equivalences in/of $(\infty,1)$-categories
An ∞-group is a group object in ∞Grpd.
Equivalently (by the delooping hypothesis) it is a pointed connected $\infty$-groupoid.
Under the identification of ∞Grpd with Top this is known as an $A_\infty$-space, for instance.
An $\infty$-Lie group is accordingly a group object in ∞-Lie groupoids. And so on.
For details see groupoid object in an (∞,1)-category.
$\infty$-group, braided ∞-group
The homotopy theory of $\infty$-groups that are n-connected and r-truncated for $r \leq n$ is discussed in