Yang-Baxter Equation

Yang-Baxter Object

Suppose $(\mathsf{C},\otimes ,\text{one},\alpha ,\lambda ,\eta )$ is a monoidal category. A Yang-Baxter object is an object $X$ equipped with an automorphism $r:X\otimes X\to X\otimes X$ called a Yang-Baxter operator such that

commutes.

Theorem

In a braided monoidal category $(\mathsf{C},\otimes ,\text{one},\alpha ,\lambda ,\rho ,\tau )$, we have the following diagram commutes for all objects $X,Y,Z$:

Proof We cut the diagram into three parts as illustrated below. The left (red) and right (blue) parts are exactly the hexagon axioms for the braiding, and the middle square commutes by naturality of the braiding. Therefore, the whole diagram commutes. $\square$