Any braided monoidal category has a natural isomorphism

B x,y:xyyxB_{x,y} : x \otimes y \to y \otimes x

called the braiding.

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

For example, in Vect, the braiding maps aba \otimes b (a typical generator of xyx \otimes y) to bab \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 (