Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site.


One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start with the projective model structure on simplicial presheaves and then we take the Left Bousefield localization with respect to the map of the form $j \hookrightarrow x$ where $j$ is a covering sieve of $x$ for the topology $J$, where they are both seen as 'constant' simplicial presheaves.


This is a model for the $(\infty,1)$-topos of (possibly non Hypercomplete) $\infty$-sheaves over $(C,J)$.


It is proved in the appendix of [Hypercovers and simplicial presheaves, by Dugger, Hollander and Isaksen](http://www.math.uiuc.edu/K-theory/0563/) that for any simplicial presheaves $S$ the map from $S$ to its levelwise sheafication is a weak equivalence. In particular, any map of simplicial presheaves is a weak equivalence for this model structure if an only if its level sheafication is a weak equivalence.


This mean that one get a nice notion of weak equivalence for maps between simplicial objects of the topos $\mathcal{T}$, that we will call "Cech weak equivalence" (by opposition to the Jardin-Joyal weak equivalence which are easily describe internally as the maps inducing isomorphism on all $\pi\_n$).


My question is:


*To what extent this notion of "Cech weak equivalence" depends on the choice of a site of definition of the topos $\mathcal{T}$ ?*


I'm completely willing to assume that $C$ has all finite limite if it change something (see the next remarks)


The key result which I think is relevant to this question is Lurie's Proposition 6.4.5.7 of Higher topos theory, which (assuming $C$ has finite limits) describe the universal property of the $(\infty,1)$-topos of $\infty$-sheaves over $(C,J)$ purely in terms of $\mathcal{T}$, and hence proves that the model categories of simplicial presheaves over different sites of definitions of $\mathcal{T}$ (admiting finite product) are all canonically Quillen equivalent.
This allows to says a few thing, but unfortunately this is not completely enough to conclude that the weak equivalence between simplicial sheaves are the same (only between those which are fibrant/cofibrant) ! 


My area of expertise being more topos theory than model category I might be missing something, that is why I'm asking this question here.


PS: it also appears that starting with the injective model structure instead of the projective one does not change the weak equivalences at all (even after localization).



Site dependance of the Cech weak equivalences on simplicial sheaves
It is independent, provided the two sites of definition, also yield the same infinity topos (so e.g. when they both have finite limits, or the infinity-topos is hypercomplete). Given two simplicial objects $F$ and $G$ of the topos of $\mathcal{E}$, so $$F,G:\Delta^{op} \to \mathcal{E},$$ consider the embedding $$\theta:\mathcal{E} \hookrightarrow Sh\_\infty\left(\mathcal{E}\right),$$ where I am writing $Sh\_\infty\left(\mathcal{E}\right)$ for the $1$-localic infinity-topos corresponding to $\mathcal{E}.$ I claim that a map $f:F \to G$ is a weak equivalences if and only if the induced map between $\operatorname{colim} \theta \circ F \to \operatorname{colim} \theta \circ G$ is an equivalence in $Sh\_\infty\left(\mathcal{E}\right)$.


Proof: $f$ is a weak equivalence if and only if it is is one when considered as a map of simplicial presheaves, which is if and only if it becomes one in the associated infinity-topos. Choose a site $C$. In simplicial presheaves on $C$ with the global model structure, the homotopy colimit of $\theta \circ F$ is $F$ itself, which can be computed as the diagonal of the bisimplicial presheaf which is constant in one direction. Hence, the colimit of $\theta \circ F$ is the infinity sheaf associated to $F$.


