nLab
braiding

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

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).

For more see at signs in supergeometry.

Last revised on July 26, 2018 at 12:24:34. See the history of this page for a list of all contributions to it.