppe|1980|p=88}}</ref> * 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 <math>\mathbb{R}^n</math> (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.<ref name="traces"/> * The category <math>\mathrm{End}(\mathbf{C})</math> of endofunctors of a category <math>\mathbf{C}</math> is a monoidal category under composition of functors. 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 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== * 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 | 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}}

{{categorytheory-stub}} Category:Monoidal categories