Okay, I see. So it’s good that you are looking into this in detail!

Have adjusted the footnote at Ho(CombModCat) and copied it also to derivator.

But please feel invited to further edit as need be.

]]>Re #100: Dugger’s theorem does prove homotopical essential surjectivity, equivalently, essential surjectivity for 2-localizations like in Renaudin’s paper.

But what do you want to do for 1-morphisms and 2-morphisms? More generally, how do you intend to prove homotopical fully faithfulness?

In terms of hammock localizations, one would need to show that the induced inclusion of hammock localizations is a weak equivalence.

This is hardly obvious since a hammock between two left proper combinatorial model categories can pass through non-left proper combinatorial model categories.

]]>Thanks, sounds good. Looking forward to having a look!

Yes, I was going by Renaudin’s arXiv version – have now added a remark on the discrepancies to the references here.

Regarding left properness: I’d say, since by Dugger’s theorem every combinatorial model category is Quillen equivalent to a left proper one, and since the statement is about localization at the Quillen equivalences, it should follow immediately that one can include this condition according to taste, without changing the statement.

But I have added a footnote, here, on this point.

]]>Wrote 20 pages so far, hoping to finish soon.

With respect to this, in the articles derivator and Ho(CombModCat) you systematically omit the left properness conditions from Renaudin’s results, whereas Renaudin appears to have added the left properness condition in the journal version of his article.

See, in particular, Theorem 3.4.4 in the journal version, which replaces ThQ^c from the arXiv version with ThQ^cpg, where “pg” means “propre à gauche”, i.e., left proper.

(By the way, the theorem numbers are different in the arXiv and journal versions, and you seem to be citing the arXiv version.)

With the machinery developed in my upcoming paper, it is actually quite easy to show that the inclusion of left proper combinatorial model categories into combinatorial model categories is a Dwyer–Kan equivalence of relative categories, but I cannot find such an equivalence statement in Renaudin’s paper.

So unless I am missing something, left properness conditions should be added to the statements of Renaudin’s results.

]]>Excellent, thanks! Looking forward to it.

]]>Re #96: Okay, I started a draft, let’s see where it goes.

]]>Thanks for the reply! Sorry for losing sight of this for a while.

It sounds all plausible, thanks. If this is a proof with all the nitty-gritty i-s dotted, then it would be great to have it written up, as this seems to be regarded an open question.

(E.g. David White, apparently having discussion with Mark Hovey in mind, reacted to the question on MO:q/304399 with the words: ” isn’t this question one that has been known to experts for years without a conclusive answer?”)

I guess I could make a full writeup of a proof with the hints you provided, though I’d rather not, as I am busy with some other proofs. But maybe you rather want to do it?

]]>Re #94: Yes, and see also Proposition A.3.7.6.

we’d get the commuting square in #92 for every left adjoint functor of presentable ∞\infty-categories that can be written as induced from a functor of presenting objects, under passage to presheaves and then localization. That’s probably always the case – what would be a reference?

Observe that the construction holds in a rather large generality: Given any presentable ∞-category C, choose any small ∞-subcategory C’⊂C such that the restricted Yoneda embedding C→PreSh(C’) is fully faithful. Then the induced functor PreSh(C’)[S^{−1}]→C given by the inclusion C’→C, where S is the accessible subcategory comprising morphisms inverted by the ∞-cocontinuous functor PreSh(C’)→C, is an equivalence of ∞-categories.

In particular, C’ can be harmlessly enlarged to a bigger small ∞-subcategory of C.

Furthermore, for sufficiently large regular cardinals λ (namely, ones for which C is λ-presentable), the small subcategory C_λ of λ-presentable objects does have this property.

Hence, we can proceed as follows. Choose a regular cardinal λ such that C and D are λ-presentable, and the right adjoint D→C is λ-accessible.

Then the left adjoint C→D preserves λ-presentable objects. Increasing λ as necessary, we also ensure that the right adjoint D→C preserves λ-presentable objects.

