This question arose through reading "Interactions between homotopy theory and algebra" (the first chapter by Goerss and Schemmerhorn). In particular, I am struggling with the proof of Proposition 4.32, the cofiber sequence of cotangent complexes associated to two composable ring maps.
We fix a ring $R$ and take a map $A \to B$ in the category of $R$-algebras. We now take a cofibrant replacement $X \to A$ in the category of simplicial $R$-algebras and factorise $X \to B$ as $X \to Y \to B$, where $X \to Y$ is a cofibration and $Y \to B$ is an acyclic fibration.
There is a clear map $A \otimes _X Y \to B$ by the universal property of the colimit and then they claim that this must be a weak equivalence. I cannot figure out why this is the case. Of course, it is enough to show that $Y \to A \otimes_X Y$ is a weak equivalence, and this intuitively makes sense since we are pushing out along a weak equivalence, but I can't seem to produce a proof.
Would this be true in a general model category?
NOTE: wherever I wrote "cofibration", the original text says "free", and perhaps this is used, but I thought this might be true in greater generality.