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I have recently started approaching Stable Homotopy Theory and came up with what is probably a rather naive question to ask, though it looks like I can not find references on it around.

Let $\mathcal{S}:=\mathsf{Ho}(\mathbf{Sp})$ be the stable homotopy category and let $F\colon\mathcal{S}\rightarrow\mathcal{S}$ be a triangulated self-equivalence of $\mathcal{S}$ such that $F(\mathbb{S})\cong\mathbb{S}$, where $\mathbb{S}$ is the sphere spectrum. Suppose also that $F$ is $\pi^s_\ast-$linear, where $\pi^s_\ast$ is the non-negatively graded ring given by the stable homotopy groups of spheres (see Schwede and Shipley's ``An uniqueness theorem for stable homotopy theory'', Definition 2.2).

  1. Is $F$ naturally isomorphic to $\text{Id}_\mathcal{S}$? If not, are there meaningful (sufficient and necessary) conditions on $F$ so that this becomes true?
  2. Does the fact of knowing that $F$ is the total left derived functor of a left Quillen functor $\mathbf{Sp}\rightarrow\mathbf{Sp}$ help in any way?

I originally thought that a positive answer to the first question came from Schwede and Shipley's paper (Theorems 5.1 and 5.3), but my reasoning was flawed and careless. Unfortunately, I don't know enough stable homotopy theory to really try to give any insight on the question or seriously think about it, so, please, feel free to be very verbose in answering.

Thanks in advance!

EDIT: Added a further hypothesis on $F$ that I missed to start with.

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    $\begingroup$ I'm not sure about automorphisms of the homotopy category of spectra, but if you would consider instead the automorphisms of the $\infty$-category ${\bf Sp}$ of spectra then the answer is pleasing: ${\bf Sp}$ is the free stable $\infty$-category generated by a single object (the sphere spectrum). As a result, the automorphisms of ${\bf Sp}$ are exactly the invertible spectra. Furthermore, any equivalence $f:F(\mathbb{S}) \stackrel{\simeq}{\to} \mathbb{S}$ extends to an equivalence $T_f: F \stackrel{\simeq}{\to} {\rm Id}$ in an essentially unique way. $\endgroup$ – Yonatan Harpaz Jun 6 '17 at 19:58
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    $\begingroup$ What is $-Id_S$? $\endgroup$ – Tom Goodwillie Jun 6 '17 at 20:00
  • $\begingroup$ @YonatanHarpaz Thanks for your answer! I'd still be interested in an answer about the homotopy category, but the infinity category approach is still interesting. Do you have references to suggest? Also, wouldn't your functor $F$ need to be colimit-preserving rather than just exact for the result about extending an equivalence $F(\mathbb{S})\rightarrow \mathbb{S}$ to hold? $\endgroup$ – Marco Vergura Jun 6 '17 at 20:12
  • $\begingroup$ @TomGoodwillie Good point, I do not know what I was thinking. I will edit the question in a second. $\endgroup$ – Marco Vergura Jun 6 '17 at 20:12
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    $\begingroup$ @MarcoVergura, indeed I was inaccurate: the free stable $\infty$-category on a single object is the $\infty$-category of finite spectra. However, ${\bf Sp}$ itself is the free stable presentable $\infty$-category. This means that (as you said) the above uniqueness holds only for colimit preserving endo-functors $F: {\bf S} \to {\bf S}$: for such functors any equivalence $F(\mathbb{S}) \to \mathbb{S}$ extends to an equivalence $F \to {\rm Id}$ in an essentially unique way. These universal properties of spectra are described in the first chapter of Jacob Lurie's Higher Algebra. $\endgroup$ – Yonatan Harpaz Jun 6 '17 at 20:49

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