# When does every $\infty$-localization correspond to a Bousfield localization?

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?

• The reference in Higher Topos Theory for the positive results is Section A.3.7. – AAK Jul 20 '16 at 4:08
• 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... – Tim Campion Jul 20 '16 at 6:37

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.
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.