nLab balanced monoidal category

Definition

A twist, or balance, in a braided monoidal category $B$ is a natural transformation from the identity functor on $B$ 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 $\theta$ is the balance and $\beta$ is the braiding, is that $\theta_{x \otimes y}$ should be the composite of $\beta_{x,y}$, $\theta_y \otimes \theta_x$, and $\beta_{y,x}$.

A balanced monoidal category is a special case of a balanced pseudomonoid in a balanced monoidal bicategory?.

Properties

Every symmetric monoidal category is balanced in a canonical way; in fact, the identity natural transformation (on the identity functor of $B$) is a balance on $B$ if and only if $B$ 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.

References

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:

• A. Joyal, R. Street, The geometry of tensor calculus I, Adv. Math. 88(1991), no. 1, 55–112, doi.