Suppose $\mathfrak{M}$ is a left proper celluar model category and $S$ is a set of cofibrations in $\mathfrak{M}$. What are the generating trivial cofibrations of $L_S\mathfrak{M}$? Are they $J\cup S$, where $J$ is the set of generating trivial cofibrations of $\mathfrak{M}$?
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No, definitely not! They are much more complicated to characterize. You need to take horns on the set of morphisms you just wrote down. This is all detailed carefully in Hirschhorn's book, summarized here. Search in there for "generating acyclic cofibrations" and you'll see what I mean by "horns".
answered Apr 20, 2020 at 10:57
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$\begingroup$ Also, FYI, it's no real restriction that S is a set of cofibrations. If S were any set of maps, you could use cofibrant replacement to find a set $S'$ of cofibrations such that the localization with respect to $S'$ is the same as the localization with respect to S (by the two out of three property). So, I pretty much always assume S is a set of cofibrations and that doesn't help gain control over the generating trivial cofibrations. $\endgroup$ Commented Apr 20, 2020 at 10:59
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$\begingroup$ I think we could also use Ayoub's $\nabla_{\infty}$ construction. $\endgroup$ Commented Apr 22, 2020 at 2:11