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
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The model structure $sPSh(C)$ on sSet-enriched presheaves is supposed to be a presentation of the (∞,1)-category of (∞,1)-sheaves on an sSet-site $C$.
It generalizes the model structure on simplicial presheaves which is the special case obtained when $C$ happens to be just an ordinary category.
This means that in as far as the model structure on simplicial presheaves models ∞-stacks, the model structure on sSet-enriched categories model derived stacks.
The construction of the model structure on sSet-enriched categories closely follows the discussion of the model structure on simplicial presheaves, only that everything now takes place in enriched category theory.
Regard the closed monoidal category sSet as a simplicially enriched category in the canonical way.
For $C$ any $sSet$-category, write $sk_1 C$ for its underlying ordinary category.
Write $[C^{op}, sSet]$ for the enriched functor category.
Hence the ordinary category $sk_1 [C^{op}, sSet]$ has as objects enriched functors $C^{op} \to sSet$ and a morphism $f : A \to B$ in $sk_1 [C^{op}, sSet]$ is a natural transformation given by a collection of morphisms $f_c : A(c) \to B(c)$ in sSet, for each object $c \in C$.
(global model structure)
Let be $C$ a simplicially enriched category.
The global projective model structure $sk_1[C^{op},sSet]_{proj}$ on $sk_1[C^{op}, sSet]$
The global projective model structure on $sk_1 [C^{op}, sSet]$ makes the $sSet$-category $[C^{op}, sSet]$ a combinatorial simplicial model category.
For $C$ an sSet-site, the local projective model structure on $sk_1 [C^{op}, sSet]$ is the left Bousfield localization of $sk_1 [C^{op}, sSet]_{proj}$ at…
This appears on (ToënVezzosi, page 14).
It seems that the claim is that, indeed, in the special case that $C$ happens to be an ordinary category, the model structure on $sPSh(C)_{proj}^{l loc}$ reproduces the projective local model structure on simplicial presheaves.
For an sSet-site $C$ regarded as an (∞,1)-site, the local model structure on $[C^{op}, sSet]$ is a presentation of the (∞,1)-category of (∞,1)-sheaves on $C$, in that there is an equivalence of (∞,1)-categories
This is Lurie, prop. 6.5.2.14, remark 6.5.2.15.
Where a topos or (∞,1)-topos over an ordinary site encodes higher geometry, over a genuine sSet-site one speaks of derived geometry. An ∞-stack on such a higher site is also called a derived stack.
Therefore the model structure on $sSet$-presheaves serves to model contexts of derived geometry. For instance over the etale (∞,1)-site.
(…)
The theory of model structures on $sSet$-enriched presheaf categories was developed in
The relation to intrinsically defined (∞,1)-topos theory is around remark 6.5.2.15 of