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Let $\mathrm{CombModCat}$ be the category of combinatorial model categories with left Quillen functors between them. By Dugger's theorem and the appendix of Lurie's "Higher Topos Theory" it ought to be true that its localization at the class of Quillen equivalences is equivalent to the homotopy category of presentable $\infty$-categories with left adjoint $\infty$-functors between them:

$$ \mathrm{CombModCat}\big[\text{QuillenEquivs}^{-1}\big] \;\simeq\; \mathrm{Ho}\big( \mathrm{Presentable}\infty\mathrm{Cat} \big) $$

Has this been made explicit anywhere, in citable form?

Something close is made explicit in

  • Olivier Renaudin, "Theories homotopiques de Quillen combinatoires et derivateurs de Grothendieck" (arXiv:math/0603339)

(thanks to Mike Shulman for the pointer!), where it is shown that the 2-categorical localization of the 2-category version of $\mathrm{CombModCat}$ is equivalent to the 2-category of presentable derivators with left adjoints between them.

[edit:] By corollary 2.3.8, this implies that the 1-categorical localization of $\mathrm{CombModCat}$ is equivalent to the 1-categorical homotopy category of presentable derivators with left adjoints between them.

The latter clearly ought to be equivalent to the homotopy category of $\mathrm{Presentable}\infty\mathrm{Cat}$, but is that made explicit anywhere?

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  • $\begingroup$ Will be collecting details at ncatlab.org/nlab/show/Ho(CombModCat) $\endgroup$
    – Urs Schreiber
    Commented Jul 6, 2018 at 18:54
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    $\begingroup$ Possible duplicate: mathoverflow.net/questions/299365/… $\endgroup$
    – David White
    Commented Jul 7, 2018 at 10:25
  • $\begingroup$ @DavidWhite thanks for the pointer! I hadn't seen that. Somebody should then point to Renaudin's article in the discussion there. $\endgroup$
    – Urs Schreiber
    Commented Jul 7, 2018 at 12:30
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    $\begingroup$ Sure, Urs. But, more pressingly, isn't this question one that has been known to experts for years without a conclusive answer? I remember talking to Mark Hovey about this question 8 years ago. If I recall correctly, there were some concerns about the set theory involved. I know it's popular nowadays to ignore such concerns, and maybe then the question becomes obvious. Anyway, I'm not aware of a reference for the claim in the question, but of course it must be true once the statement can be made precise. And now I brace for a litany of people coming on to tell me not to worry about set theory. $\endgroup$
    – David White
    Commented Jul 7, 2018 at 13:01
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    $\begingroup$ @DavidWhite, If you know more about what the experts know, you might want to make that an answer to let us all know. My impression is that Hovey has been asking for much more than just the homotopy category, namely for a "2-model category" structure on the category of model categories. And what's wrong with Renaudin's answer? $\endgroup$
    – Urs Schreiber
    Commented Jul 7, 2018 at 14:38

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The answer is affirmative and is provided by the paper Combinatorial model categories are equivalent to presentable quasicategories.

Among other things, it proves that the relative categories of combinatorial model categories, left Quillen functors, and left Quillen equivalences; presentable quasicategories, cocontinuous functors, and equivalences; and other models for homotopy locally presentable categories and homotopy cocontinuous functors are all weakly equivalent to each other as relative categories. This also implies the equivalence of underlying quasicategories.

Additionally, combinatorial model categories can be assumed to be left proper and/or simplicial. There is also an analogous statement for derivators, which must be formulated as an equivalence of (2,1)-categories due to the truncated homotopical nature of derivators.

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