Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization

Let $$L_{Nis}(sPre(Sm_S))$$ be the Nisnevich localization of the category of simplicial presheaves, how to see that whether $$\mathbb{A}^1$$-projections $$\mathbb{A}^1\times_S X\to X$$ are closed under homotopy pullback in $$L_{Nis}(sPre(Sm_S))$$? How to compute the homotopy pullback of maps $$\mathbb{A}^1\times_S X\to X$$ along itself in $$L_{Nis}(sPre(Sm_S))$$?

In your question, you are missing a step. Generally, $$Sm_S$$ is the category of smooth and finite type schemes over a scheme S of finite dimension. This category doesn't have good categorical behavior (it's not bicomplete, for example), so Voevodsky embedded it into simplicial presheaves $$SPre(Sm_S)$$, i.e. functors into simplicial sets. This has a model structure and you can do the localization $$L_{Nis} SPre(Sm_S)$$ that you wanted. To get the statement about $$\mathbb{A}^1$$-weak equivalences, you do a further localization to make $$\mathbb{A}^1$$ contractible, and it's this model structure that tells you homotopy pullbacks preserve weak equivalences. See Morel-Voevodsky.
The resulting model category is proper, meaning that if you look at a diagram $$A \to B \gets C$$ where one of those maps is a fibration, then you can compute homotopy pullback as just the regular old pullback, which is easy in presheaf categories. For a proof, see Jardine. If neither of those maps is a fibration, you can do fibrant replacement to replace one by a fibration.
• The $A_1$-localization is for the class $S$ of all maps $A_1\times X\to X$. What will happen if localizing $L_{Nis}sPre$ at the homotopy pullback closure of $S$? – L. Xie Aug 20 '19 at 10:40
• If just localizing with respect to $\bar{S}$ once, I expect them not to be the same. $L_{\bar{S}}$ is left exact. But it's a fact that motivic homotopy theory is not a model topos. What's the problem here? – L. Xie Aug 20 '19 at 14:44
• @Alexis The homotopy pullback of $A^1\times X\to X$ along any map $F\to X$ is $A^1\times F\to F$, which is still an $A^1$-weak equivalence, so indeed the localization does not change. On the other hand, the class of all $A^1$-weak equivalences in $L_{Nis}sPre$ is not closed under homotopy pullbacks, so if you localize at the closure you will get a different localization (which still may or may not be an ∞-topos!). To actually get a left exact localization, you have to localize at the simultaneous closure of $A^1$-weak equivalences under homotopy colimits and homotopy pullbacks. – Marc Hoyois Aug 20 '19 at 16:18