equivalences in/of $(\infty,1)$-categories
An ∞-group $G$ is braided if it is equipped with the following equivalent structure
Regarded as a monoidal (∞,1)-category, $G$ is a braided monoidal (∞,1)-category.
The delooping ∞-groupoid $\mathbf{B}G$ has the structure of an ∞-group.
The double delooping ∞-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 (∞,0)-category.
See the examples at braided 2-group, braided 3-group.
braided ∞-group,