Site dependance of the Cech weak equivalences on simplicial sheaves

Question

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 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).

Answer 1

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 $F$.