category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
A dualizable object in a symmetric monoidal (∞,n)-category is called fully dualizable if the structure maps of the duality unit and counit each themselves have adjoints, which have adjoints, and so on.
By the cobordism hypothesis-theorem, symmetric monoidal (∞,n)-functors out of the (∞,n)-category of cobordisms are characterized by their value on the point, which is a fully dualizable object.
In the symmetric monoidal category Vect of vector spaces (over some field), the fully dualizable objects are the finite-dimensional vector spaces.
In the symmetric monoidal (infinity,3)-category of monoidal categories and bimodule categories between them, the fully dualizable objects are (or at least contain) the fusion categories. See there for details.
geometry | monoidal category theory | category theory |
---|---|---|
perfect module | (fully-)dualizable object | compact object |
The definition appears around claim 2.3.19 of