Models for equivariant genuine commutative ring spectra

Question

The following question is an attempt at understanding various flavours of equivariant commutative ring spectra; it may not be suitable level for this forum.

Let $\mathcal{C}(G)$ be a symmetric monoidal homotopical category such that $Ho(\mathcal{C}(G))$ is the category $SH(G)$ of genuine $G$-equivariant spectra. Here $G$ is a finite group. The examples I have in mind are the category $\mathcal{C}_O(G)$ of $G$-objects in orthogonal spectra, with the G-equvariant stable weak equivalences (Schwede) or the category $\mathcal{C}_\Sigma(G)$ of T-symmetric spectra in the category of $G$-simplicial sets, with $T$ an appropriate simplicial version of the regular representation sphere (Mandell).

Going back to general $\mathcal{C}(G)$, we may then form the homotopical category $CMon(\mathcal{C}(G))$ of commutative, unital monoid objects in $\mathcal{C}(G)$ (with weak equivalences the underlying weak equivalences). Furthermore, given an operad $O$ with values in $\mathcal{C}(G)$ we can talk about the homotopical category $O-Alg(\mathcal{C}(G))$.

Note that $CMon(\mathcal{C}(G))$ is just $O-Alg(\mathcal{C}(G))$ for $O$ the commutative operad (all spaces just the tensor unit).

Any choice $\mathcal{C}(G)$ should admit a symmetric monoidal functor from the category $sSet(G)$ of $G$-simplicial sets. Then, given any operad in $G$-simplicial sets, we can talk about the induced operad in $\mathcal{C}(G)$. In particular we have "the" classical E-infinity operad $E \in Operads(sSet) \subset Operads(sSet(G))$ (consisting of spaces with trivial action). As far as I understand there is also the "genuine G-equivariant E-infinity operad" $E_G \in Operads(sSet(G))$, and in fact many operads between those two (see Blumberg-Hill).

So we now have six homotopical categories: $E-Alg(\mathcal{C}_O(G)), E_G-Alg(\mathcal{C}_O(G)), CMon(\mathcal{C}_O(G))$ and $E-Alg(\mathcal{C}_\Sigma(G)), E_G-Alg(\mathcal{C}_\Sigma(G)), CMon(\mathcal{C}_\Sigma(G))$. My question is, which of these are known (or expected) to have equivalent homotopy categories?

Remark 1

I'm phrasing this question in terms of homotopical categories because there are many different model structures, and there are many subtle issues regarding these, but as far as I can tell my question is not about this.

Remark 2

As far as I understand, if $G$ is the trivial group, all of these categories model commutative ring spectra and have equivalent homotopy categories.

Remark 3

If I understand correctly the work of Blumberg-Hill (and many others), then $Ho(E-Alg(\mathcal{C}_O(G))) \ne Ho(E_G-Alg(\mathcal{C}_O(G)))$ because the objects on the right have "norm maps" but on the left not. A related thing to say is that the homotopy groups on the right are tambara functors and on the left they have less structure.

I think it is also mentioned in loc. cit. that $Ho(E_G-Alg(\mathcal{C}_O(G))) = Ho(CMon(\mathcal{C}_O(G)))$.

Remark 4

There are various articles proving that $Ho(E-Alg(Spt(\mathcal{M},T))) = Ho(CMon(Spt(\mathcal{M},T)))$ for rather general model categories $\mathcal{M}$, see e.g. Pavlov-Scholbach. This seems to suggest to me that $Ho(E-Alg(\mathcal{C}_\Sigma(G))) = Ho(E_G-Alg(\mathcal{C}_\Sigma(G))) = Ho(CMon(\mathcal{C}_\Sigma(G)))$ (but checking the detailed list of requirements for their theorem is non-trivial). Note that this would be in stark contrast to the case of orthogonal spectra! So my best guess is that loc. cit. does not apply in our situation?

Some references

Schwede on equivariant orthogonal spectra: http://www.math.uni-bonn.de/people/schwede/equivariant.pdf

Mandell on equivariant symmetric spectra: http://pages.iu.edu/~mmandell/papers/gssfinal.dvi

Blumberg-Hill on equivariant E-infinity operads: https://arxiv.org/pdf/1309.1750v3.pdf

Pavlov-Scholbach: http://wwwmath.uni-muenster.de/sfb878/publications/files/phpimKZBl5582.pdf

Answer 1

For the questions asked here, there is no difference between orthogonal $G$-spectra, symmetric $G$-spectra of either $G$-spaces or $G$-sSets, or EKMM $G$-spectra. For the first two, the nonequivariant arguments in Mandell-May-Schwede-Shipley http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf generalize directly, once one corrects Lemma III.8.4 in Mandell-May http://www.math.uchicago.edu/~may/PAPERS/MMMFinal.pdf, as is done in HHR. That lemma should say that for positive cofibrant $G$-spectra $X$ (in any good category of $G$-spectra), the natural map $E(G,\Sigma_n)\wedge_G X^n \to X^n/\Sigma_n$ is a weak equivalence, where $E(G,\Sigma_n)$ is the universal principal $(G,\Sigma_n)$-bundle. This implies that the homotopy categories of $E_G$-algebras and of commutative monoids are equivalent for any genuine $E_{\infty}$ $G$-operad $E_G$. As David White says, his thesis improves these equivalences to Quillen equivalences of model categories. The "canonical" choice of $E_G$ is in the eyes of the beholder: there are many different natural choices with different applications of each: infinite Steiner or little discs $G$-operads, linear isometries $G$-operads, permutativity (or Barratt-Eccles) $G$-operads, etc. There are analogous equivalences using naive $E_{\infty}$-$G$-operads, which are nonequivariant $E_{\infty}$ operads regarded as $G$-trivial. That is the weakest in the hierarchy of point-set level kinds of genuine $G$-spectra with commutative ring structures given by Blumberg-Hill,whereas commutative monoids is the strongest.

Answer 2

I proved in my thesis that $CMon(C_O(G))$ admits a transferred model structure, if you work with the positive or positive flat stable model structure on $C_O(G)$ (for the former, see Mandell-May, for the latter, see Stolz's thesis). I then proved that $CMon(C_O(G))$ is Quillen equivalent to $E_G-Alg(C_O(G))$, in the section on rectification. A reference is arXiv:1403.6759. I don't think you can use the Pavlov-Scholbach stuff for this. As I recall, they don't touch anything equivariant.

My work on preservation under localization, together with Mike Hill's example of a localization that destroys $E_G-Alg$ structure, provides a concrete example that $E-Alg$ and $E_G-Alg$ are not Quillen equivalent. A reference is arXiv:1404.5197. Mike's example is reproduced there as Example 5.7. I just realized I never uploaded the submitted version of this paper to arxiv, so the arxiv version still has a mistake in Theorem 5.9. It's not true that such localizations preserve genuine commutativity, precisely because $E-Alg$ is not equivalent to $CMon$ (Justin Noel emailed me about this ages ago; embarrassing that I forgot to update arxiv). What's true is that preservation for CMon is the same as for $E_G$-Alg.

I never studied $C_\Sigma$, but Mark Hovey and I had some vague plans to think about it if we ever had time. It seemed like the approach we took in arXiv:1312.3846 (by now, a better reference is probably the HHR appendix of the Kervaire paper) would make it easy to prove such an equivalence. If you plan to go down that route, feel free to email me and we can talk more.