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?



What are the naive fixed points of a non-naive smash product of a spectrum with itself?
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.


