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Suppose that $\mathscr{P}$ is a (locally) presentable $\left(\infty,1\right)$ category (which we can assume WLOG is infinity presheaves on some small $\left(\infty,1\right)$ category) , and $R$ and $S$ are classes of morphisms of $\mathscr{P}$ with respect to which we want to localize. Let us not assume that either $R$ or $S$ are a priori saturated classes. Let $\mathscr{P}_S$ and $\mathscr{P}_R$ denote the full subcategory on $S$-local objects and $R$-local object respectively. Their respective inclusions into $\mathscr{P}$ have left adjoints $L_R$ and $L_S$.

Question: On what conditions on $S$ and $R$ (besides $S \subset R$) will $L_R$ map $\mathscr{P}_S$ to itself?

(An answer even for $1$-categories would be very helpful).

Remark: I'm most interested in the case where $\mathscr{P}$ is $\infty$-presheaves on a Grothendieck site, $S$ consists of the associated covering sieves, and $R$ is in the essential image of the Yoneda embedding.

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  • $\begingroup$ One sufficient condition is to have $S \subseteq R$. No? $\endgroup$
    – Zhen Lin
    Jul 3, 2014 at 19:02
  • $\begingroup$ Ok, maybe I should rephrase my question, because I'm not looking for when this is trivially true, but good point. $\endgroup$ Jul 3, 2014 at 19:10
  • $\begingroup$ Do you have answers for the special cases where $\mathscr{P}$ is just a 1 or 2-category? I am just trying to figure out what sort of sufficient conditions would appeal to you. $\endgroup$ Jul 3, 2014 at 19:21
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    $\begingroup$ This is a non-trivial thing to consider in general. The only not to difficult conditions I know are of the form `$S$-local objects are preserved by homotopy colimits and $L_R$ can be expressed as an homotopy colimits of functors which preserve $S$-local objects'. The other examples I know are very far from being formal and I would not be able to give a general (and workable) criterium out of them. Maybe it might help if you could be more specific! $\endgroup$ Jul 3, 2014 at 22:38
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    $\begingroup$ I mean false with this level of generality. In fact, if we restrict to the smooth Nisnevich site over a perfect field, we can prove that this works for presheaves with additional structures (e.g. presheaves of abelian groups with transfers) but these are non-trivial results. To be more specific, I just mean that all the examples and counter-examples I know (and alluded to in my previous comments) fit in your framework. $\endgroup$ Jul 4, 2014 at 0:26

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