Kerodon

Typo: in part a) of the statement, and also in the proof, the following formula appears: $(B_{\bullet } \times \Lambda ^2_1) \coprod _{A_{\bullet } \times \Lambda ^2_1 } (A_{\bullet } \times \Delta ^2) \subseteq A_{\bullet } \times \Delta ^2$. I think it should be: $(B_{\bullet } \times \Lambda ^2_1) \coprod _{A_{\bullet } \times \Lambda ^2_1 } (A_{\bullet } \times \Delta ^2) \subseteq B_{\bullet } \times \Delta ^2$ with $B_\bullet$ instead of $A_\bullet$ in the right hand side.

Yep. Thanks!

Typos: in the first chain of inclusions, $\subset$ needs to be replaced by $\subseteq$ to account for the case $j=0$; three instances of $Y_j$ need to be replaced by $Y(j)$; in the last sentence of the proof, $Y(m+2)$ needs to be replaced by $Y(m+1)$.

Yep. Thanks!

Typo: Towards the end of the proof it reads "andoyne".

Yep. Thanks!

Why is $S$ weakly saturated?

I guess one can prove $S$ is weakly saturated using factorizations, whose existence follows from the small object argument. But this has not been introduced at this point.

I tried proving it directly and I can see why it's closed under retracts and pushouts, but I am stuck on composition.

I have also tried proving (directly) that $S$ is weakly saturated, and I'm also stuck.