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I'm sorry if this is well-known. I'm new to stable homotopy theory.

There are many models of spectra, and an answer to this MO post says most of them are equivalent, even equivalent while taking the symmetric monoidal structure (smash product) into account.

My question is motivated by wanting to know how these equivalences are established. There's a clean development of the stable $\infty$-category of spectra and the smash product thereof in Lurie's Higher Algebra, and both are characterized by universal properties, but it seems anachronistic to say that checking these universal properties is the way to establish the equivalence of different models.

Finally, even if one is able to establish that the different model categories of spectra with smash product (symmetric, orthogonal, what have have you) are equivalent, it seems like this doesn't easily construct a functor from one model to another--for instance, it's not obvious how to take two arbitrary Omega spectra (more or less the model in Higher Algebra) and spit out a pair of symmetric spectra while verifying that their smash products in either model are equivalent.

Is there a written account of:

  1. Historical efforts that failed, and in particular, attempted models for spectra that are not equivalent to modern models,
  2. A list of equivalent models,
  3. How we know the models are equivalent, and
  4. When known, functors between the models?
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  • $\begingroup$ The paper Peter May is referring to in his answer is implicitly the Mandell, May, Schwede, Shipley paper Denis mentions below. $\endgroup$ – Sean Tilson Dec 1 '16 at 11:18
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The paper you're looking for is "The stable homotopy category is rigid" by Stefan Schwede. It shows that an equivalence on the homotopy category level implies a Quillen equivalence on the model category level.

For monoidal results, check out "Monoidal Uniqueness of Stable Homotopy Theory" by Brooke Shipley. This paper has exactly the same universal property for the model category of spectra that you mention from Lurie, but of course many years before his work. I think it's safe to say these papers of Schwede and Shipley were what Lurie had in mind when he wrote the $\infty$-category version.

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    $\begingroup$ Actually that paper by Schwede proves something much stronger than what Lurie does in Higher Algebra. It is so strong that if I did not know that to be true I'd swear it is false. Really a great result. $\endgroup$ – Denis Nardin Nov 26 '16 at 22:18
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    $\begingroup$ Other papers that may be relevant is Model categories of diagram spectra by Mandell, May, Schwede and Shipley, and S-Modules and symmetric spectra by Schwede. Between those two you should find proofs of the equivalence of all the major models for spectra. $\endgroup$ – Denis Nardin Nov 26 '16 at 22:25

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