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Jan 14, 2020 at 14:43 comment added David White I finished the paper, so a reference for the claims in this answer can now be found at arxiv.org/abs/2001.03764
Nov 17, 2019 at 13:07 vote accept Valery Isaev
Nov 17, 2019 at 12:49 comment added David White @ValeryIsaev: I emailed you, so we don't have to discuss here in the comments.
Nov 17, 2019 at 12:41 history edited David White CC BY-SA 4.0
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Nov 17, 2019 at 12:12 comment added Valery Isaev After some thought, I can solve the second problem I mentioned if the domains of generating cofibrations are cofibrant. Do you know how to do this in general? Also, the first problem is easy to solve. Weak equivalences actually do form an accessible subcategory. The rest of the proof seems to work just fine if we assume that a transfinite composition of pushouts of maps in $\mathrm{cof}(I) \cap \mathcal{W}$ is a weak equivalence as long as it has cofibrant domain. Is this correct?
Nov 16, 2019 at 17:41 comment added Valery Isaev That's good! I actually need this theorem to construct a particular left semi-model category. Could you write down the conditions of the theorem, so I could check that it applies in my case?
Nov 16, 2019 at 17:18 history answered David White CC BY-SA 4.0