More on categories of modules over the algebraic cobordism spectrum I have the following questions on monoidal model structure(s) for the motivic stable homotopy category $SH(k)$ (where $k$ is a field); certainly, I am also interested in general statements concerning these matters.


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*Which "models" for $SH(k)$ are monoidal model categories (with the extra monoid axiom fulfilled) such that the unit object (sphere spectrum) is cofibrant?

*For $MGl'$ being a cofibrant ring spectrum in this category that is weakly equivalent to the Voevodsky algebraic cobordism spectrum $MGl$ (over $k$) I would like to have a criterion that ensures that a cohomological functor from $SH(k)$ (into abelian groups) factors through the (triangulated) homotopy category $D^{MGl}$ of highly structured  $MGl'$-modules. Any suggestions?
3.[My own idea concerning 2]. The spectrum  $MGl$ possesses a universality property (see https://projecteuclid.org/euclid.hha/1251811074) that ensures that various forms of K-theory are represented by spectra that are algebras over $MGl$ in $SH(k)$. To prove that these "K-theories" factor through $D^{MGl}$ I would like to prove that certain "replacements" of these K-theory spectra are highly structured  $MGl'$-modules. It seems reasonable to consider the model structure for ring spectra in the model for $SH(k)$ that is provided by http://www.math.uni-bonn.de/~schwede/AlgebrasModules.pdf
Yet to use this model structure I probably need a calculation of morphisms in the corresponding homotopy category; is anything known about this?


*Proposition 1.10 of Hovey's unpublished http://mhovey.web.wesleyan.edu/papers/mon-mod.pdf says that for a left proper cellular monoidal model category $C$ the category of modules over a cofibrant monoid object $R$ of $C$ is (cellular and) left proper also. Does a (published) proof of this fact or something similar exist? Cf. the discussion at Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

*I believe that the category of $R$-modules is a $C$-module category in the sense of Definition 4.2.18 of Hovey's book. Does there exist a reference for this statement?
 A: For (1), I recommend the paper Motivic Functors by Dundas-Rondigs-Ostvaer. They compare many different models for SH(k), with an eye towards monoidal properties (i.e. pushout product axiom, monoid axiom, axiom that cofibrant objects are flat). 
For (2), you probably know that these categories of modules are Quillen equivalent, going back to Schwede-Shipley's "algebras and modules" paper. Another reference is Fresse's book "Modules over operads and functors" Theorem 12.5.A (which avoids an assumption Schwede-Shipley had to make about $-\otimes_R S$ when you're changing ring from $R$ to $S$), since both categories are algebras over a $\Sigma$-cofibrant operad. Dmitri also has a version of rectification (in his paper #1 from the comments above) that does not require $\Sigma$-cofibrancy, and works for any weak equivalence of colored operads. I've got a version with Donald Yau ("Homotopical Algebraic Lifting Theorem") which is meant to study a more general phenomena than rectification (regarding lifting Quillen equivalences to categories of algebras), and provide easier-to-check conditions for levelwise cofibrant operads, but in your case the Fresse reference already does the job. 
For (4), a lot of those results ended up in Spitzweck's thesis, and later in Fresse's book. For those that didn't (including the one you cite), I included the version for commutative monoids in my thesis (see Theorem 4.17 in "Model structures on commutative monoids in general model categories", now accepted to JPAA). In your case, you probably don't care that the category of modules is cellular, because it's actually combinatorial, and that's even better. A reference is Beke, "Sheafifiable homotopy model categories", 2.3.
For (5) this is Proposition 5.3 in Spitzweck's thesis. The operad in question (whose algebras are $R$-modules) is $\Sigma$-cofibrant. When Spitzweck says that the $J$-semi model category $Alg(O)$ is a $C$-module, this implies that, if there is a full model structure (which there is in this case, by Schwede-Shipley), then it's a $C$-module. 
