equivalences in/of $(\infty,1)$-categories
A symmetric 2-group is a 2-group equipped with the following equivalent structure:
Regarded as a monoidal category, $G$ is a symmetric monoidal category.
Regarded as a 1-truncated ∞-group it has the structure of an abelian ∞-group.
The delooping 2-groupoid $\mathbf{B}G$ is a braided 3-group.
The double delooping 3-groupoid $\mathbf{B}^2 G$ is a 4-group.
The triple delooping 4-groupoid $\mathbf{B}^4 G$ exists.
The A-∞ algebra/E1-algebra structure on $G$ refines to an E3-algebra structure.
$G$ is a 3-tuply monoidal groupoid.
$G$ is a groupal 3-tuply monoidal (1,0)-category.
symmetric 2-group