# nLab braided 3-group

group theory

### Cohomology and Extensions

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Definition

A braided 3-group is a braided ∞-group which is a 3-group. For $G$ a 3-group, a braiding on it is the following equivalent structure

1. the structure of a 2-group on the delooping $\mathbf{B}G$;

2. a doudle delooping $\mathbf{B}^2 G$;

3. a lift of tha A-∞=E-1-algebra structure on $G$ to an E-2 algebra structure.

## Examples

For $R$ a commutative ring, and $Alg_R \simeq 2 Vect_R$ the braided monoidal 2-category of $R$-algebras, bimodules and bimodule homomorphism, the maximal 3-group

$\mathbf{Br}(R) \hookrightarrow Core(Alg_R)$

inside is a braided 3-group. Its homotopy groups are the Brauer group, the Picard group and the group of units of $R$. See at Brauer group – Relation to category of modules for more on this.

Revised on December 12, 2012 16:47:33 by Urs Schreiber (71.195.68.239)