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In Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Ayoub defined the notion of a stable homotopical algebraic derivators; roughly, for a diagram of quasi-projective $S$-schemes ($S$: fixed base scheme) $(\mathscr{F},I)$, we associate a triangulated category $\mathbb{H}(\mathscr{F},I)$ satisfying various base change conditions. As far as I can understand, one of the main reason Ayoub introduces such notion is to define the motivic nearby cycles functor as a functor (originally, the motivic nearby cycles functor does not involve diagrams of schemes, it is defined in The motivic nearby cycles and the conservation conjecture as a homotopy colimit - in the sense of triangulated categories - of so-called motivic unipotent nearby cycles functors, so it is not really a functor at first).

At the end of chapter III, he wants to define the Log specialization system, so that he proves that the unipotent nearby cycles functor can be expressed in terms of the standard (see the beginning of chapter III) specialization system and the tensor product. This is clearly not true arbitrarily. The extra hypotheses he added is Hypothèse 3.6.40; one of the involved condition is $$(p_{\Delta})_{\#}f_* \overset{\sim}{\longrightarrow} f_*(p_{\Delta})_{\#}$$ that is, push-forwards commute with $\Delta$-homotopy colimits and he noted that this condition is often satisfied. I think that it should be the case for $\mathbf{SH}$ (the motivic stable homotopy category of Morel and Voevodsky, studied intensively in the next chapter of loc.cit). In fact, I suppose that it should hold true for all diagrams instead of just $\Delta$ but I do not see how this should be true.

Thank you in advance for your help.

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    $\begingroup$ It does preserve colimits for SH because its left adjoint f^* preserves compact objects, which boils down to the fact that the Nisnevich sheaf condition can be expressed with finite limits instead of Cech nerves (at least in the qcqs case). See eg Hoyois’s Lefschetz formula paper, Appendix C. Note that the statement is false for the Betti sheaf theory for non-proper morphisms though. $\endgroup$
    – crystalline
    Commented May 25, 2023 at 5:04
  • $\begingroup$ @crystalline, thanks, noetherian schemes of finite dimension are enough for my purpose. By the way, I think proposition C.12 part (3) of Hoyois has a mistake, if $f^*$ preserves compact generators, then its right adjoint $f_*$ only preserves direct sums. Combining with the fact that $\mathbf{SH}$ is stable, $f_*$ preserves finite limits, hence finite colimits, we deduce that $f_*$ preserves all colimits. $\endgroup$
    – Alexey Do
    Commented May 25, 2023 at 9:41

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