# nLab braiding

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

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

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).

See the references at braided monoidal category