nLab
dualizable object

Idea

Being dualizable is often thought of as a category-theoretic notion of finiteness for objects in a monoidal category. For instance, a vector space is dualizable in Vect with its tensor product just when it is finite-dimensional, and a spectrum is dualizable in the stable homotopy category with its smash product just when it is a finite cell spectrum.

A more precise intuition is that an object is dualizable if its “size” is no larger than the “additivity” of the monoidal category. Since Vect and the stable homotopy category are finitely additive, but not infinitely so, dualizability there is a notion of finiteness. This is the case for many monoidal categories in which one considers dualizability. However, in a monoidal category which is not additive at all, such as Set (or any cartesian monoidal category), only the terminal object is dualizable—whereas in an “infinitely additive” monoidal category such as Rel or Sup?, many “infinite” objects are dualizable. (In $\mathrm{Rel}$, all objects are dualizable.)

In a monoidal category

Definition

Definition

An object $A$ in a monoidal category $C$ is dualizable if it has an adjoint when regarded as a morphism in the one-object delooping bicategory $BC$ corresponding to $C$. Its adjoint in $BC$ is called its dual in $C$ and often written as $A^{*}$.

If $C$ is braided then left and right adjoints in $BC$ are equivalent; otherwise one speaks of $A$ being left dualizable or right dualizable.

Remark

Unfortunately, conventions on left and right vary and sometimes contradict their use for adjoints. But if we define right duals to be the same as right adjoints in $BC$, then a right dual of $A$ is an object $A^{*}$ equipped with a unit (or coevaluation)

$$i:I\to A^{*}\otimes A$$i: I \to A^* \otimes A

and counit (or evaluation)

$$e:A\otimes A^{*}\to I$$e : A \otimes A^* \to I

satisfying the ‘triangle identities’ familiar from the concept of adjunction. Of course, in a symmetric monoidal category, there is no difference between left and right duals.

Definition

If every object of $C$ has a left and right dual, then $C$ is called a rigid monoidal category or an autonomous monoidal category. If it is additionally symmetric, it is called a compact closed category.

See category with duals for more discussion.

Examples

• Let $V$ be a finite-dimensional vector space over a field $k$, and let $V^{*}=\mathrm{Hom}(V,k)$ be its usual dual vector space. We can define $\epsilon :V^{*}\otimes V\to k$ to be the obvious pairing. If we also choose a finite basis $\{v_i\}$ of $V$, and let $\{v_i^{*}\}$ be the dual basis? of $V^{*}$, then we can define $\eta :k\to V\otimes V^{*}$ by sending $1$ to ${\sum }_i{v}_i\otimes v_i^{*}$. It is easy to check the triangle identities, so $V^{*}$ is a dual of $V$ in ${\mathrm{Vect}}_k$.

• Let $M$ be a finite-dimensional manifold, choose an embedding $M\hookrightarrow {ℝ}^n$ for some $n$, and let $\mathrm{Th}(NX)$ be the Thom spectrum of the normal bundle of this embedding. Then the Thom collapse map defines an $\eta$ which exhibits $\mathrm{Th}(NX)$ as a dual of ${\Sigma }_{+}^{\mathrm{\infty }}M$ in the stable homotopy category. This is a version of Spanier-Whitehead duality.

Properties

Dualizable objects support a good abstract notion of trace.

In a symmetric monoidal $(\mathrm{\infty },n)$-category

Definition

An object in a symmetric monoidal (∞,n)-category $C$ is called dualizable if it is so as an object in the ordinary symmetric monoidal homotopy category $\mathrm{Ho}(C)$.

This appears as (Lurie, def. 2.3.5).

Remark

This means that an object in $C$ is dualizable if there exists unit and counit 1-morphism that satisfy the triangle identity up to homotopy. The definition does not demand that this homotopy is coherent (that it satisfies itself higher order relations up to higher order k-morphisms).

If the structure morphisms of the adjunction of a dualizable object have themselves all adjoints, then the object is called a fully dualizable object.

Remark

As before, we may equivalently state this after delooping the monoidal structure and passing to the $(\mathrm{\infty },n+1)$-category $BC$. Then $C$ has duals for objects precisely if $BC$ has all adjoints.

References

Duals in a monoidal category are a very classical notion. A large number of examples can be found in

• Kate Ponto and Mike Shulman, Traces in symmetric monoidal categories, PDF.

The notion of duals in a symmetric monoidal $(\mathrm{\infty },n)$-category is due to section 2.3 of

• Jacob Lurie, On the Classification of Topological Field Theories