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People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask for help in "navigating" in this matter. So, which advantages does each of the model have? Does any "guide" for this subject exist?

In particular, is the category of symmetric (motivic) $T$-spectra endowed with the injective (="levelwise"?) model structure a "reasonable" monoidal model category?

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The injective model structure is monoidal (satisfies the pushout product axiom), see Hornbostel's paper "Localizations in motivic homotopy theory", Thm 1.9 and Lemma 1.10. The projective model structure is also monoidal. See Hovey's "Spectra and symmetric spectra in general model categories", where he introduces the projective model structure.

Later, Hornbostel introduced the positive model structure, a modification of the projective model structure that, like the case for ordinary (non-motivic) symmetric spectra, avoids Gaunce Lewis's obstruction regarding commutative monoids. Specifically, this model structure is Quillen equivalent to the projective (and injective), but now the unit (sphere spectrum) is not cofibrant. Furthermore, there is now a transferred model structure on commutative ring spectra, which Lewis proved cannot exist for the projective model structure. For non-motivic spectra, the positive model structure was introduced in Mandell-May-Schwede-Shipley ("diagram spectra"). If you google, you can find old drafts online where I thought I'd done the positive analogue of Hovey's general-spectra paper, but Hornbostel beat me to it by several years.

There is also a positive flat model structure on symmetric spectra (see Shipley's paper "A convenient model category for commutative ring spectra") that has the extra property that cofibrant commutative monoids forget to cofibrant objects in the underlying category. This property is generally not true for the non-flat variant. I am not sure if this has been constructed for motivic symmetric spectra, but it surely must exist. An analogous program for equivariant spectra was carried out in the thesis of Martin Stolz. If this has been done for motivic, it would probably be in the paper of Pavlov and Scholbach.

If you have other questions not answered here, please let me know and I'll edit to try and answer. I agree it would be good to have a unified place with lists of pros/cons of the various structures, and that may as well be here. All these model structures are combinatorial, stable, and proper. What other properties do you care about?

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  • $\begingroup$ Thank you very much! This appears to imply that I have to use some cofibrant replacements for my purposes. So, I would like these replacements and related matters to "behave nicely". Probably, properness will help me. Are there any properties of model categories of similar nature?:) I will possibly modify my question later when I will understand what I need in more detail. $\endgroup$ – Mikhail Bondarko Apr 28 '17 at 13:08
  • $\begingroup$ Sometimes it has been useful for me to know that the domains of the generating (trivial) cofibrations are cofibrant. When the model category is combinatorial, Barwick coined the term "tractable" for this condition. I believe it is satisfied in all the examples above (the only one I'm not 100% sure on is the positive model structure, since the sphere is not cofibrant). Another property that Hovey often uses, for a monoidal model category, is that "cofibrant objects are flat", i.e. for every cofibrant $X$, $X\wedge -$ preserves weak equivalences. This should be true in all the models above. $\endgroup$ – David White Apr 29 '17 at 14:30
  • $\begingroup$ Oh, and I have though that this flatness property is one of the axioms.(: $\endgroup$ – Mikhail Bondarko Apr 29 '17 at 15:19
  • $\begingroup$ It's related to the unit axiom, but stronger. The unit axiom is only about the weak equivalence $QS \to S$ where $S$ is the unit, and $QS$ its cofibrant replacement. It implies that the unit on the model category level is also a unit on the homotopy category level. Many have been able to get away without the unit axiom, e.g. Schwede-Shipley in "Algebras and Modules in monoidal model categories." Fernando Muro has also done a lot of work about units in monoidal model categories. See also mathoverflow.net/questions/85329/… $\endgroup$ – David White Apr 29 '17 at 19:29
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    $\begingroup$ @HeleneSigloch: Actually, it turns out I was wrong. I'll edit my answer. Hornbostel proves it's monoidal. The paper is "Localizations in motivic homotopy theory", and this result is mentioned on page 2, and in thm 1.9 and lemma 1.10 $\endgroup$ – David White Jul 14 '17 at 6:22

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