on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
on strict ∞-categories?
The model structure on simplicial groups is a presentation of the ∞-groups in ∞Grpd $\simeq$ Top. See group object in an (∞,1)-category.
There is a model category structure on the category $sGrp$ of simplicial groups where a morphism is
is a weak equivalence if the underlying morphism is a weak equivalence in the standard model structure on simplicial sets;
is a fibration if the underlying morphism is a Kan fibration of simplicial sets;
is a cofibration if it has the left lifting property with respect to all trivial fibrations.
Forming loop space objects and classifying spaces provides a Quillen equivalence
with the model structure on reduced simplicial sets.
The general theory is in chapter V of
The Quillen equivalence is in proposition 6.3.