Let $X$ a smooth manifold. Is the pullback morphism
$\Omega^\bullet(X)\to\Omega^\bullet(X\times \mathbb{R}^n)$ an acyclic cofibration of differential graded commutative algebras? I guess so, and even that this should be the basic example to have in mind, but being no expert, I don't trust myself too much.



Cofibrations of differential graded commutative algebras
It depends completely on what you mean by cofibrations. The choice is
not quite simple to make as the homotopy category of real commutative dga's is
anti-equivalent to "real homotopy" which would suggest that the cofibrations
should correspond very roughly to fibrations of spaces (judging from your
example this looks like the notion you are searching for). Then the proper notion
would seem to be a pseudofree extension algebra (i.e., the extension algebra
forgetting the differential) should be a polynomial algebra over the base. In
that case the map $\Omega^\bullet(X)\rightarrow\Omega^\bullet(X\times\mathbb
R^n)$ is not a cofibration. I find it difficult to imagine an interesting model structure on
commutative dga's which would make it a cofibration.


