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 perfect field, fully dualizable objects separable algebras which are a projective module over $k$ SchommerPries 11, section 3.8.3.
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)