Dual object

<h2 id="motivation">Motivation</h2>
Let V be a finite-dimensional vector space over some <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K. The standard notion of a <a href="/facts/Dual_vector_space/4Uw4Knz1">dual vector space</a> V∗ has the following property: for any K-vector spaces U and W there is an <a href="/facts/Adjoint_functors/bJ1P7AN2">adjunction</a> HomK(U ⊗ V,W) = HomK(U, V∗ ⊗ W), and this characterizes V∗ up to a unique <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>. This expression makes sense in any category with an appropriate replacement for the <a href="/facts/Tensor_product/7K3pR1ap">tensor product</a> of vector spaces. For any <a href="/facts/Monoidal_category/F2l5oKMD">monoidal category</a> (C, ⊗) one may attempt to define a dual of an object V to be an object V∗ ∈ C with a <a href="/facts/Natural_isomorphism/xwPumTHt">natural isomorphism</a> of <a href="/facts/Bifunctor/l2u3rIBE">bifunctors</a>

HomC((–)1 ⊗ V, (–)2) → HomC((–)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.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> An actual definition of a dual object is thus more complicated.
In a <a href="/facts/Closed_monoidal_category/FBe3ecNJ">closed monoidal category</a> C, i.e. a monoidal category with an <a href="/facts/Internal_Hom/H76BLKCD">internal Hom</a> functor, an alternative approach is to simulate the standard definition of a dual vector space as a space of <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functionals</a>. For an object V ∈ C define V∗ to be 
 
 
 
 
 
 
 
 H
 o
 m
 
 _
 
 
 
 C
 
 
 (
 V
 ,
 
 
 1
 
 
 C
 
 
 )
 
 
 {\displaystyle {\underline {\mathrm {Hom} }}_{C}(V,\mathbb {1} _{C})}
 
, where 1C 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.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="definition">Definition</h2>
Consider an object 
 
 
 
 X
 
 
 {\displaystyle X}
 
 in a <a href="/facts/Monoidal_category/F2l5oKMD">monoidal category</a> 
 
 
 
 (
 
 C
 
 ,
 ⊗
 ,
 I
 ,
 α
 ,
 λ
 ,
 ρ
 )
 
 
 {\displaystyle (\mathbf {C} ,\otimes ,I,\alpha ,\lambda ,\rho )}
 
. The object 
 
 
 
 
 X
 
 ∗
 
 
 
 
 {\displaystyle X^{*}}
 
 is called a left dual of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 if there exist two morphisms

η
 :
 I
 →
 X
 ⊗
 
 X
 
 ∗
 
 
 
 
 {\displaystyle \eta :I\to X\otimes X^{*}}
 
, called the coevaluation, and 
 
 
 
 ε
 :
 
 X
 
 ∗
 
 
 ⊗
 X
 →
 I
 
 
 {\displaystyle \varepsilon :X^{*}\otimes X\to I}
 
, called the evaluation,
such that the following two diagrams commute:

<table><tbody><tr><td></td><td>and</td><td></td></tr></tbody></table>
The object 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is called the right dual of 
 
 
 
 
 X
 
 ∗
 
 
 
 
 {\displaystyle X^{*}}
 
. 
This definition is due to Dold & Puppe (1980).
Left duals are canonically isomorphic when they exist, as are right duals. When C is <a href="/facts/Braided_monoidal_category/u20sctLa">braided</a> (or <a href="/facts/Symmetric_monoidal_category/AxhumNny">symmetric</a>), every left dual is also a right dual, and vice versa. 
If we consider a monoidal category as a <a href="/facts/Bicategory/vaYsrIbg">bicategory</a> with one object, a dual pair is exactly an <a href="/facts/Adjoint_pair/bJ1P7AN2">adjoint pair</a>.

<h2 id="examples">Examples</h2>
<ul><li>Consider a monoidal category (VectK, ⊗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 <a href="/facts/Dual_vector_space/4Uw4Knz1">dual vector space</a>.</li>
<li>Consider a monoidal category (ModR, ⊗R) of <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> R with the standard <a href="/facts/Tensor_product_of_modules/K3oaFIS4">tensor product</a>. A module M is dualizable if and only if it is a <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> <a href="/facts/Projective_module/lHymqhNC">projective module</a>. In that case the dual object M∗ is also given by the module of <a href="/facts/Module_homomorphism/SCSzvgxx">homomorphisms</a> HomR(M, R).</li>
<li>Consider a <a href="/facts/Homotopy_category/JaEnPakj">homotopy category</a> of <a href="/facts/Pointed_space/V4DTgRCd">pointed</a> <a href="/facts/Spectrum_(topology)/mWho3roy">spectra</a> Ho(Sp) with the <a href="/facts/Smash_product/3gLz9z6f">smash product</a> as the monoidal structure. If M is a <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Deformation_retract/477Gm67f">neighborhood retract</a> in 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 (for example, a compact smooth <a href="/facts/Manifold/rXPERJtu">manifold</a>), then the corresponding pointed spectrum Σ∞(M+) is dualizable. This is a consequence of <a href="/facts/Spanier%E2%80%93Whitehead_duality/a3W4ADYQ">Spanier–Whitehead duality</a>, which implies in particular <a href="/facts/Poincar%C3%A9_duality/lwNjUCIa">Poincaré duality</a> for compact manifolds.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>The category 
 
