Is there Jeff Smith's theorem for left semi-model structures? Jeff Smith's theorem gives a simple criterion for the existence of a combinatorial model category. Is there a similar theorem for combinatorial left semi-model categories? I see two problems that occur when you try to apply the theorem in this context.
First, I don't think that the class of weak equivalences is necessarily an accessibly embedded accessible full subcategory of the category of morphisms. We can try to fix this by modifying this condition so that it says something about weak equivalences with cofibrant domains only, but I don't know what this modification might be.
The second problem I see is that we cannot prove that $\mathrm{inj}(J) \cap \mathcal{W} \subseteq \mathrm{inj}(I)$. Usually, this follows from the fact that $\mathrm{cof}(I) \cap \mathcal{W} \subseteq \mathrm{cof}(J)$, but in a semi-model category the latter condition only holds for maps with cofibrant domains while the former condition should be true for all maps.
 A: Yes, and this is one of the main results in a paper I hope to put on arxiv very soon. I wrote about this result in a previous mathoverflow answer here. You are right that the way to do it is to focus on maps between cofibrant objects. This result was also known to Clark Barwick, and probably to Cisinski, as I discuss in my other answer. That's one of the reasons it took me so long to write it up. I didn't know why anyone would care. But, in the paper I'm finishing, there are tons of applications of this result. If you want to correspond more, I would be happy to.
Edit in response to request from the OP:
The conditions of the theorem state that $M$ is locally presentable, and 


*

*$W$ is $\kappa$-accessible for some $\kappa$, 

*$W$ is closed under retracts, 

*morphisms in inj$(I)$ are weak equivalences, 

*within cof$(I)\cap W$, morphisms with cofibrant domain are closed under pushouts of diagrams of cofibrant objects, and are closed under transfinite compositions,

*The maps of $I$ have cofibrant domain and the initial object of $M$ is cofibrant.


Then you have a combinatorial semi-model structure with generating cofibrations $I$, generating trivial cofibrations $J$ constructed as in Barwick or Beke's papers, cofibrations cof$(I)$, and fibrations rlp$(J)$.
I should be done with this paper by the end of 2019. Right now I'm just adding lots of examples.
