equivalences in/of $(\infty,1)$-categories
A 3-group is equivalently
a 2-groupoid $G$ equipped with the structure of a loop space object of a connected 3-groupoid $\mathbf{B}G$ (its delooping);
a monoidal 2-category in which every object has an weak inverse under the tensor product.
Some classes of 3-groups are modeled by 2-crossed modules or crossed squares.