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 product 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 products) 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).

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).

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

One endows the category of simplicial presheaf of $C$ of the "Rezk-Lurie" model structure: that is we start by 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 $\infty$-sheaves over $\mathcal{T}$.

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 map between simplicial object 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 product 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 products) describe the universal property of the $(\infty,1)$-topos of $\infty$-sheaves over $\mathcal{T}$ 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).

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 product 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 products) 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).