category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A twist, or balance, in a braided monoidal category is a natural transformation from the identity functor on to itself satisfying a certain condition that links it to the braiding. A balanced monoidal category is a braided monoidal category equipped with such a balance.
The condition linking the balancing to the braiding, where is the balance and is the braiding, is that should be the composite of , , and .
A balanced monoidal category is a special case of a balanced pseudomonoid in a balanced monoidal bicategory?.
Every symmetric monoidal category is balanced in a canonical way; in fact, the identity natural transformation (on the identity functor of ) is a balance on if and only if is symmetric. Thus balanced monoidal categories fall between braided monoidal categories and symmetric monoidal categories. (They should not be confused with balanced categories, which are unrelated.)
In the string diagram calculus for ribbon categories, the twist is represented by a 360-degree twist in a ribbon.
This definition is taken from Jeff Egger (Appendix C), but the original definition can be found in chapter 4 of this paper by Joyal and Street:
Last revised on January 17, 2020 at 23:09:14. See the history of this page for a list of all contributions to it.