# Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper?

Since weak equivalences and fibrations are detected via the forgetful functor $R-mod\to C$, right properness seems to be obvious. On the other hand, I do not understand how to check the left properness. Looking at the case of complexes of modules over a ring, I suspect that the forgetful functor respects pushouts; possibly $R-mod$ is proper if $R$ is cofibrant in $C$. Is this true? Does it make sense to pass to the category of cofibrant objects in $R-mod$?

Now a very concrete motivic question. I would like to prove that the category of modules over the Voevodsky's algebraic cobordism spectrum is proper. This assertion should probably very similar to its analogue for the case of modules over the (motivic Eilenber-MacLane spectrum) $MZ$. Yet in the paper Oliver R¨ondigs, Paul Arne Østvær, Modules over Motivic Cohomology, http://www.math.uni-bielefeld.de/~oroendig/MZfinal.pdf, I was not able to find the discussion of properness for the latter case (see Proposition 2.36). It is not stated that $MZ$ is cofibrant; one does not pass to cofibrant objects in $MZ-mod$.

Upd. It seems that Proposition 2.9 of Hovey's http://arxiv.org/abs/math/9803002 confirms my idea: everything is ok if MGl is cofibrant. Still I wonder whether the latter is known (for some model of $MGl$), and whether this result of Hovey was published somewhere.

This question is answered very nicely in the recent preprint Homotopy theory for algebras over polynomial monads by Michael Batanin and Clemens Berger. Schwede and Shipley's "Algebras and Modules in Monoidal Model Categories" paper gives a general machine for transferring a model structure from a monoidal model category $M$ to $R$-mod where $R$ is a monoid in $M$. The condition you need on the base model category $M$ is known as the monoid axiom. Some smallness hypotheses are also needed, which Batanin and Berger formalize under the name "compactly generated model category'' and which Hovey calls "strongly cofibrantly generated" in his recent paper about the Eilenberg-Watts Theorem.

Batanin and Berger take a careful look at the Schwede-Shipley machine and figure out how to make the output left proper. Their solution is to strengthen what it means to be monoidal, and insist the model category is $h$-monoidal. They give a number of equivalent statements for this condition. I discovered one such condition independently in my thesis: a map $f$ is an $h$-cofibration if every pushout square with $f$ on top is a homotopy pushout square (Batanin and Berger define $h$-cofibrations in a different way on page 5 of their paper and prove that it's equivalent to this definition if $M$ is left proper). A model category is $h$-monoidal if for any object $X$ and for any cofibration $f$ we have $X \otimes f$ is an $h$-cofibration. I think of this as saying that $M$ behaves a bit like $Top$, since we're requiring the cofibrations to behave like neighborhood deformation retracts. Batanin and Berger check this condition for a number of examples. They also check that $M$ is $h$-monoidal if weak equivalences are closed under product (a condition which they call strongly $h$-monoidal). For a monoidal model category, being $h$-monoidal implies left proper. Theorem 3.1 proves that if $M$ is strongly $h$-monoidal and satisfies the monoid axiom then $R$-mod is left proper. Note that you do not need any cofibrancy hypothesis on $R$.

Batanin and Berger do not discuss examples coming from motivic homotopy theory, but in my thesis I check that the $h$-monoidal condition is satisfied. I've checked this for the model category of motivic functors and I think it should also hold for motivic symmetric spectra (you have to use Hovey's approach rather than Jardine's if you want the pushout product axiom to be satisfied). The compactly generated hypotheses are known in these examples. I don't know enough about Po Hu's motivic S-modules to know if the axiom holds there or not, but if you work in the other models then the answer to your question is yes. My thesis is not yet available for public consumption, but I could email you with the argument that motivic functors is $h$-monoidal if you wanted. The model category of motivic functors was developed by Dundas-Rondigs-Ostvaer and it's a very readable paper. I also have a condition on $M$ so that these hypotheses are preserved under Bousfield localization (this was my motivation for considering this problem), but since you didn't ask about that I'll hold myself back from commenting on it.

• Thank you! It would certainly interesting to read your argument; my e-mail is mbondarko et gmail.com. Yet I wonder whether R¨ondigs and Østvær had a more simple reasoning in mind. – Mikhail Bondarko Dec 5 '13 at 4:25
• Okay, I'll try to send you something this weekend. Right now I'm a bit swamped with job applications. In the meanwhile, which model category of motivic spectra were you planning to work in? – David White Dec 5 '13 at 19:54
• Thank you! Probably I will ask Rondings about this matter, so there is no need for you to hurry. Currently I prefer to work with the injective model structure for T-spectra; see mathoverflow.net/questions/150555/… – Mikhail Bondarko Dec 5 '13 at 21:22
• It seems that Lemma 2.18 in the 'motivic functor' paper www.math.uiuc.edu/documenta/vol-08/14.pdf has something to do with this matter. – Mikhail Bondarko Dec 5 '13 at 21:37
• Besides, Proposition 2.9 of Hovey's arxiv.org/abs/math/9803002 confirms my idea: everything is ok if MGl is cofibrant. – Mikhail Bondarko Dec 6 '13 at 10:22

Warning! This post may contain self-promotion.

If you only want to apply the left properness property to push-out squares where the underlying $R$-modules are cofibrant in $C$, then you can use Theorem 1.13 in this paper. This situation is actually rather common. The result is for operads rather than rings, but rings are operads concentrated in arity 1, so it applies. It assumes that $R$ is cofibrant in $C$ and that the tensor unit $I$ is cofibrant. You also have a version not assuming that the tensor unit is cofibrant, Theorem D.13. In that case it is enough that the unit $I\rightarrow R$ be a cofibration in $C$.

• Thank you! I am definitely interested in self-promotion of this sort! Yet the situation seems to be rather strange: some authors say something about a cofibrant model for $MGl$, but nobody states that such a model exists. – Mikhail Bondarko Dec 7 '13 at 9:47
• Cofibrant models exist for abstract reasons. Models with underlying cofibrant objects are even easier to get, I dare say they are ubiquitous. Left properness is unfortunately dependent on the point set models. – Fernando Muro Dec 7 '13 at 11:43
• So, any ring object has a cofibrant model (which is a ring object also)? Could you give some reference for this fact? – Mikhail Bondarko Dec 7 '13 at 11:59
• the category of ring objects has a model structure by Schwede-Shipley, hence any ring has a cofibrant resolution there. That doesn't mean that the underlying object is cofibrant in C but at least the unit map is a cofibration, so it satisfies the hypotheses in my comment – Fernando Muro Dec 7 '13 at 12:41