What are the naive fixed points of a non-naive smash product of a spectrum with itself? Let $X$ be an object in one of the well-known symmetric monoidal model categories of spectra. E.g., an $\mathtt S$-module in the sense of EKMM, or an orthogonal spectrum, or a $\Gamma$-space, etc.
One can form the spectrum $X\wedge X$, which is a "naive" spectrum with an action of $\Sigma_2$, and one can take the categorical fixed points to obtain the spectrum $(X\wedge X)^{\Sigma_2}$. What can be said about the homotopy type of this spectrum as a functor of $X$?
More specifically I would like to know the following:

*

*Does this construction have good homotopical properties. For example, does it preserve weak equivalences between cofibrant spectra?

*Does the homotopy type of $(X\wedge X)^{\Sigma_2}$ (say for cofibrant $X$) depend on the specific model of smash product?

*Is it true that if $X$ is a suspension spectrum then $(X\wedge X)^{\Sigma_2}\simeq X$? If it is not always true, is it true in some cases, i.e., for some models of smash product?

*Can the homotopy type of $(X\wedge X)^{\Sigma_2}$ be described in other cases?

 A: In both the orthogonal world and the EKMM world one can set things up so that $G$-spectra are just spectra with an action of $G$, and the naive and genuine equivariant stable categories are the homotopy categories for two different model structures on the same underlying geometric category.  I will take that point of view.
I will write $\kappa^G(X)$ for the categorical fixed points of a spectrum $X$ with $G$-action, i.e. the largest subspectrum on which the action is trivial.    We also have the Lewis-May fixed point functor which I will denote by $\lambda^G(\cdot)$, and the geometric fixed point functor $\phi^G(\cdot)$.  There are natural maps $\kappa^G\to\lambda^G\to\phi^G$.
Here are some facts about the EKMM category:

*

*The functor $\Sigma^\infty$ preserves smash products on the nose, but $\Sigma^\infty X$ is not cofibrant unless $X$ is a point.

*The functor $\Sigma^\infty$ also commutes with $\kappa^G$, so $\kappa^{C_2}(X\wedge X)=X$ whenever $X$ is a suspension spectrum.

*Now let $T$ be the cofibrant replacement of $S^0$.  This is an $\mathbb{L}$-spectrum in the sense of EKMM, so it has a kind of action of the monoid $\mathcal{L}(1)=\mathcal{L}(\mathcal{U},\mathcal{U})$ of linear isometric embeddings of the universe $\mathcal{U}=\mathbb{R}^\infty$ in itself.  If we just consider the underlying Lewis-May spectrum, it can be identified with $\Sigma^\infty\mathcal{L}(\mathcal{U},\mathcal{U})_+$.  Similarly, $T\wedge T$ is just $\Sigma^\infty\mathcal{L}(\mathcal{U}^2,\mathcal{U})_+$.  From this it is easy to see that $\kappa^{C_2}(T\wedge T)=0$.

*Similarly, I am pretty sure that $\kappa^{C_2}(X\wedge X)=0$ whenever $X$ is cofibrant.

*We can also define $T_G$ to be $\Sigma^\infty\mathcal{L}(\mathcal{U}\otimes\mathbb{R}[G],\mathcal{U})_+$, equipped with appropriate action of $\mathcal{L}(1)$.  This is the cofibrant replacement of $S^0$ in the genuine model structure, and in the case $G=C_2$ we just have $T_G=T\wedge T$.  The Lewis-May fixed point functor can be described as $\lambda^G(X)=\kappa^G(F(T_G,X))$.  John Rognes's answer describes $\lambda^{C_2}(X\wedge X)$ in various cases.

I think that the story in the orthogonal world is similar, but the analysis is a bit more complicated and I do not have the details to hand.
A: In the context of functors with smash product (FSP), or symmetric spectra, or orthogonal spectra, the spectrum with $\Sigma_2$-action $X \wedge X$ prolongs essentially uniquely to a $\Sigma_2$-spectrum indexed on any fixed choice of complete $\Sigma_2$-universe.  In the FSP setting this goes back to Boekstedt around 1987, cf. Hesselholt-Madsen (1997).  In the orthogonal setting this is the Hill-Hopkins-Ravenel (2016) norm $N_e^{\Sigma_2} X$.  See also Schwede's Barcelona lecture notes.  If X is cofibrant (= q-cofibrant), or more generally flat (= S-cofibrant), then this construction is homotopy invariant.  There is then a natural homotopy cofiber sequence
$$
(E\Sigma_{2+} \wedge X \wedge X)^{\Sigma_2} \to (X \wedge X)^{\Sigma_2} \to (\tilde E\Sigma_2 \wedge X \wedge X)^{\Sigma_2}
$$
and equivalences
$$
(E\Sigma_{2+} \wedge X \wedge X)^{\Sigma_2} \simeq (X \wedge X)_{h\Sigma_2} = D_2(X)
$$
and ([HM97, Prop. 2.5], [HHR16, Prop. B.209])
$$
(\tilde E\Sigma_2 \wedge X \wedge X)^{\Sigma_2} \simeq \Phi^{\Sigma_2}(X \wedge X) \simeq X .
$$
In particular, $(X \wedge X)^{\Sigma_2} \to X$ is an equivalence if and only if $D_2(X) \simeq *$, which is only rarely true for suspension spectra $X$.
If you are interested in $G$-fold smash powers, viewed as $G$-spectra, then perhaps https://arxiv.org/abs/2207.13408 is relevant.