Now the full subcategories of λ-presentable objects C_λ and D_λ are the ∞-subcategories of C and D that we seek.

Indeed, we already know that the diagram

```
PSh(C_λ)_S → PSh(D_λ)_T
↓ ↓
C → D
```

commutes by construction since the horizontal maps are left adjoints and the vertical maps are equivalences of ∞-categories.

Rectifying C_λ and D_λ to E and F respectively completes the proof.

]]>We have a commutative square… as shown by Lurie

Okay, I guess you are referring to Prop. 4.2.4.4 (which is stated more explicitly for our situation on the bottom of p. 462). You probably want to use that the map shown there is natural in $S$, under defining $\mathcal{C} \coloneqq \mathfrak{C}(S)$.

Once we have that this is natural in $S$, we’d get the commuting square in #92 for every left adjoint functor of presentable $\infty$-categories that can be written as induced from a functor of presenting objects, under passage to presheaves and then localization. That’s probably always the case – what would be a reference?

]]>the left hom is the derived mapping simplicial set whose vertices are relative functors between relative categories

How did you get there from the simplicial set whose vertices are left Quillen functors? I suppose you mean ∞-colimit-preserving relative functors, but I don’t see what your argument is to derive this.

Sorry, forgot to say that the relative functors also must be homotopy left adjoint.

The point is that any left adjoint functor C→D that belongs to the λth step in the filtration restricts to a left adjoint functor C_λ→D_λ.

Vice versa, any left adjoint functor C_λ→D_λ can be promoted to a left adjoint functor C→D by taking λ-Ind objects on both sides.

This correspondence defines an equivalence of ∞-groupoids of invertible natural transformations between left adjoint functors C→D and left adjoint functors C_λ→D_λ.

The argument for Loc(C)→Loc(D) and Loc(C)_λ→Loc(D)_λ is analogous.

]]>G induces a simplicial Quillen adjunction sPSh(E)

{W_E} → sPSh(F){W_F} of left Bousfield localizations.This Quillen adjunction represents the original adjunction of presentable quasicategories.

That sounds plausible, but what’s the proof?

We have a commutative square of quasicategories (for the top row, pass to the underlying quasicategories first):

sPSh(E)_{W_E} → sPSh(F)_{W_F}

↓ ↓

C → D

The vertical maps are equivalences, as shown by Lurie in Higher Topos Theory (that’s how he proves that any presentable quasicategory can be presented by a simplicial model category).

But this means precisely that the top map is equivalent to the bottom map, i.e., the exhibited Quillen adjunction presents the functor C→D.

]]>Thanks!

In #89 you write:

This Quillen adjunction represents the original adjunction of presentable quasicategories.

That sounds plausible, but what’s the proof?

In #90 you say:

the left hom is the derived mapping simplicial set whose vertices are relative functors between relative categories

How did you get there from the simplicial set whose vertices are left Quillen functors? I suppose you mean $\infty$-colimit-preserving relative functors, but I don’t see what your argument is to derive this.

]]>Three years ago (#79), after the statement of Renaudin’s theorem, I had left a comment that this suggests that (at least) the homotopy 2-categories of (a) locally presentable derivators and (b) locally presentable (infinity,1)-categories (and, I suppose (c) of combinatorial model categories) are equivalent, but that a full proof remains to be given.

Any progress on this, since?

I would suggest the following direct proof.

First, define a functor Loc: CMC→PQC by sending a combinatorial model category to the relative category of its cofibrant objects and weak equivalences between cofibrant objects, which is then converted to a presentable quasicategory using any known functorial construction of a quasicategory from a relative category, e.g., the one in Meier’s paper.

As shown by Lurie, this functor is essentially surjective on objects.

