category with duals (list of them)
dualizable object (what they have)
abstract duality: opposite category,
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, up to level .
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