The notion of a (left, say) Bousfield localization of a model category doesn't seem to be invariant under Quillen equivalence. There are a lot of things that could go wrong. But I don't know any examples. So if anyone could help me out with even one of the below questions, I'd appreciate it.

Let $M$ be a model category and $\mathrm{Ho} M$ its homotopy category.

At the most basic level, what is an example where there exists a localization of $\mathrm{Ho} M$ which isn't the homotopy category of a left Bousfield localization of $M$? This isn't usually how left Bousfield localizations are presented (although it seems the most invariant way I can think of), so, as a bonus: what is an example where there is a localization of $\mathrm{Ho} M$ where the inverted objects aren't of the form $\{S-\text{local objects}\}$ for some class $S$?

More specifically, what is an example of (1) where the implied acyclic cofibrations aren't generated by a small set (actually I'd be happy to know just this, but of course what I'm driving at is that they shouldn't admit factorizations)?

Alternatively, what is an example of (1) where the implied acyclic cofibrations are perhaps generated by a small set, but not one which admits the small object argument (and they don't have factorizations)?

Pressing forward, if $F: M \overset{\to}{\to} N: U$ is a Quillen equivalence and $M'$ is a left Bousfield localization of $M$, then if the model structure $N$' induced by $M'$ along $F$ exists, it is a left Bousfield localization of $N$ and $F: M' \overset{\to}{\to} N': U$ is a Quillen equivalence. What's an example where $N'$ fails to be a model structure? Again it's interesting to ask about the various ways this could go wrong.

In the other direction, it seems dicier to try to induce a left Bousfield localization along a right Quillen equivalence. I guess that dualizing the argument from (4) will show that if the induced model structure exists, it is a left Bousfield localization and the Quillen equivalence restricts to a Quillen equivalence between the localizations. But it must be not unusual that this induced model structure fails to exist, right?

What is an example of a Bousfield localization that can't be induced from a Bousfield localization along some particular Quillen equivalence? How about two different Bousfield localizations that induce the same Bousfield localization along a Quillen equivalence?

Are there any model categories where

*all*Bousfield localizations are known to exist, not just those generated by a small set?

adjunctionbetween $\infty$-categories. But not every localization of an $\infty$-category (in the sense of (1) above) will be anadjointlocalization. This is an obstruction to the existence of Bousfield localizations in the sense of (1). The lesson is that the notion of localization in (1) is too broad: we usually only care about localizations which are adjoint ones in the $\infty$-categorical sense -- a fact which is hard to even state without using $\infty$-categories. $\endgroup$adjointlocalizations, see HTT Prop A.3.7.8. $\endgroup$