# Bousfield Localization and Quillen Equivalence

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.

1. 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 $S$?

2. 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)?

3. 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)?

4. 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.

5. 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?

6. 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?

7. Are there any model categories where all Bousfield localizations are known to exist, not just those generated by a small set?

• Concerning the 7th question, any model category such that all maps are weak equivalences. All Bousfield localizations exist. Commented Mar 2, 2016 at 10:18
• Note to self: A Bousfield localization, or more generally any Quillen adjunction, induces an adjunction between $\infty$-categories. But not every localization of an $\infty$-category (in the sense of (1) above) will be an adjoint localization. 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. Commented May 5, 2018 at 16:54
• For existence of all adjoint localizations, see HTT Prop A.3.7.8. Commented May 5, 2018 at 16:55

For (1)-(2), look at work of Carles Casacuberta. He has lots of good examples. His paper with Chorny on the orthogonal subcategory problem has an example for your (2), on the last page. This paper of Casacuberta-Chorny goes into great depth about (3) as well, and it led to Chorny's work on class combinatorial model categories and localizations (when the model structure is generated by a class, and you have a generalized small object argument). This paper may also include an example for (1). If it doesn't, I can dig out an example Casacuberta showed me last summer and send you a sketch by email (it's unpublished and not mine, so I don't want to post it here).

Part of my thesis can answer (4). You could have a Quillen equivalence of monoidal model categories $M_1$ and $M_2$, and suppose both satisfy the commutative monoid axiom so that commutative monoids inherit transferred model structures. There was a condition in my thesis that if the free commutative monoid functor Sym(-) preserves $C$-local equivalences then a localization exists at the level of commutative monoids. Since this condition need not be preserved by a Quillen equivalence, you can have a left Bousfield localization of $M = CMon(M_1)$ that does not transfer to a left Bousfield localization of $N = CMon(M_2)$.

For (5), I don't have an example off the top of my head, but I agree with you that I would not expect this to come for free. This would be a strange example, because on the infinity category level there's no difference between left Quillen equivalence and right Quillen equivalence, and there I do expect localizations to be induced (assuming everything in sight is presentable and accessible). So maybe look at some very simple model structures with a lot going on with the cofibrations and fibrations.

For the first part of (6), I see two ways to interpret this. Every Bousfield localization is induced by the identity functor, so I assume you didn't mean that. For the other interpretation, if someone hands me a Quillen equivalence and a Bousfield localization of one of the two model categories I see no reason at all to expect it to be induced by a Bousfield localization on the other piece. For the other part of (6), what do you mean by "two different Bousfield localizations"? If they provide Quillen equivalent model structures most people would not say they are different.

• Thanks so much, David! I'll definitely take a look at some of Casacuberta and Chorny's work, and yours. If I'm reading you right, your condition for the localization of commutative monoids to exist is not a necessary condition, so it might take some work to show that the model structure actually doesn't transfer even if your condition holds in $M_1$ but not in $M_2$, right? As for (7), I guess I was looking for a nontrivial example. Commented Mar 2, 2016 at 23:08
• As for my meaning in (6), I meant, that if $M_1,M_2$ are two distinct Bousfield localizations of $M$ (i.e. with different homotopy categories), can we find a Quillen equivalence to $N$, say, such that $M_1$ and $M_2$ induce Bousfield localizations $N_1,N_2$ of $N$ and it turns out that $N_1$ and $N_2$ are the same? More broadly, I guess one thing that would be nice to have to work with would be a concrete example of a Bousfield localizaiton of $sSet$, say, which doesn't transfer to $Top$. Or vice versa. Commented Mar 2, 2016 at 23:09