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I am needing a reference for the following statement (in case it is true): Quillen functor between stable model categories preserve finite limits iff it preserves finite colimits.

For stable $\infty$-categories, I found this one, but for stable model categories, I couldn't find such a statement in standard references like those of Hovey and Hirschorn.

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    $\begingroup$ It's certainly not true if by "(co)limit" you mean a 1-categorical one in the model category. If you mean homotopy (co)limits, then it should follow from the $\infty$-categorical statement by passing to homotopy categories, right? $\endgroup$
    – Mike Shulman
    Commented Sep 27, 2023 at 16:39
  • $\begingroup$ I am sorry since my knowledge of $\infty$-categories is very limited: how are these two contexts related? $\endgroup$
    – Alexey Do
    Commented Sep 27, 2023 at 17:29
  • $\begingroup$ Every model category has a homotopy $\infty$-category, indeed that is sort of the point of a model category. And every Quillen functor between model categories induces a functor between homotopy $\infty$-categories. $\endgroup$
    – Mike Shulman
    Commented Sep 28, 2023 at 0:41

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The statement is false in its current form: there are left Quillen functors between stable model categories that do not preserve finite limits.

However, since ∞-categories are mentioned, presumably what is being meant is that the derived functor of a Quillen functor preserves finite homotopy (co)limits.

Without loss of generality we can treat the case of left Quillen functors. The left derived functor of a left Quillen functor preserves all (small) homotopy colimits.

Thus, it remains to show that the left derived functor of a left Quillen functor between stable model categories preserves finite homotopy limits. The latter property in its turn boils down to the preservation of the zero object and homotopy fibers.

The zero object is also the initial object, which is preserved by any left Quillen functor. The homotopy fiber of $f\colon X→Y$ can be computed as the (derived) loop object of the homotopy cofiber of $f$. The left derived functor of a left Quillen functor preserves homotopy cofibers, so it remains to show that it commutes with the derived loop object functor.

By definition of a stable model model category, the derived loop object functor is the inverse of the derived suspension functor, and the latter functor is computed using a homotopy pushout, which is preserved by the left derived functor of a left Quillen functor. This completes the proof.

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    $\begingroup$ Why does it remain to treat the case of finite homotopy limits? And a relevant question: in case our model category is stable, do homotopy colimits defined in a model-theoretic way) coincide with homotopy colimits in the context of triangulated cat, defined as "mapping telescope" (possibly by Neeman)? $\endgroup$
    – Alexey Do
    Commented Sep 27, 2023 at 18:31
  • $\begingroup$ @AlexeyDo: (1) Not sure if I understood the question. Small homotopy colimits were treated in the previous paragraph; for any left Quillen functor its left derived functor preserves all small homotopy colimits. (2) You cannot define homotopy colimits in triangulated categories in general. In some very special cases, such as (for example) homotopy coproducts and homotopy cofibers, the notions supplied by triangulated categories (via ordinary coproducts and distinguished triangles) do indeed coincide with the model-categorical notions. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 27, 2023 at 19:56
  • $\begingroup$ (1) you wrote "Thus, it remains to show that the left derived functor of a left Quillen functor between stable model categories preserves finite homotopy limits." $\endgroup$
    – Alexey Do
    Commented Sep 27, 2023 at 22:26
  • $\begingroup$ (2) ok I should say that under the assumption that our category admits arbitrary direct sums. Do you have any reference for this? $\endgroup$
    – Alexey Do
    Commented Sep 27, 2023 at 22:28
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    $\begingroup$ @AlexeyDo: (1) If we already proved that arbitrary small homotopy colimits are preserved by a derived left Quillen functor, then all what remains to be shown is that a derived left Quillen functor preserves finite homotopy limits. This will imply the statement in the main post: a derived Quillen functor between stable model categories preserves finite homotopy limits if and only if it preserves finite homotopy colimits. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 27, 2023 at 23:03

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