With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
A twist, or balance, in a braided monoidal category $\mathscr{C}$ is a natural isomorphism $\theta$ from the identity functor on $\mathscr{C}$ to itself satisfying the following compatibility condition with the braiding:
where $\beta$ is the braiding on $\mathscr{C}$. A balanced monoidal category is a braided monoidal category equipped with such a balance. Equivalently, a balanced monoidal category can be described as a braided pivotal category.
Balanced monoidal categories should not be confused with the other unrelated notation of a balanced category.
Every symmetric monoidal category is balanced in a canonical way. In fact, the identity natural transformation on the identity functor of $\mathscr{C}$ is a balance on $\mathscr{C}$. In this way, the twist can be seen as a way of βcontrollingβ the non-symmetric behavior of the braiding.
In the language of string diagrams, the balancing is represented by a 360-degree twist:
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 May 8, 2023 at 00:10:19. See the history of this page for a list of all contributions to it.