Pushout along weak equivalence gives weakly equivalent object 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.
 A: This is true in general in any left proper model category. To be left proper means the pushout of a weak equivalence (for you, $X\to A$) along a cofibration (for you, $X\to Y$) is again a weak equivalence (for you, $Y \to A\otimes_X Y$).
It is not true that any pushout of a weak equivalence is a weak equivalence (there are many easy counterexamples, e.g., in the category of topological spaces, because pushouts are not homotopy pushouts), but the point of left properness is that the pushout square formed by a span, where one leg is a cofibration and one leg is a weak equivalence, is a homotopy pushout square.
The other property Goerss and Schemmerhorn need in that proof you referenced, that $A\otimes_X Y$ is cofibrant as an $A$-algebra, is true because the map $A\to A\otimes_X Y$ is a cofibration (since it's a pushout of the cofibration $X\to Y$), and $A$ is the initial $A$-algebra.
A: I'm answering the question Sofía Marlasca Aparicio asked in a comment below my previous answer. My previous answer was to the question "would this [pushouts of weak equivalences are weak equivalences] be true in a general model category?"
A great paper regarding left properness of categories of algebras is Rezk's Every homotopy theory of simplicial algebras admits a proper model. It proves (as a consequence of a much more general result) that if $R$ is cofibrant, then the category of $R$-algebras is left proper. It also proves, in Example 2.11, that if $R$ has Tor-dimension greater than zero, then the category of simplicial $R$-algebras is NOT left proper.
Fortunately, in the Goerss-Schemmerhorn paper, they are working in the context of commutative $R$-algebras, as Definition 4.28 makes clear (plus, the whole discussion from page 31-35 is about Andre-Quillen homology, for commutative algebras over a commutative ring). Note as well that Theorem 4.31 identifies a pushout as a tensor product, demonstrating again that the context is commutative rings.
The category of simplicial commutative $R$-algebras is indeed left proper. A reference is Model structures on commutative monoids in general model categories, Theorem 4.17 (and Section 5.2 plus the paper Homotopy theory for algebras over polynomial monads to see that the category of simplicial sets satisfies the hypotheses).
