nLab braiding

Idea

The structure of a braided monoidal category on top of the underlying monoidal category with tensor product “$\otimes$” is a natural isomorphism of the form

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

called the braiding.

A braided monoidal category is called symmetric if and only if $B_{x,y}$ and $B_{y,x}$ are inverse morphisms to each other (while they are isomorphisms in any case).

Examples

Example

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

Example

The braiding for the tensor product of chain complexes and that of super vector spaces is as in Exp. up to multiplication by sign $a \otimes b \mapsto (-1)^{deg(a) deg(b)} (b \otimes a)$ (see at signs in supergeometry).

Related concepts

• braid group

References

See the references at braided monoidal category