braiding

Any braided monoidal category has a natural isomorphism

$B_{x,y} : x \otimes y \to y \otimes x$

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).

For example, in Vect, the braiding maps $a \otimes b$ (a typical generator of $x \otimes y$) to $b \otimes a$. But a braiding is most interesting when it does *not* look like something trivial like that.

Revised on April 26, 2010 00:04:25
by Toby Bartels
(98.19.56.65)