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Let $\mathcal{M}$ be a model category presenting an $\infty$-category $\mathcal{C}$. I believe that every left Bousfield localization $\widetilde{\mathcal{M}}$ of $\mathcal{M}$ corresponds to a reflective $\infty$-subcategory $\widetilde{\mathcal{C}}$ in $\mathcal{C}$ -- perhaps some hypotheses are needed for this.

What about the converse? Suppose I have a reflective $\infty$-subcategory $\widetilde{\mathcal{C}} \subseteq \mathcal{C}$. Under what conditions is this induced by a left Bousfield localization $\widetilde{\mathcal{M}} \subseteq \mathcal{M}$? Does it suffice that every left Bousfield localization exist in $\mathcal{M}$? Is this a necessary condition? Now is probably the time for me to admit that I don't even know an example of a model category that doesn't admit all left Bousfield localizations -- what is an example?

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    $\begingroup$ The reference in Higher Topos Theory for the positive results is Section A.3.7. $\endgroup$ – AAK Jul 20 '16 at 4:08
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    $\begingroup$ Thanks! Apparently it's A.3.7.8 to be precise. Of course this is in the locally presentable context. I don't know why I keep expecting more to be known outside this context... $\endgroup$ – Tim Campion Jul 20 '16 at 6:37
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You definitely need $M$ to be combinatorial for these types of statements. I believe Lurie has shown that every accessible localization of a presentable infinity category can be expressed as a left Bousfield localization. See chapter 5 section 5 of HTT. He uses strongly reflective to mean it comes from an accessible localization. Without accessibility, it breaks down, as the next example shows.

Example: There are plenty of model categories that don't admit all left Bousfield localizations. The most famous unsolved problem in this vein is cohomological localization for sSet. Very likely, the existence of this localization is equivalent to Vopenka's principle. See work of Casacuberta for evidence to this effect. Right now, it's not known that there is a SET of maps you can invert to do cohomological localization. In Lurie's language, no one can prove it's an accessible localization. Related is the theorem that, if you assume Vopenka's principle, then any left Bousfield localization of any combinatorial model category exists. It's not known if Vopenka can be proven within ZFC, so it would imply a big result in set theory if you found a left Bousfield localization of a combinatorial model category that provably did not exist.

However, as soon as you leave the shelter of combinatorial model categories, there are all sorts of counterexamples. See Chorny's work on class cofibrantly generated model categories. Similarly, for non presentable infinity categories you do not expect all localizations to exist.

If, for some reason, every left Bousfield localization of $M$ was known to exist, then I would expect that any localization of the $\infty$-category of $M$ comes from a localization of $M$. I'd prove this using the universal principle. Since the cofibrations are the same in any left localization, only the weak equivalences matter, and under the assumption about localizations existing, every class of weak equivalences corresponds to a left localization. But this assumption is ridiculously strong, and certainly not necessary for the result you want, as chapter 5 of HTT shows.

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    $\begingroup$ Thanks! Regarding Vopenka's principle (VP), it's known that VP can't be proven in ZFC unless ZFC itself is inconsistent (essentially by Godel's theorem) -- it's the negation of VP that you'd prove if you found a localization that provably didn't exist, which as you say would still be a big result in set theory. When you say "this assumption is ridiculously strong" you mean the assumption that all Bousfield localizations exist, not just small ones, right? Because 5.5 deals with presentable things, which do have combinatorial models, where all small Bousfield localizations exist. $\endgroup$ – Tim Campion Jul 20 '16 at 7:02
  • $\begingroup$ According to arxiv.org/abs/1101.2792, the existence of cohomological localizations follows from the existence of arbitrarily large supercompact cardinals, which is significantly weaker than Vopenka's principle. $\endgroup$ – Mike Shulman Aug 31 '19 at 13:06
  • $\begingroup$ @MikeShulman: yes, that's one of my favorite papers. It makes a strong case that the existence of cohomological localizations should be equivalent to the existence of arbitrarily large supercompact cardinals. I'd love to get your thoughts on that at some point. $\endgroup$ – David White Sep 1 '19 at 11:23
  • $\begingroup$ Ok - I mentioned it because your answer above suggested that cohomological localization is probably equivalent to Vopenka's principle instead. I haven't looked at this enough to have any thoughts myself. $\endgroup$ – Mike Shulman Sep 1 '19 at 12:22

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