braided monoidal category

Idea

• : a category with a tensor product and an isomorphism called the braiding, which lets us swap the order of objects in a tensor product like $X \otimes Y$.
• thus the tensor product is commutative in a sense, but not as coherently commutative as in a symmetric monoidal category.

Definition

• : a braided monoidal category is a monoidal category $C$ equipped with a natural isomorphism
  • $B_{x, y}: x \otimes y \to y \otimes x$
  • called the braiding, such that the following two kinds of diagrams commute for all objects involved (called the hexagon identities encoding the compatibility of the braiding with the associator for the tensor product):