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.
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$\begingroup$ If $X$ is a point and $n = 1$, this is asking whether the ring of smooth functions on $\mathbb{R}$ is a retract of a polynomial algebra. That seems unlikely, but I don't have a proof. $\endgroup$– Tyler LawsonCommented Oct 8, 2010 at 17:41
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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.
answered Oct 8, 2010 at 17:46
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$\begingroup$ so if instead of $\Omega^\bullet(X\otimes \mathbb{R}^n)$ I would use $\Omega^\bullet(X)\otimes \Omega^\bullet_{poly}(\mathbb{R}^n)$ would that work? (here $\Omega^\bullet_{poly}(\mathbb{R}^n$ means polynomial differential forms on $\mathbb{R}^n$) $\endgroup$ Commented Oct 8, 2010 at 19:30
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$\begingroup$ I haven't really thought about what the exact conditions needed to get a model category structure (though I am sure that people have done it). It is quite possible however that there is a model structure where these would be cofibrations. $\endgroup$ Commented Oct 9, 2010 at 7:53
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$\begingroup$ Those would both be cofibrations (in particular, they are coproduct with a cofibrant object) in the standard model structure, which has fibrations being levelwise surjections. $\endgroup$ Commented Oct 11, 2010 at 5:06