category with duals (list of them)
dualizable object (what they have)
abstract duality: opposite category,
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 SupLat, many “infinite” objects are dualizable. (In , all objects are dualizable.)
An object in a monoidal category is dualizable if it has an adjoint when regarded as a morphism in the one-object delooping bicategory corresponding to . Its adjoint in is called its dual in and often written as .
If is braided then left and right adjoints in are equivalent; otherwise one speaks of being left dualizable or right dualizable.
i: I \to A \otimes A^*
and counit (or evaluation)
e : A^* \otimes A \to I
satisfying the ‘triangle identities’ familiar from the concept of adjunction. With this convention, if in is interpreted as composition in in diagrammatic order, then right duals in are the same as right adjoints in — whereas if in is interpreted as composition in in classical ‘Leibnizian’ order, then right duals in are the same as left adjoints in .
Of course, in a symmetric monoidal category, there is no difference between left and right duals.
See category with duals for more discussion.
Let be a finite-dimensional vector space over a field , and let be its usual dual vector space. We can define to be the obvious pairing. If we also choose a finite basis of , and let be the dual basis of , then we can define by sending to . It is easy to check the triangle identities, so is a dual of in .
Let be a finite-dimensional manifold, choose an embedding for some , and let be the Thom spectrum of the normal bundle of this embedding. Then the Thom collapse map defines an which exhibits as a dual of in the stable homotopy category. This is a version of Spanier-Whitehead duality.
Dualizable objects support a good abstract notion of trace.
This appears as (Lurie, def. 2.3.5).
This means that an object in 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.
As before, we may equivalently state this after delooping the monoidal structure and passing to the -category . Then has duals for objects precisely if has all adjoints.
Duals in a monoidal category are a very classical notion. A large number of examples can be found in
The notion of duals in a symmetric monoidal -category is due to section 2.3 of