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's 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. A category of finite-dimensional vector spaces with a 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. A standard notion of a dual vector space V has the following property. For any vector spaces U and W there is an adjunction Homk(UV,W) = Homk(U, VW), 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 VC with a natural isomorphism of bifunctors

HomC((–)1V, (–)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 VC 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 morphsims

, called the coevaluation, and , called the evaluation,

such that the following two diagrams commute

and

The object is called the right dual of . 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 a standard tensor product. A space V is dualizable if and only if it's a finite-dimensional vector space. In that 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 a standard tensor product. A module M is dualizable if and only if it's a finitely generated projective module. In that case the dual object R is also given by the module of homomorphisms HomR(M, R).
  • Consider a homotopy category of pointed spectra Ho(Sp) with a smash product as a 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 is a monoidal category under functor composition. It is true that a functor is a left dual of a functor iff , that is 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

  1. Ponto, Kate; Shulman, Michael (2014). "Traces in symmetric monoidal categories". Expositiones Mathematicae. 32 (3): 248–273. arXiv:1107.6032. Bibcode:2011arXiv1107.6032P.
  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 9780444823755.
  3. "dual object in a closed category in nLab". ncatlab.org. Retrieved 11 December 2017.
  4. See for example exercise 2.10.4 in Pavel Etingof "Tensor Categories"
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