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In a left Bousfield localization of the projective model structure on the category of simplicial presheaves, what is the condition that transfinite composition preserves weak equivalences? How about for transfinite composition of fibrant simplicial presheaves?

I apologize if the question doesn't make sense. My question comes from the proof here for Theorem 4.27 showing that the $A^1$ localization functor $L_{A^1}L_{Nis}$ is equivalent to the transfinite composition of $L_{Nis}Sing^{A^1}$, where they show that $L_{Nis}Sing^{A^1}$ takes values in $A^1$-fibrant objects and preserves $A^1$-weak equivalence then conclude the following. So I think the reason for this is that transfinite composition of $A^1$-weak equivalences between fibrant objects are $A^1$-weak equivalence. Is this true and how to prove it?

Edit: The idea of the proof comes as following: Let $\Phi=L_{A^1}L_{Nis}$, they first show that $\Phi^{\circ \mathbb{N}}X$ is fibrant in $L_{A^1}L_{Nis} sPre(Sm_S)$ for any $X$. Then they show that $\Phi$ preserves $A^1$-local weak equivalences. Since $X\simeq \Phi (X)$ when $X$ is $A^1$-fibrant, applying $\Phi$ tranfinitely to this equivalence should give the following

$\Phi^{\circ \mathbb{N}}(X)\simeq \Phi ^{\circ\mathbb{N}}(L_{A^1}L_{Nis} X)\simeq L_{A^1}L_{Nis} X$.

The first equivalence comes from that $\Phi^{\circ \mathbb{N}}$ preserves weak equivalences if $\Phi$ does, the second should come from applying $\Phi$ tranfinitely to $X\simeq \Phi (X)$. But if weak equivalences are already closed under transfinite composition. Why do they need to show that $\Phi^{\circ \mathbb{N}}X$ is $A^1$-fibrant?

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Yes, weak equivalences are always closed under transfinite compositions in this model category.

The standard set of generating cofibrations of (a left Bousfield localization of) the projective model structure has compact domains and codomains. In any model category with such a property, weak equivalences are closed under transfinite compositions. See, for example, Lemma 2.0.3(iv) or Proposition 6.1.3(i) in the paper https://arxiv.org/abs/1510.04969, but this is a fairly standard fact.

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  • $\begingroup$ Thanks! But in the proof I mentioned, why do they show that the functor takes value in fibrant objects? Could you please have a look at the place I edited? $\endgroup$
    – L. Xie
    Aug 25, 2019 at 4:28
  • $\begingroup$ @Alexis: If X→LX is a local weak equivalence and LX is fibrant in the local model structure, then L must be the localization functor, i.e., LX is weakly equivalent to the left Bousfield localization of X, which is what they're proving in this proposition. $\endgroup$ Aug 25, 2019 at 5:50

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