related by the Dold-Kan correspondence
The model structure on presheaves of simplicial groupoids is one of the models for ∞-stack (∞,1)-toposes. It is a slight variant on the model structure on simplicial presheaves. (At that link more general information is collected).
For various applications it is useful to
use a model structure on (pre)sheaves with values in simplicial groupoids instead of the model structure on simplicial presheaves (see there for motivation on why to consider these models in the first place).
The transferred model structure along on of the global injective model structure on simplicial presheaves exists on : fibrations and weak equivalences are those that become global injective fibrations and weak equivalences, respectively, under .
This appears as (LBK, theorem 3.10).
A model structure on sheaves with values in simplicial groupoids is due to
A model structure on presheaves with values in simplicial groupoids is due to
A model structure on simplicial sheaves of groupoids is discussed in