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Apr 22, 2020 at 2:11 comment added Nanjun Yang I think we could also use Ayoub's $\nabla_{\infty}$ construction.
Apr 22, 2020 at 2:09 vote accept Nanjun Yang
Apr 20, 2020 at 10:59 comment added David White 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.
Apr 20, 2020 at 10:57 history answered David White CC BY-SA 4.0