equivalences in/of $(\infty,1)$-categories
A 2-group $G$ is braided if it is equipped with the following equivalent structure:
Regarded as a monoidal category, $G$ is a braided monoidal category.
The delooping 2-groupoid $\mathbf{B}G$ is a 3-group.
The double delooping 3-groupoid $\mathbf{B}^2 G$ exists.
The groupal A-∞ algebra/E1-algebra structure on $G$ refines to an E2-algebra structure.
$G$ is a doubly groupal groupoid.
$G$ is a groupal doubly monoidal (1,0)-category.
A discussion of ∞-group extensions by braided 2-groups is in