nLab
braided 3-group

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 $BG$;

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

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 ${\mathrm{Alg}}_R\simeq 2{\mathrm{Vect}}_R$ the braided monoidal 2-category of $R$-algebras, bimodules and bimodule homomorphism, the maximal 3-group

$$\mathrm{Br}(R)\hookrightarrow \mathrm{Core}({\mathrm{Alg}}_R)$$\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.

Related concepts

• ∞-group, k-tuply groupal n-groupoid

• braided ∞-group,

  • braided 2-group

  • braided 3-group

• sylleptic ∞-group

  • sylleptic 3-group

• abelian ∞-group

  • symmetric 2-group