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Suppose $p$ is a prime, and consider the stabilization of the category $\text{Pro}(\mathcal{S}^{p-fc})$, using the notation of Lurie. This gives us a stable $\infty$-category $\text{Sp}(\text{Pro}(\mathcal{S}^{p-fc}))$. I would like to know if in the literature there is a description of stable homotopy theory in this setting. I am in particular interested in knowing what the appropriate notion of stable homotopy groups of p-profinite spaces is. I want such a notion to send a prespectrum object $\{X(n)\}$ over p-profinite spaces to the stable homotopy group of the levelwise materialization $\{\text{Mat}(X(n))\}$ of $\{X(n)\}$. However, would this behave well with respect to fibrations and their long-exact sequence? Ultimately, I want to know if $\text{Sp}(\text{Pro}(\mathcal{S}^{p-fc}))$ has all the basic properties of $\text{Sp}$ that we know and love!

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