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 natural model category structure on the 1-category of groupoids and functors is a presentation of the (2,1)-category of groupoids, functors and natural isomorphisms.
This is one flavor of the various natural model structures on categories and higher categories.
Let Grpd be the 1-category of small groupoids and functors between them. Say a morphism in $Grpd$ – a functor $f : C \to D$ – is
a weak equivalence if it is an equivalence of categories;
a fibrations if it is an isofibration;
a cofibration if it is an injection on objects.
Equipped with this structure $Grpd_{nat}$ is a model category which is
a simplicial model category with respect to the natural sSet-enriched category structure induced by the canonical enrichment over itself, under the nerve.
This is originally due to (Anderson) and (Bousfield). A detailed discussion is in section 6 of (Strickland). In the context of the model structure for (2,1)-sheaves it appears as (Hollander, theorem 2.1).
The model structure $Grpd_{nat}$ is the restriction of the canonical model structure on Cat from categories to groupoids.
See natural model structure for more.
Let
be the pair of adjoint functors, where $N$ is the nerve of groupoids with values in sSet.
With the natural model structure on $Grpd$ and the standard model structure on simplicial sets this is a Quillen adjunction
and $Grpd_{nat}$ is the transferred model structure obtained from $sSet_{Quillen}$ under this adjunction.
canonical model structure on $Grpd$
This model category structure appears originally in
and
A detailed description is in section 6 of
The model structure on functors with values in $Grpd_{nat}$ (a model structure for (2,1)-sheaves) is discussed in