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 $\mathfrak{M}$?



Generating trivial cofibrations of Bousfield localization
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](https://ncatlab.org/nlab/show/Bousfield+localization+of+model+categories). Search in there for "generating acyclic cofibrations" and you'll see what I mean by "horns".


