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Let $\mathcal{M}$ be a model category and $S$ a set of cofibrations between cofibrant objects. Then every $S$-local object has the right lifting property with respect to $S$. The converse does not hold in general, but it does under suitable closure conditions on $S$. I think the following should be enough:

Proposition: Suppose that, for every map $f : A \to B$ in $S$, there are cylinder objects $C(A)$ and $C(B)$ and a morphism of cylinder objects $C(A) \to C(B)$ such that the map $C(A) \amalg_{(A \amalg A)} (B \amalg B) \to C(B)$ belongs to $\mathrm{cof}(J \cup S)$, where $J$ is the class of trivial cofibrations of $\mathcal{M}$. Then a fibrant object of $\mathcal{M}$ is $S$-local if and only if it has the right lifting property with respect to $S$.

I can propably prove this proposition myself, but I'd prefer not to write down the proof. So, I'd like to know whether such a proposition appears in the literature (I could not find such a result in Hirschhorn's book). Alternatively, I will accept a proof of this proposition if it is short and slick.

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  • $\begingroup$ It appears that you are asking what conditions should be imposed on S so that S is a set of generating acyclic cofibrations for the left Bousfield localization of M with respect to S. (This is not quite true literally, since you do not require M to be left proper and are also inquiring about fibrant objects, not fibrations in general.) This is difficult and there is no simple answer. That being said, you could use Definition 5.5.4.5 in Lurie's Higher Topos Theory to get such conditions (you need to adjust his definition to the case of cofibrations between cofibrant objects, though). $\endgroup$ Nov 27, 2019 at 18:42
  • $\begingroup$ @DmitriPavlov A set that recognizes local objects is easier to describe than a set of generating trivial cofibrations. We can always take the set $\widetilde{A} \otimes \Delta^n \amalg_{\widetilde{A} \otimes \partial \Delta^n} \widetilde{B} \otimes \partial \Delta^n \to \widetilde{B} \otimes \Delta^n$ for all $A \to B$ in $S$ (using Hirschhorn's notation). Moreover, if $S$ is closed under this construction, then it already recognizes $S$-local objects. But this condition is too strong. The one I gave in the question is essentially a special case with $n = 1$. I believe it should be enough. $\endgroup$ Nov 27, 2019 at 19:01
  • $\begingroup$ If you are indeed only interested in fibrant objects and assume the existence of a simplicial enrichment (or use framings as a substitute), then the above formula with a pushout product will suffice. But sticking to the case n=1 essentially amounts to demanding surjectivity of RHom(B,X)→RHom(A,X) (where X is an S-local object) on π_0 and π_1, and it seems unlikely that surjectivity for π_n with n>1 can somehow follow from the case n=1. $\endgroup$ Nov 27, 2019 at 19:14
  • $\begingroup$ @DmitriPavlov Well, I assume that $S$ is closed under this construction. If we iterate it, we should get surjectivity for all $n$. Basically, the iteration of this construction should give us cubical framings. So, this might be one way to prove this, but then we need to compare cubical $\mathrm{Hom}$-sets with simplicial ones and this looks like a rather tedious proof. I also think that it might be possible to proof this directly without appealing to cubical framings, but I didn't try this since I hope it was already done. $\endgroup$ Nov 27, 2019 at 19:19
  • $\begingroup$ Also, I don't literally assume that $S$ is closed under construction. I assume a weaker condition that the map belongs to $\mathrm{cof}(J \cup S)$. I'm not sure about that, it might be too weak. If this is the case, it'd be fine to replace the condition with the stronger one. $\endgroup$ Nov 27, 2019 at 19:26

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