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 $L_{Nis}(sPre(Sm_S))$?


First, in any model category, weak equivalences are closed under homotopy pullback. That's the whole point of the "homotopy" in "homotopy pullback."

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.

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  • $\begingroup$ 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 $S$? $\endgroup$ – L. Xie Aug 20 '19 at 10:40
  • $\begingroup$ You would get the same result. In general, if you localize a model category with respect to a class of maps contained in the weak equivalences, nothing changes. So, if you first localize with respect to S, then with respect to the homotopy pullback closure of S (or just do both together, with this one localization), you get the same as if you just did it with respect to S. $\endgroup$ – David White Aug 20 '19 at 13:20
  • $\begingroup$ 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? $\endgroup$ – L. Xie Aug 20 '19 at 14:44
  • $\begingroup$ I think we have different definitions of what it means to be "closed under homotopy pullbacks" - in my answer, it means that if you have a map of diagrams that is an objectwise weak equivalence then it induces a weak equivalence on homotopy limits. Googling your comment about "model topos" I find this paper by Raptis and Strunk, where it closure seems to mean something else arxiv.org/pdf/1704.08467.pdf $\endgroup$ – David White Aug 20 '19 at 15:28
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    $\begingroup$ @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. $\endgroup$ – Marc Hoyois Aug 20 '19 at 16:18

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