The term twist or twisted is one of the hugely overloaded terms in math. Among the various meaning it may have is

• twisted cohomology

  • twisted bundle

  • twisted K-theory

• twisting cochain

• twisted complex

• twisted tensor product

• twisted module of homomorphisms

• etc….

• This page here is about the notion of twist in a braided monoidal category that is part of the structure of a balanced monoidal category.

Twists in braided monoidal categories

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 ${\beta }_{y,x}$.

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 is due to Joyal and Street.