TITLE: Direct proof that the model category of cdgas is left proper
QUESTION [6 upvotes]: Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred from the projective model structure on chain complexes, it follows formally that the projective model structure on $cdga$ is right proper. However the only reference for it being left proper I know is Toen &  Vezzosi's HAG II, and this proof is roundabout, in that they show that $cdga$ forms a HAG context, which hence implies it must be left proper. Does someone know a more direct proof, or a reference with one? Thanks!

REPLY [3 votes]: If you want a reference other than Toen-Vezzosi, this statement is proven as Theorem 4.17 in my paper Commutative Monoids in General Model Categories, plus example 5.1 to see that the commutative monoid axiom holds for chain complexes in characteristic zero, plus the table on page 3 of Batanin-Berger Homotopy theory for algebras over polynomial monads, which checks that chain complexes in characteristic zero are h-monoidal and compactly generated. By the way, this proof specializes to Tyler's proof (i.e. using the same diagram plus the same filtration), but is phrased to hold in much more general settings. My proof is just a riff on the similar proof which is one of the main results in the Batanin-Berger paper.