I'm sure this must be well known, but I could not find any references.

*My basic question is:* Are there "abstract" descriptions of the geometric fixed point functors in equivariant stable homotopy theory, not tied to the Lewis-May construction of the stable quivariant homotopy category using a complete universe?

Of course this is rather vague. I will give a bit of background, and then formulate some more concrete questions at the end.

**Background**

Let $G$ be a finite group. I denote the stable equivariant homotopy category with respect to $G$ by $SH(G).$ Without getting into too many details, it relates to the category of spaces with $G$-action in a similar way that the ordinary stable homotopy category $SH$ relates to ordinary spaces. It is a symmetric monoidal, triangulated category. One of its most important features is that all the representation spheres of $G$ are invertible, not just the ordinary ones. Of course, $SH(G)$ by itself is "not good enough", we also need point set level models for it, and there are several.

For any subgroup $H$ of $G,$ there exists a functor $\Phi^H: SH(G) \to SH,$ called the "geometric fixed points functor". These functors have the following properties (not an exhaustive list I'm sure):

- Monoidality: the functors $\Phi^H$ are monoidal (and triangulated).
- Compatiblity with space-level fixed points: if $X$ is a $G$-space and $\Sigma^\infty_G$ denotes the suspension spectrum, then we have $\Sigma^\infty X^H \simeq \Phi^H \Sigma^\infty_G X$ (where $X^H$ is the subspace of fixed points, viewed as an ordinary space, and $\Sigma^\infty$ denotes the ordinary suspension spectrum).
- Whitehead theorem: a map $f: E \to F$ in $SH(G)$ is a weak equivalence if and only if all of the $\Phi^H(f)$ are.

As I said before, the only construction of these functors I could find is using a specific model for $SH(G),$ and it is not clear to me how to transfer this to others.

**Here are some concrete questions:**

The space-level fixed point functor is given by $X^H = S(G/H, X),$ where $S$ denotes the enrichement of $G$-spaces in ordinary spaces (i.e. if $FX$ denotes the $G$-space $X$ with $G$-action forgotten, then $S(X, Y)$ is the subspace of $S(FX, FY)$ consisting of equivariant maps, where $S(FX, FY)$ is the usual mapping space). As it so happens, $SH(G)$ is a spectral category (provided with mapping spectra), let $\mathcal{S}$ denote the enrichment. Then the most analogous definition would be $\Phi'^H(E) = \mathcal{S}(\Sigma^\infty_G G/H_+, E),$ but I suspect this is not the same as $\Phi^H.$ Indeed it satisfies

~~(2) and~~^{edit: that's not actually true}(3) but I do not see why it should satisfy (1). However, this is the kind of description I would hope for.A different description of $SH(G)$ is in terms of symmetric spectra on $G$-spaces, with the regular representation being inverted (I believe). How can the geometric fixed point functors be described in this language?

Yet another description is as follows: let $\mathcal{O}$ denote the full spectral subcategory of $SH(G)$ with objects $\Sigma^\infty_G G/H_+$ (this is slightly sloppy). One can consider the category of "spectral presheaves on $\mathcal{O}$", $\mathbf{Pre}(\mathcal{O}, SH)$ (this notation is even sloppier). As it turns out, this category is equivalent to $SH(G)$. Again, how does the geometric fixed points functor look like in this framework? (Of course this question isn't really well-posed, since the answer depends on how one describes $\mathcal{O}$ and $SH$ in the first place.)

guessis that it is characterized by the properties: (i) it preserves homotopy colimits, and (ii) it makes the diagram commute between G-spaces, spaces, G-spectra, and spectra- i.e. geometric fixed points of a suspension spectrum are suspension spectra of geometric fixed points $\endgroup$ – Dylan Wilson Apr 24 '14 at 14:08