Dual object

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.[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 ${\displaystyle {\underline {\mathrm {Hom} }}_{C}(V,\mathbb {1} _{C})}$, 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.[3]

Consider an object ${\displaystyle X}$ in a monoidal category ${\displaystyle (\mathbf {C} ,\otimes ,I,\alpha ,\lambda ,\rho )}$. The object ${\displaystyle X^{*}}$ is called a left dual of ${\displaystyle X}$ if there exist two morphisms

${\displaystyle \eta :I\to X\otimes X^{*}}$, called the coevaluation, and ${\displaystyle \varepsilon :X^{*}\otimes X\to I}$, called the evaluation,

such that the following two diagrams commute:

and

The object ${\displaystyle X}$ is called the right dual of ${\displaystyle X^{*}}$. 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.

• 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 ${\displaystyle \mathbb {R} ^{n}}$ (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.[1]
• The category ${\displaystyle \mathrm {End} (\mathbf {C} )}$ of endofunctors of a category ${\displaystyle \mathbf {C} }$ is a monoidal category under composition of functors. A functor ${\displaystyle F}$ is a left dual of a functor ${\displaystyle G}$ if and only if ${\displaystyle F}$ is left adjoint to ${\displaystyle G}$.[4]

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.

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.

• Dualizing object