model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A model structure of $(2,1)$-sheaves is a model category presentation of the (2,1)-category of (2,1)-sheaves over some site or (2,1)-site.
There are several equivalent ways to set up a model category structure for $(2,1)$-sheaves.
Suppose first that the (2,1)-site $C$ is just a 1-category, hence just a site.
The following definition first defines a model presentation for (2,1)-presheaves (1-truncated (∞,1)-presheaves) and then localizes at the covering morphisms in order to obtain the $(2,1)$-sheaves.
Write Grpd for the category of small category groupoids and functors between them. Write $Grpd_{nat}$ for the natural model structure on groupoids.
Write $[C^{op}, Grpd_{nat}]_{proj}$ for the projective model structure on functors on the functor category $[C^{op}, Grpd]$.
Let $W = \{ C(\{U_i\})\to j(U) \}$ be the set of Cech nerve projections in $[C^{op}, Grpd]$ for each covering family $\{U_i \to U\}$ in the site $C$.
Then let finally
be the left Bousfield localization at the set of morphisms $W$.
The following definition first gives the presentations for (∞,1)-sheaves and then further restricts the 1-truncated objects in there, preseting the (n,1)-topos inside the full (∞,1)-topos over $C$, for $n = 2$.
Write $[C^{op}, sSet_{Quillen}]_{loc}$ for a local model structure on simplicial presheaves on $C$, the one which presents the (∞,1)-category of (∞,1)-sheaves on $C$.
Let $W = \{\partial \Delta[n] \cdot U \to \Delta[n] \cdot U| n \geq 2 \in \mathbb{N}, U \in C\}$ be the set of generating morphisms of weak equivalences on homotopy 1-types.
Write
for the left Bousfield localization of the model structure for (∞,1)-sheaves at the morphisms $W$. Then this is a model structure for $(2,1)$-sheaves on $C$.
These two model structures are equivalent:
Let
be the nerve functor and its left adjoint $\tau$. Postcomposition with this induces a Quillen adjunction
that is a Quillen equivalence.
This appears as (Hollander, theorem 5.4).
model structure for $(2,1)$-sheaves
model structures for $(\infty,1)$-sheaves
A model structure on presheaves of groupoids Quillen equivalent to the left Bousfield localization of the local model structure for (∞,1)-sheaves at morphisms that are weak equivalences of homtopy 1-types is in.
A discussion of $(2,1)$-sheaves/stacks as 1-truncated objects in the full model structure for (∞,1)-sheaves is in