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  • Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences.

  • Let $\mathbf Q$ be the corresponding $\infty$-category (in whatever foundations you prefer).

  • Let $Pres^L$ be the $\infty$-category of presentable $\infty$-categories and left adjoint functors.

  • The usual functor from relative categories to $\infty$-categories (modeled however you prefer) descends to a functor $N: \mathbf Q \to Pres^L$.

Variation A: I'm happy to work with simplicial combinatorial model categories rather than ordinary ones.

Variation B: It would also be interesting to know the answer to the following questions for combinatorial model $\infty$-categories and left $\infty$-Quillen functors. This is probably easier because one has more flexibility in this setting.

Question 1: Is $N: \mathbf Q \to Pres^L$ an equivalence of $\infty$-categories?

This functor is known to be essentially surjective -- for simplicial model categories this is in HTT, I think.

Question 2: Can the homotopical category $Q$ be refined to a model category?

There are size issues here; I'm happy with any way of handling them. I might suspect that something like Dugger's universal homotopy theories provide cofibrant resolutions. More specifically, I'd like to know:

Question 3: Are there model categories $C, D$ such that every left adjoint functor $NC \to ND$ is modeled by a left Quillen functor?

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    $\begingroup$ Isn't question 2 a famously hard problem? Like, listed on Hovey's open problem list and in the Vistas section of his book? I feel like many, many papers have been written in that direction (e.g. Bergner's papers on homotopy (co)limits of model categories, work by Chorny, etc). Is there some reason to believe the problem has become easier? $\endgroup$ – David White May 4 '18 at 4:00
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    $\begingroup$ Proposition 1.3.4.25 in Higher Algebra and Proposition A.3.7.8 in Higher Topos Theory may be useful for Question 1. The first one compares functor categories and ∞-categories, the second compares Bousfield localizations and ∞-localizations. Any combinatorial model category admits a presentation as a left Bousfield localization of simplicial presheaves on a small category. $\endgroup$ – Dmitri Pavlov May 4 '18 at 4:00
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In Two Models for the Homotopy Theory of Cocomplete Homotopy Theories, Karol Szumiło shows that there exist a fibration category structure on the (big) category $\mathcal C$ of homotopy cocomplete cofibration categories as well as a fibration category structure on the (big 1-)category $\mathbf{Co}$ of cocomplete quasicategories. I want to remark that fibrant categories are fibrant relative categories, as Lennant Meier showed. Moreover, Szumiło builds a nerve functor $\mathbf N_{\text{f}} \colon \mathcal C \to \mathbf{Co}$, the frames nerve, and proves it is a Barwick-Kan equivalence of relative categories.

It seems therefore natural to place a homotopy theory of combinatorial model categories in this frame. With Karol, mostly using results by Lurie, we can establish a fibration category structure on the (1-)category of presentable $\infty$-categories and colimit preserving functors. Making use of Karol's technology and of the results from Cellular categories by M. Makki and J. Rosický (and some other formal fact about accessibility), we get close to show the existence of a fibration category structure for combinatorial model categories in which all objects are cofibrant.

In particular, we get that the pullback of a left Quillen functor along an isofibration with some lifting properties (see Definition 1.9 of [1]) is a locally presentable relative category endowed with two cofibrantly generated factorisation systems. In order to apply Smith's theorem and get a combinatorial model category, it suffices to show that any trivial fibration is a weak equivalence. This I do not know how to prove it. If established, it should be reasonably easy to deal with path objects and hence obtain an equivalence of homotopy theories using the same frames nerve.

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I think it shouldn't be too hard to show using Dugger's technology that $N$ is full, i.e. essentially surjective on 1-morphisms. Suppose $C,D$ are combinatorial model categories and $f : N C \to N D$ is a cocontinuous $\infty$-functor. Then there is a small full sub-$\infty$-category $C' \subseteq N C$ such that $f$ is the left Kan extension of $f|_{C'}$. Using Dugger's technology we should be able to construct a span $C \leftarrow U C' / S \to D$ in $Q$, where $U C'$ is the universal model category on a simplicial-category presentation of $C'$ and $U C' / S$ is a localization thereof making $U C' / S \to C$ a Quillen equivalence (so that $U C' / S$ is a "presentation" of $C$ in Dugger's terminology). Thus, this span represents a morphism $C\to D$ in $\mathbf{Q}$ whose image under $N$ should be equivalent to $f$. Maybe such an argument could be refined to show that $N$ is an equivalence.

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  • $\begingroup$ This seems promising -- and just the sort of argument I was hoping for! Maybe one could get faithfulness by applying the same argument to functor categories $C \to D^{\Delta[n]}$ in a suitably functorial manner... perhaps modified to ask that the 1-cells be sent to weak equivalences. $\endgroup$ – Tim Campion May 5 '18 at 16:32
  • $\begingroup$ Note that being full is much, much weaker than being fully faithful for ∞-categories (it's the same difference as between effective epimorphisms and equivalences between homotopy types). It's believable that the functor is fully faithful, but I do not believe it is so easy to prove. $\endgroup$ – Denis Nardin May 8 '18 at 19:35
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    $\begingroup$ @DenisNardin Yes, certainly; I didn't mean to suggest it would be easy. As far as I had gotten was a vague thought of some kind of path-objects like Tim suggests. $\endgroup$ – Mike Shulman May 8 '18 at 23:15
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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.

By corollary 2.3.8 there, this implies that the 1-categorical localization of $\mathrm{CombModCat}$ is equivalent to the 1-categorical homotopy category of presentable derivators with left adjoint $\infty$-functors between them.

The latter clearly ought to be equivalent to the homotopy category of presentable $\infty$-categories, but maybe that remains open?

I am collecting what I have on this at http://ncatlab.org/nlab/show/Ho(CombModCat)

(Not aware of this thread here, I had just asked the same question again here.)

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