symmetric monoidal (∞,1)-category of spectra
An ordinary group is either an abelian group or not. For an ∞-group there is an infinite tower of notions ranging from completely non-abelian to completely abelian. An abelian ∞-group is one which is maximally abelian. This is equivalently a connective spectrum object.
Write
for the (∞,1)-functor which sends a commutative ∞-ring to its ∞-group of units.
The ∞-group of units (∞,1)-functor of def. 1 is a right-adjoint (∞,1)-functor (or at least a right adjoint on homotopy categories)
This is (ABGHR 08, theorem 2.1).
A 0-truncated abelian $\infty$-group is equivalently an abelian group.
A 1-truncated abelian $\infty$-group is equivalently a symmetric 2-group.
A 2-truncated abelian $\infty$-group is equivalently a symmetric 3-group.
abelian ∞-group
General discussion is in section 5 of
Discussion in the context of E-∞ rings and twisted cohomology is in