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Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that this action is far from being free). There is a natural map from the homotopy colimit of the action (which is the extended power $E\Sigma_n \otimes_{\Sigma_n} X^n$) to the colimit $X^{\otimes n}/\Sigma_n$. In the pointed setting this map goes from $(E\Sigma_n)_+ \otimes_{\Sigma_n} X^{\otimes n}$. Let $(P)$ be the property that this map is a weak equivalence.

Does $(P)$ hold for sSet? [EDIT: Answer is no, see the comments]

Note that $(P)$ is known in at least a couple of instances. For example, Lemma 15.5 of Model Categories of Diagram Spectra by MMSS proves the pointed version for the positive model structures on Orthogonal Spectra and Symmetric Spectra, while EKMM's Theorem III.5.1 proves it for S-alg. I learned these references from Shipley's paper "Monoidal Uniqueness of Stable Homotopy Theory" which is also where I read about the property above. Shipley posits that this property is necessary in order for commutative monoids in $M$ to inherit a model structure. In my current thesis work I've found a more general way, and now I'm trying to see if $(P)$ implies my hypotheses, as well as work out lots of examples.

(1) Are there any other model categories of interest where this property is known to hold or fail?

I imagine this property fails when one knows that commutative monoids don't form a model structure, e.g. in chain complexes over a field of characteristic nonzero (does $(P)$ hold in characteristic zero?), for topological spaces (since a commutative monoid must be a product of Eilenberg-MacLane spaces), or for the usual model structure on symmetric spectra (this was known by Gaunce Lewis, and led to the positive model structure).

(2) Are there any other model categories where you can't have a model structure on commutative monoids? Do these other categories fail to satisfy $(P)$?

(3) Are there any standard hypotheses on a closed symmetric monoidal model category $M$ which will imply $(P)$?

For (3), we can certainly assume $M$ is cofibrantly generated and left proper, even combinatorial, cellular, or simplicial if you like. We can even assume all objects are cofibrant if that will lead to a proof. It's well-known that the map from the homotopy colimit of a diagram to the colimit of the diagram is a weak equivalence if the diagram is Reedy cofibrant, but I feel like this would force the action to be free, which we can't have. Perhaps I'm wrong, though, and there is some way to use Reedy cofibrancy.

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Regarding (1): what if $X = \Delta[0]$? –  Oscar Randal-Williams Jul 10 '12 at 21:57
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Oscars comment seems to generalize to any pointed model category tensored over $sSets$ or $Top$. –  Sean Tilson Jul 10 '12 at 23:03
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If you look at when $X$ is a point you are asking for $BG$ to be a point. –  Sean Tilson Jul 11 '12 at 3:02
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This modification gives $(B\Sigma_n)_+\rightarrow pt$, right? –  HenrikRüping Jul 11 '12 at 11:49
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@David: I think I went wrong in assuming that the zero object is a actually something like the point in $Top$. –  Sean Tilson Jul 11 '12 at 20:35
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1 Answer

up vote 9 down vote accepted

The equivalence (P) is a deep and subtle property of the smash product of spectra in modern symmetric monoidal models for the stable homotopy category. It is very unlikely to hold in other contexts. It was first found in EKMM [III.5.1] because the use of operads there visibly builds it into the smash product. It was later seen to hold for orthogonal and symmetric spectra in MMSS [15.5]. I doubt that it holds for the $\mathcal W$-spectra defined in MMSS.

Perhaps this is not the place for a confession, but the equivariant version of (P) as published by Mandell and myself for orthogonal $G$-spectra in "Equivariant orthogonal $G$-spectra and $S$-modules'' and by Mandell for symmetric $G$-spectra in "Equivariant symmmetric spectra'' is incorrect. The correct version replaces $E\Sigma_n$ with the universal principal $(G,\Sigma_n)$-bundle $E(G,\Sigma_n)$. For $E_{\infty}$ $G$-operads $\mathcal C$, $\mathcal C(n)$ is just such a universal principal $(G,\Sigma_n)$-bundle.

The essential point is that this is a topological result, not remotely to be expected in a general model theoretical context.

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Thanks, this is a great answer. I just love how MathOverflow lets you get expert opinions on things like this so quickly. –  David White Jul 12 '12 at 13:33
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