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>



Models for equivariant genuine commutative ring spectra
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.


