This is a follow-up question to [another](https://mathoverflow.net/questions/101892/when-is-homotopy-orbit-space-weakly-equivalent-to-orbit-space-other-than-situati) question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Quillen equivalences between them. One of the models is $W$-spaces, i.e. diagrams of shape $W$ in $Top$ where $W$ is the category of based spaces which are homeomorphic to finite CW complexes, and $Top$ means compactly generated spaces.



> 
> Does the category of commutative monoids in $W$-spaces inherit a model structure?
> 
> 
> 


By commutative here I mean strictly commutative, not $E\_\infty$. My interest is for the background section of a paper I'm working on now about when commutative monoids form a model category. If it's not known one way or the other I can try my theorem on the category of $W$-spaces, but I suspect that category won't satisfy my hypotheses. If it's known to be true, maybe I can recover it as a special case of my theorem. If it's known to be false, that would be good to know too.


Note that this question comes up on page 5 of MMSS and the authors say they don't know. On page 47 they point out the obstacle which prevented them from proving CommMon($W$-spaces) inherits a model structure, and it's the same obstacle Peter May pointed out in his answer to my previous question. They couldn't prove that every relative $Sym(K^+)$-cell complex was a stable equivalence (where $K^+$ are the positive stable generating trivial cofibrations) because they couldn't prove $W$-spaces have the property that for every cell $S$-module $M$, $(E\Sigma\_j)\_+ \wedge\_{\Sigma\_j} M^j \to M^j/\Sigma\_j$ is a weak equivalence. If someone had proven this in the past decade I'm sure Peter May would know. What I'm wondering is if someone found a different way to prove CommMon($W$-spaces) forms a model category or if someone found evidence that it can't, similar to the evidence in [this](https://mathoverflow.net/questions/23269/non-examples-of-model-structures-that-fail-for-subtle-surprising-reasons) answer that CDGA can't be a model category, or on page 13 of [this](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.5238) paper of Schwede's that CommMon($\Gamma$-spaces) can't be a model category. Even if no hard evidence exists, I'd like to hear some soft evidence, i.e. why one would or would not believe this to be true.


The reason one might hope the answer is yes is that Theorems 0.1 and 0.4 prove that $W$-spaces are Quillen equivalent to Orthogonal Spectra and Symmetric Spectra, and that this Quillen equivalence carries over when we pass to the model categories of monoids (the fact that such model structures exist follows from the fact that all these categories satisfy the pushout product and monoid axioms). Both Symmetric Spectra and Orthogonal Spectra have a notion of positive model structures, which makes the category of commutative monoids inherit a model structure. It's natural to wonder if $W$-spaces has something like that too.


MMSS also raise another natural question (on page 6): 



> 
> Is $Ho(E\_\infty$ $W$-spaces$) \cong Ho($Commutative $W$-spaces$)$?
> 
> 
> 


It seems like with all the work that has been done on operads in the last decade this should be known. Can someone confirm that and give a reference?



State of knowledge on the Commutative W-spaces which appear in "Model Categories of Diagram Spectra"
There is a very illuminating paper I did not know about when I last answered a similar question:
Tyler Lawson. ``Commutative $\Gamma$ rings do not model all commutative ring spectra". Homology,
Homotopy and Applications Vol 11, 2009, 189-194. The title says it all. Together with the monoidal functor from $\Gamma$-spaces to $W$-spaces of MMSS, Tyler's result makes clear that commutative $W$-rings cannot give a homotopy category equivalent to the categories of orthogonal, symmetric, or EKMM commutative ring spectra. Incidentally, the question has a minor slip. David writes: "By commutative here I mean strictly commutatative, not $E\_{\infty}$". As explained in EKMM, strictly commutative EKMM ring spectra are Quillen equivalent to $E\_{\infty}$ ring spectra as defined way back in the 1970's. Those still have the nicest connection to spaces, namely $E\_{\infty}$ ring spaces. See
my paper: What precisely are $E\_{\infty}$ ring spaces and $E\_{\infty}$ ring spectra.
Geometry & Topology Monographs 16(2009), 215--282 (and on my web page) for a modern overview.


