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It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist compatible proper model structures for the stable homotopy categories of motivic $S^1$-spectra and $T$-spectra, so that the natural connecting functors (between all the four categories) are left Quillen ones. Could you suggest me a reference for this fact? I do not need tensor structures for motivic spectra; I would prefer to present $T$ as $A^1/G_m$. Besides, I need simplicial sheaves only because for embeddings $X\to Y\to Z$ of smooth varieties, $j\ge 0$, I want $(Y/X)\wedge T^j\to (Z/X)\wedge T^j\to (Z/X)\wedge T^j$ to be a cofibration sequence (in the stable $S^1$-homotopy category); does there exist a reference for this fact? Also, where could I find some more or less canonical notation for these functors and for their adjoints (I also need the functor from $T$-spectra to Voevodsky's motives and its adjoint, though only on the homotopy level)? I have looked at several sources on motivic homotopy theory (including some papers of Morel and Jardine); yet I was not able to find a single reference for all these things.

So, I do not need any proofs; I just want some basic facts and notation. Does something like a survey of these matters exist at the moment? In any case, I would be deeply grateful to anyone interested in this area for providing me with those references that she or he believes to be the most appropriate for answering my questions (certainly, it would be very nice if the numbers of sections would be included in this information).

Upd. Actually, I just need the following: for any (open) embeddings $X\to Y\to Z$ of smooth varieties, $j\ge 0$, I want $(Y/X)\wedge T^j\to (Z/X)\wedge T^j\to (Z/X)\wedge T^j$ to be a cofibration sequence in the categories of $S^1$-motivic spectra, $T$-motivic spectra, and $MGl$-modules. I also need the natural connecting functors between the corresponding triangulated categories and their adjoints.

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For the unstable case (i.e. simplicial presheaves and simplicial sheaves) the Quillen adjunctions are displayed and explained in math.uni-bielefeld.de/~rost/data/severitt.pdf, page 44 –  Elden Elmanto Sep 9 at 17:20

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