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Take non-negatively graded differential graded algebras, which are vector spaces over some field (the last assumption just to simplify things). We have the usual $d^2=0$, but the product is not assumed to be graded commutative. The problem is to find model structures for DGA maps on such things. We can take the usual equivalences or quasi-isomorphisms, inducing isomorphisms in all cohomology. The problem is the cofibrations or fibrations. There are two cases, the second one is the one I am mainly interested in, for reasons I will describe. Anything non-trivial that can be said about the problem would be greatly appreciated. (Apologies, I am not an expert in model categories, so this may all be well known...)

Case 1) The cofibrations are maps surjective in all degrees of the DGAs. I would not be surprised if this case was well known.

Case 2) The fibrations are DGA maps $\iota:B^n\to A^n$ with the property that $$ \frac{\iota B^{p}\wedge A^{n-p}}{\iota B^{p+1}\wedge A^{n-p-1}} \cong \iota B^{p} \otimes \frac{ A^{n-p}}{\iota B^{1}\wedge A^{n-p-1}}\ . $$ by the obvious map $\beta\otimes [\xi]\mapsto[\beta\wedge\xi]$.

The case (2) requires more explanation, so I will explain where the problem comes from. In noncommutative differential geometry, there is a Serre-Leray spectral sequence for the de Rham cohomology of a fibration, which supposes that a fibration satisfies the condition in case (2). The problem is to see what the corresponding cofibrations are. Mild additional assumptions would be OK, but commutativity is not. All comments welcome.

[The Serre spectral sequence of a noncommutative fibration for de Rham cohomology, Beggs & Brzeziński, Acta Math 195(2), 155-196.]

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  • $\begingroup$ Do you mean that the fibrations are surjective in all degrees in case 1? $\endgroup$ Nov 3, 2014 at 15:02
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    $\begingroup$ If you take fibrations to be surjections in all positive degrees, the answer can be found, e.g., in Schwede, Shipley, Algebras and Modules in Monoidal Model Categories, p. 503: math.uni-bonn.de/people/schwede/AlgebrasModules.pdf This paper will give you general criteria how to go from a model structure on chain complexes to one of dgas. $\endgroup$ Nov 3, 2014 at 15:20
  • $\begingroup$ for non-negatively graded dgas, it will be impossible to take fibrations to be surjective in all degrees, at least with a standard kind of weak equivalence, because you won't be able to factor a map from a DGA with nothing in degree $0$ to a DGA with something in degree $0$ as a weak equivalence followed by a surjection. $\endgroup$ Nov 3, 2014 at 15:53

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