equivalences in/of $(\infty,1)$-categories
A sylleptic 3-group is a 3-group eqipped with the following equivalent structure:
Regarded as a monoidal 2-category, $G$ is a sylleptic monoidal 2-category.
The A-∞ algebra/E1-algebra structure on $G$ refines to an E3-algebra structure.
$G$ is a 3-tuply monoidal 2-groupoid.
$G$ is a groupal 3-tuply monoidal (2,0)-category.