I wonder if every simplicial $S^1$-spectrum stable equivalent to an abelian group simplicial $S^1$-spectrum? It seems that I could use the stable Dold-Kan on the heart and use Postnikov towers?

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    $\begingroup$ If I understand correctly your question, the category of simplicial $S^{1}$-spectrum is the stable category with respect to the suspension functor i.e., it is the category of spectra. The category of abelian group simplicial $S^{1}$-spectrum is the stabilization category of the model category of simplicial abelian groups with respect to the "suspension" functor, this category is the category of $HZ$-spactrum i.e the category of spectrum which have the structure of $HZ$-module. The answer to your question (if my interpretation is correct) is no. $\endgroup$ – GSM Mar 11 at 13:07
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    $\begingroup$ More precisely (if GSM's interpretation of your question is correct, I'm also not 100% sure what your objects are), what goes wrong with your argument sketch is that the objects in the heart are $HZ$-modules, but the way they're glued together (the k-invariants in the postnikov tower) doesn't respect that, i.e. there are non-$HZ$-linear maps between $HZ$-modules. $\endgroup$ – Achim Krause Mar 11 at 13:20
  • $\begingroup$ @Achim Krause I completely agree with you. $\endgroup$ – Nanjun Yang Mar 12 at 3:22

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