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. Often, an object is dualizable only when it satisfies some finiteness or compactness property.

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. 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 $\underline_C(V, \mathbb_C)$, 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.

Definition

Consider an object $X$ in a monoidal category $(\mathbf,\otimes, I, \alpha, \lambda, \rho)$. The object $X^*$ is called a left dual of $X$ if there exist two morphisms
:$\eta:I\to X\otimes X^*$, called the coevaluation, and $\varepsilon:X^*\otimes X\to I$, called the evaluation,
such that the following two diagrams commute:

The object $X$ is called the right dual of $X^*$.
This definition is due to .

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'').
* 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 $\mathbb^n$ (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.
* The category $\mathrm(\mathbf)$ of endofunctors of a category $\mathbf$ is a monoidal category under composition of functors. A functor $F$ is a left dual of a functor $G$ if and only if $F$ is left adjoint to $G$.See for example

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

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Monoidal categories