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 $\mathcal{C}$ 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 $(n-1)$.
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 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).
In the symmetric monoidal 3-category of monoidal categories and bimodule categories between them, the fully dualizable objects are (or at least contain) the fusion categories. (DSPS 13).
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
Chris Schommer-Pries, The Classification of Two-Dimensional Extended Topological Field Theories (arXiv:1112.1000)
Chris Douglas, Chris Schommer-Pries, Noah Snyder, Dualizable tensor categories (arXiv:1312.7188)