 
 
 
 E
 n
 d
 
 (
 
 C
 
 )
 
 
 {\displaystyle \mathrm {End} (\mathbf {C} )}
 
 of <a href="/facts/Endofunctor/l2u3rIBE">endofunctors</a> of a category 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbf {C} }
 
 is a monoidal category under composition of <a href="/facts/Functor/l2u3rIBE">functors</a>. A functor 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is a left dual of a functor 
 
 
 
 G
 
 
 {\displaystyle G}
 
 if and only if 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is left adjoint to 
 
 
 
 G
 
 
 {\displaystyle G}
 
.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li></ul>
<h2 id="categories-with-duals">Categories with duals</h2>
A monoidal category where every object has a left (respectively right) dual is sometimes called a left (respectively right) autonomous category. <a href="/facts/Algebraic_geometry/rh7RHVp3">Algebraic geometers</a> call it a left (respectively right) <a href="/facts/Rigid_category/GMVCBYeq">rigid category</a>. A monoidal category where every object has both a left and a right dual is called an <a href="/facts/Autonomous_category/ryzE4N44">autonomous category</a>. An autonomous category that is also <a href="/facts/Symmetric_monoidal_category/AxhumNny">symmetric</a> is called a <a href="/facts/Compact_closed_category/zrRPGzao">compact closed category</a>.

<h2 id="traces">Traces</h2>
Any endomorphism f of a dualizable object admits a <a href="/facts/Categorical_trace/cf731ABA">trace</a>, which is a certain endomorphism of the monoidal unit of C. This notion includes, as very special cases, the <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace in linear algebra</a> and the <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a> of a <a href="/facts/Chain_complex/YAlchtCT">chain complex</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Dualizing_object/kF0mrSrI">Dualizing object</a></li></ul>

<ul><li><a href="/facts/Albrecht_Dold/JWc768sX">Dold, Albrecht</a>; <a href="/facts/Dieter_Puppe/tfsSDWDV">Puppe, Dieter</a> (1980), "Duality, trace, and transfer", Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), PWN-Polish Scientific Publishers, pp. 81–102, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9788301017873, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0656721">0656721</a>, <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/681088710">681088710</a></li>
<li><a href="/facts/Peter_Freyd/jbQAncXD">Freyd, Peter</a>; Yetter, David (1989). <a href="https://doi.org/10.1016%2F0001-8708%2889%2990018-2">"Braided Compact Closed Categories with Applications to Low-Dimensional Topology"</a>. <a href="/facts/Advances_in_Mathematics/kXtcl4pM">Advances in Mathematics</a>. 77 (2): 156–182. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0001-8708%2889%2990018-2">10.1016/0001-8708(89)90018-2</a>.</li>
<li><a href="/facts/Andr%C3%A9_Joyal/Xc0dKuXB">Joyal, André</a>; <a href="/facts/Ross_Street/1wXoGIte">Street, Ross</a>. <a href="http://www.math.mq.edu.au/~street/GTCII.pdf">"The Geometry of Tensor calculus II"</a> (PDF). Synthese Library. 259: 29–68. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.532.1533">10.1.1.532.1533</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">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. <a href="/wiki/Michael_Shulman_(mathematician)" target="_blank">/wiki/Michael_Shulman_(mathematician)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">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.
 <a href="978-0-444-82375-5" target="_blank">978-0-444-82375-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">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. <a href="/wiki/Michael_Shulman_(mathematician)" target="_blank">/wiki/Michael_Shulman_(mathematician)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">dual object in a closed category at the nLab <a href="https://ncatlab.org/nlab/show/dual+object+in+a+closed+category" target="_blank">https://ncatlab.org/nlab/show/dual+object+in+a+closed+category</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">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. <a href="/wiki/Michael_Shulman_(mathematician)" target="_blank">/wiki/Michael_Shulman_(mathematician)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">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. <a href="978-1-4704-3441-0" target="_blank">978-1-4704-3441-0</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Dual object open-in-new

Dual object