nLab
braiding

Context

Monoidal categories

Contents

Idea

Any braided monoidal category has a natural isomorphism

B x,y:xyyxB_{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,yB_{x,y} and B y,xB_{y,x} are inverses (although they are isomorphisms regardless).

Examples

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

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

Revised on March 1, 2016 06:48:56 by Urs Schreiber (31.55.12.174)