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):


1. Monoidality: the functors $\Phi^H$ are monoidal (and triangulated).
2. 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).
3. 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:**


1. 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.
2. 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?
3. 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.)



"abstract" description of geometric fixed points functor
Specifically addressing question (2), the category of equivariant symmetric spectra for $G$ with respect to the sphere $S^{\rho\_G}$ for the regular representation is put together systematically in Mandell's paper "Equivariant symmetric spectra," Contemporary Mathematics, vol 346 (a preprint is available [on his webpage](http://pages.iu.edu/~mmandell/papers/).) In section 9.5 of the preprint version he discusses the geometric fixed point functor.


It is pretty straightforward in this framework. If your object $X$ consists of a sequence of $G \times \Sigma\_n$-spaces $X\_n$ with structure maps $S^{\rho\_G} \wedge X\_n \to X\_{1+n}$ (with appropriate equivariance conditions), one model for the $H$-geometric fixed points is as follows. Noting that we have a fixed-point identification $(S^{\rho\_G})^H = S^{\rho\_H}$, we take $(\Phi^H X)\_n = (X\_n)^H$, with structure maps
$$
S^{\rho\_H} \wedge (X\_n)^H = (S^{\rho\_G} \wedge X\_n)^H \to (X\_{1+n})^H.
$$
This makes it a very literal fixed-point application in this framework, and it is very different than the function spectrum $F\left(\Sigma^\infty\_G G/H\_+, X\right)$.