Next, fix some C,D∈CMC; we want to show that hom(C,D)→hom(Loc(C),Loc(D)) is a weak equivalence. It suffices to show this for invertible 2-morphisms only, so we may take hom(-,-) to be the derived mapping simplicial set whose vertices are left Quillen functors respectively functors of quasicategories.

Both hom(C,D) and hom(Loc(C),Loc(D)) are filtered homotopy colimits of hom_λ(C,D) and hom_λ(Loc(C),Loc(D)) (where λ ranges over sufficiently large regular cardinals), comprising left adjoint functors that induce an adjunction between λ-compact objects.

Thus, it suffices to show that hom_λ(C,D)→hom_λ(Loc(C),Loc(D)) is a weak equivalence.

Passing to the induced adjunctions between λ-compact objects, we might as well show that hom(C_λ, D_λ) → hom(Loc(C)_λ, Loc(D)_λ) is a weak equivalence. Here the left hom is the derived mapping simplicial set whose vertices are relative functors between relative categories C_λ and D_λ.

For sufficiently large regular cardinals λ, Loc(C)_λ is weakly equivalent to Loc(C_λ).

Thus, the map becomes hom(C_λ, D_λ) → hom(Loc(C_λ), Loc(D_λ)) and is a weak equivalence because Loc induces a weak equivalence between derived mapping simplicial sets.

]]>For this functor to be full would mean that every adjunction of presentable quasicategories comes from a simplicial Quillen adjunction of simplicial model categories, up to equivalence.

Somebody must have thought about whether that’s true or not?

It is true. Suppose L: C→D and R: D→C is an adjunction between presentable quasicategories.

Choose a regular cardinal λ such that L and R restrict to an adjunction L_λ: C_λ→D_λ, R_λ: C_λ→D_λ between the categories of λ-presentable objects.

Next, rectify L_λ to a simplicial functor G: E→F between small simplicial categories.

G induces a simplicial Quillen adjunction sPSh(E)→sPSh(F) between the categories of simplicial presheaves on E and F.

Consider the classes of weak equivalences W_E⊂sPSh(E) and W_F⊂sPSh(F) given by those morphisms of presheaves that become equivalences in C respectively D once we apply the homotopy cocontinuous functor sPSh(E)→C (converting the domain to a quasicategory first), and likewise for F and D.

G induces a simplicial Quillen adjunction sPSh(E)_{W_E} → sPSh(F)_{W_F} of left Bousfield localizations.

This Quillen adjunction represents the original adjunction of presentable quasicategories.

]]>Ah, in #86 it’s of course wrong that the construction of adjunctions of quasi-categories from simplicial Quillen adjunctions is strictly 1-functorial, due to the (co)fibrant repacement involved.

So it’s a little more tricky than that.

]]>For this functor to be full would mean that every adjunction of presentable quasicategories comes from a simplicial Quillen adjunction of simplicial model categories, up to equivalence.

Somebody must have thought about whether that’s true or not?

If we drop the requirement that quasi-categories be presentable, then I see that Mazel-Gee 15 comments on this question on p. 2, saying it is ” presumably false”.

It wouldn’t surprise me if it were indeed false (without the presentability condition), but the reason given there – that simplicial adjunctions are “extremely rigid” – is not convincing.

(Namely: Sure they are rigid, but so are simplicial categories, and yet they present all presentable $\infty$-categories, up to equivalence. That’s the whole point of sSet-enriched model building, that it provides rectification, and rectifications are all “extremely rigid” by design, and yet they exist in large parts of homotopy theory.)

I haven’t yet found any text discussing the Quillen-presentability of adjunctions between presentable $\infty$-categories. But that may be just my ignorance of the literature.

