category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-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 cobordismss 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.
The definition appears around claim 2.3.19 of