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
The model structure on reduced simplicial sets is a presentation of the full sub-(∞,1)-category
∞Grpd${}^{*/}_{\geq 1} \hookrightarrow$ ∞Grpd${}^{*/}$ $\simeq$ Top${}^{*/}$
of pointed ∞-groupoids on those that are connected.
By the looping and delooping-equivalence, this is equivalent to the (∞,1)-category of ∞-groups and this equivalence is presented by a Quillen equivalence to the model structure on simplicial groups.
A reduced simplicial set is a simplicial set $S$ with a single vertex:
Write $sSet_0 \subset$ sSet for the full subcategory of the category of simplicial sets on those that are reduced.
There is a model category structure on $sSet_0$ whose
weak equivalences
and cofibrations
are those in the standard model structure on simplicial sets.
This appears as (GoerssJardine, ch V, prop. 6.2).
The simplicial loop space functor $G$ and the delooping functor $\bar W(-)$ (discussed at simplicial group) constitute a Quillen equivalence
with the model structure on simplicial groups.
This appears as (GoerssJardine, ch. V prop. 6.3).
Under the forgetful functor $U : sSet_0 \hookrightarrow sSet$
In particular
The first statment appears as (GoerssJardine, ch. V, lemma 6.6.). The second (an immediate consequence) appears as (GoerssJardine, ch. V, corollary 6.8).
model structure on reduced simplicial sets
groupoid object in an (∞,1)-category, ∞-group, looping and delooping
A standard textbook reference is chapter V of