nLab braiding

Idea

Any braided monoidal category has a natural isomorphism

$$B_{x,y} \;\colon\; x \otimes y \to y \otimes x$$

called the braiding.

A braided monoidal category is symmetric if and only if $B_{x,y}$ and $B_{y,x}$ are inverses (although they are isomorphisms regardless).

Examples

In Vect or Mod, the braiding maps elements $a\otimes b$ of a tensor product of modules $X \otimes Y$ to $b \otimes a$.

For the tensor product of chain complexes or that of super vector spaces there is in addition a sign $a \otimes b \mapsto (-1)^{deg(a) deg(b)} (b \otimes a)$.

For more see at signs in supergeometry.