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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 HomK(U ⊗ V,W) = HomK(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

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.³ 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 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.⁴

Definition

Consider an object X {\displaystyle X} in a monoidal category ( 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:

and

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 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 (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 dual vector space.
• Consider a monoidal category (ModR, ⊗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 HomR(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 R n {\displaystyle \mathbb {R} ^{n}} (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.⁵
• The category E n d ( C ) {\displaystyle \mathrm {End} (\mathbf {C} )} of endofunctors of a category C {\displaystyle \mathbf {C} } is a monoidal category under composition of functors. 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} .⁶

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

• 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.

References

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. /wiki/Michael_Shulman_(mathematician) ↩

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. 978-0-444-82375-5 ↩

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. /wiki/Michael_Shulman_(mathematician) ↩

4. dual object in a closed category at the nLab https://ncatlab.org/nlab/show/dual+object+in+a+closed+category ↩

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. /wiki/Michael_Shulman_(mathematician) ↩

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. 978-1-4704-3441-0 ↩