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 Joyal- model structure on simplicial sheaves over a site is a model category presentation of the hypercomplete (∞,1)-topos over that site.
It is Quillen equivalent to the Jardine-local model structure on simplicial presheaves.
There is also a Čech model structure on simplicial sheaves (see there) modelling not the hypercompletion but just the topological localization of the (∞,1)-category of (∞,1)-presheaves.
Let $C$ be a small site. Write $Sh(C)$ for the category of sheaves on $C$ and $Sh(C)^{\Delta^{op}}$ for the category of simplicial objects in $Sh(C)$: the category of simplicial sheaves over $C$.
There is a left proper simplicially enriched model category structure on $Sh(C)^{\Delta^{op}}$ such that
cofibrations are precisely the objectwise cofibrations (monomorphisms) of simplicial sets;
weak equivalences are the local weak equivalences of the underlying simplicial presheaves as defined at model structure on simplicial presheaves.
This is (Jardine, theorem 5).
Call this the local injective model structure on simplicial sheaves.
The geometric embedding of a category of sheaves into its category of presheaves
with $L$ given by sheafification extends to a Quillen equivalence
between the above local model structure on simplicial sheaves and the injective hyperlocal Jardine-model structure on simplicial presheaves.
This is (Jardine, theorem 5).
The simplicial combinatorial model category $Sh(C)^{\Delta^{op}}_{loc}$ is a presentation for the hypercompletion $\hat Sh_{(\infty,1)}(c)$ of the (∞,1)-category of (∞,1)-sheaves on $C$:
The proof is spelled out at hypercomplete (∞,1)-topos.
For $D$ a dense sub-site of $C$ we have an equivalence of (∞,1)-categories
By the comparison lemma at dense sub-site we have already an equivalence of categories
This implies the claim with the above proposition.
The local model structure on simplicial sheaves was proposed in
This is with BrownAHT among the first proposals for models for ∞-stack.
A discussion can be found in
Jardine’s lectures
discuss the Quillen equivalence between the model structure on simplicial sheaves and the model structure on simplicial presheaves.
Wendt discusses “the construction of classifying spaces of fibre sequences in model categories of simplicial sheaves” in