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 2-category of associative algebras with bimodules between them as morphisms, over a commutative ring, fully dualizable objects are separable algebras which are projective modules over the base ring (these conditions are given in SchommerPries 11, Definition 3.70 and attributed to Lurie’s paper cited below).
|geometry||monoidal category theory||category theory|
|perfect module||(fully-)dualizable object||compact object|
The definition appears around claim 2.3.19 of
Detailed discussion in degree 2 and 3 appears in