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The motivic $S^1$-stable homotopy category $SH^{S^1}(k)$ (where $k$ is a field that is often assumed to be perfect; yet one can probably take quite general base schemes here) is "intermediate" between the unstable motivic category $H(k)$ and $SH(k)$, and it appears to be rather "unpopular". What are the main references for it and for Quillen model structures underlying it? I have only find a mentioning of one model in Morel's "An introduction to $\mathbb{A}^1$-homotopy theory" (after Definition 4.2.2) and I don't know how to "work" with this definition.

One can probably define $SH^{S^1}(k)$ as the localization of the homotopy category of the corresponding spectra of simplicial Nisnevich sheaves (since the latter category is cellular); however, did anybody write this down before me? This approach is also related to the study of the functor $H(k)\to SH^{S^1}(k)$; cf. Connecting Quillen functors between motivic homotopy categories (of different "types"): references?

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  • $\begingroup$ Theorem 3.2.5 in arxiv.org/abs/1410.5699 constructs such a model category for free and proves many properties, including the existence of model structures on algebras over operads in such spectra. $\endgroup$
    – Dmitri Pavlov
    Commented Jul 18, 2017 at 18:11
  • $\begingroup$ I am sorry; this is an abstract statement that does not mention motivic spectra. Do you explain that it can be applied to $SH^{S^1}(k)$ somewhere? $\endgroup$
    – Mikhail Bondarko
    Commented Jul 18, 2017 at 18:36
  • $\begingroup$ Motivic spectra are considered in Example 3.4.2, which applies to R_n=(S^1)^{\wedge n} and constructs a model structure for your case. $\endgroup$
    – Dmitri Pavlov
    Commented Jul 18, 2017 at 18:59
  • $\begingroup$ I wonder whether anybody compares "their" definitions of motivic stable homotopy categories with any "standard" ones.:) It appears that you ignore this matter. $\endgroup$
    – Mikhail Bondarko
    Commented Aug 13, 2017 at 8:01
  • $\begingroup$ The definition that we use is the same as that of Morel and Voevodsky, which is as standard as it could ever get. What definition do you have in mind? $\endgroup$
    – Dmitri Pavlov
    Commented Aug 13, 2017 at 11:21

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