model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
The model structure on simplicial groups is a presentation of the (∞,1)-category of ∞-groups in ∞Grpd $\simeq$ Top. See group object in an (∞,1)-category.
There is a projective 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.
(Quillen 67, II 3.7, see also Goerss-Jardine 99, V)
Every object in the projective model structure (Prop. ) is fibrant.
This statement amounts to saying that the underlying simplicial set of any simplicial group is a Kan complex. That this is the case is Moore’s theorem (here).
(Quillen equivalence between simplicial groups and reduced simplicial sets)
Forming simplicial loop space objects and simplicial classifying spaces gives a Quillen equivalence
(Goerss-Jardine 99, Chapter V, Prop. 6.3)
The model structure on simplicial groups is due to
Further discussion is in