]]>Let’s see. First to check that some model of simplicial localization gives a comparison 2-functor of the form

$2Loc_{QE}(CombModCat) \longrightarrow 2Ho(PresQuasiCat)$in the first place:

First, by Dugger’s theorem we know we can equivalently use left proper combinatorial model categories

$2Loc_{QE}(CombModCat) \;\simeq\; 2Loc_{QE}(LPropCombModCat)$and hence it is sufficient to produce a 2-functor of the form

$2Loc_{QE}(LPropCombModCat) \longrightarrow 2Ho(PresQuasiCat) \,.$Moreover, if we manage to make this 2-functor really be a model for simplicial localization, then it will respect Quillen equivalences, so that it’s sufficient to produce a 2-functor

$LPropCombModCat \longrightarrow 2Ho(PresQuasiCat) \,.$With the standard model for the right hand side, both sides are strict 2-categories. So let’s first see that we can get a 1-functor.

This follows, I’d think, by combining Lurie’s functor from simplicial Quillen adjunctions between simplicial model categories to adjunctions between the corresponding quasi-categories (here) with the functor from Prop. A.3 in Blumberg & Riehl 14.

So that’s the 1-functor part.

Now it should be immediate that this extends to a 2-functor in an evident way, taking natural transformations between left Quillen functors to natural transformations of left adjoints between quasi-category. Checking this requires sitting down and concentrating on how these 1-cells pass through these two functors. Haven’t done this yet.

If this last step works, then we have obtained, in conclusion, a model of simplicial localization that makes a 2-functor of the form

$2Loc_{QE}(CombModCat) \overset{\;L_{\mathrm{W}}}{\longrightarrow\;} 2Ho(PresQuasiCat) \,.$Moreover, we know this is essentially surjective on objects. Also, we know that it’s conservative, for what it’s worth. Now to show that its fully faithful. Hm…

]]>Yes. (Either that or, equivalently, an identification of the homotopy 2-categories of locally presentable derivators and locally presentable $(\infinity,1)$-categories.)

From “Higher Topos Theory” we know, I suppose, that the would-be comparison 2-functor $Ho(CombModCat) \overset{L_{\mathrm{W}}}{\rightarrow} Ho(Pres\infty Cat)$ is essentially surjective on objects and reflects equivalences (in that every equivalence on the right comes from a zig-zag of Quillen equivalences, according to Lurie’s Rem. A.3.7.7 in HTT.)

Anything else known, since then?

]]>What exactly is missing? An identification of the homotopy 2-category of combinatorial model categories with that of locally presentable $(\infty,1)$-categories?

]]>Three years ago (#79), after the statement of Renaudin’s theorem, I had left a comment that this suggests that (at least) the homotopy 2-categories of (a) locally presentable derivators and (b) locally presentable (infinity,1)-categories (and, I suppose (c) of combinatorial model categories) are equivalent, but that a full proof remains to be given.

Any progress on this, since?

(Checking, I see that Renaudin’s remarkable article has received no substantial citation yet. )

]]>Mention that sometimes (Der5) is omitted from the definition, instead a derivator is called “strong” if it satisfies (Der5).

]]>Added reference

- Coley, Ian,
*The theory of half derivators*, arxiv:2010.12057, 2020

I have added the remark that, as a corollary of Renaudin’s theorem, we at least have a comparison functor of the form

$Ho(PrDer) \longrightarrow Ho(Pr(\infty,1)Cat) \,.$ ]]>I changed the wording of suggested implications in the discussion around Renaudin’s theorem.

The section used to be titled

As a presentation of (∞,1)-categories

and I have changed that now to

Presentable derivators and combinatorial model categories

Then I expanded the concluding sentence

Notice that combinatorial model categories model precisely the locally presentable (∞,1)-categories, as discussed there.

as follows:

]]>Notice that locally presentable (∞,1)-categories are precisely those (∞,1)-categories that arise, up to equivalence of (∞,1)-categories as simplicial localiations of combinatorial model categories. Hence this theorem suggests that there is, at least, an equivalence of 2-categories between the 2-category of presentable derivators and the homotopy 2-category of the (∞,2)-category Pr(∞,1)Cat However, an actual proof of this seems to be missing.