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.[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.
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.[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 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.[3]
Definition
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:
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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.
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 (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]
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
References
- ^ 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 nLab
- ^ 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.
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