equivalences in/of $(\infty,1)$-categories
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
a doudle delooping $\mathbf{B}^2 G$;
a lift of tha A-∞=E-1-algebra structure on $G$ to an E-2 algebra structure.
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
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.
braided 3-group