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Added to derivator the explanation that Denis-Charles Cisinski had posted to the blog.
Zoran, I have made the material you had here the section "References", as this was mainly pointers to the literature. Please move material that you think you should go into other sections.
@Urs: Could you explain the last new part
One might argue that the notion of derivator is much weaker than the one of (?,1)-category (the latter providing much more interesting structures, whatever model you choose), but that is precisely the point; derivators provide a truncated version of higher category theory which gives us the language to characterize higher category theory using only usual category theory, without any emphasis on any particular model (in fact, without assuming we even know any).
a little further? Specifically the last sentence. Does that mean that we can define (oo,1)-categories using derivators rather than simplicial sets, or am I misreading it?
Possibly it's the same as Quillen model categories, Brown's cofibration categories or the like: they are presentations of (oo,1)-categories. But Urs probably has a better idea of what he meant.
I have a feeling that it's probably more sophisticated than Quillen's model categories, since why would Grothendieck come up with them if they're just the same thing as model categories?
They aren't the same thing as model categories, I meant to say the situation is the same, in that Urs is viewing them as presentations of (oo,1)-categories.
Model categories have a lot of cruft to ensure that the localisation of the category at the weak equivalences exists - and is locally small (this last point being the most important, because the general localisation of a large category (like sSet or Top) is not so). Derivators just say 'we don't care about locally large categories' and push on with using the useful maps that exist in that setup for applications, like the so-called six operations. (This latter was talked about by Grothendieck a lot in Pursuing Stacks, as a criterion for his developing theory. Note that PS was a few years before derivators were introduced by AG in 'Les Dérivateurs')
Ah, that makes sense. I'm still wondering if Urs meant something more profound with that statement though.
First of all, notice that this was written by Denis-Charles Cisinsik. I just moved it to the lab and edited it a bit.
But to reply to the question: these derivators see category with weak equivalences. They don't even see model categories. I think the statement is only that if the category with weak equivalence has any refinement to a model category, then the prederivator represented by it is actually a derivator.
@Harry If you can find a copy of PS it is worth glancing at the section where Ag talks about derivators. Denis-Charles mentions Heller's memoir and that is excellent. (It amuses me that Heller mentions AG's work but more or less says that he does not think it is related.) Both views could be that you need not just a nice homotopy structure on a category but also on ALL the categories of diagrams in that category AND there should be homotopy Kan extension like things linking everything together.
I added some material to derivator, including a way of motivating them which I think is more natural to someone who is already accustomed to higher category theory. I also added the definition, and a couple of questions.
Thanks, very nice.
I added a bunch of hyperlinks. Notably to homotopy Kan extension! :-)
I would also like to understand, as you indicate in your query boxes, how one comes up with the exact fine-tuning of the definition the way it is. I mean, the underlying idea is dead simple: we understand homotopy categories, we want homotopy limits, but these are just constructions in homotopy categories of diagram categories, so we simply carry all diagram categories around.
But what remains a bit mysterious (to me) is that the definition of derivator somehow manages to characterizes functorial families of homotopy categories of diagram categories without ever saying “localization” or the like.
Eventually I would like to include a lot of the theory of derivators on the lab, with proofs of some of the basic results. This seems especially desirable to me because so many of the existing references are in French, and thus more time-consuming to read for people like me who only learned enough French to squeak through their Ph.D. language exam. Of course this requires someone actually reading the French and writing the lab pages!
I don’t think it’s quite right to say that derivators characterize families of homotopy categories of diagram categories – I don’t know any reason why every derivator must be of the form Ho(C), although I also don’t know of any other examples. Rather it just characterizes some important aspects of the behavior of such families.
Does anybody knows on the relation between stable infinity categories of Lurie and derivators ? I mean when one takes an abelian category then one can make its derived category, for example the usual triangulated one, or one can take dg-version due Drinfel’d and Keller, or A-infinity derived category of Lyubashenko or one can take the derivator made out of triangulated categories for all possible diagrams (for a point one gets just the usual derived category). Some people say that in ccharacteristics zero all 4 approaches to enhanced derived categories are equivalent in characteristic zero. Now I understand that Lurie’s stable inf category, earlier stable mdoel categories, dg and A infinity give the same, but I do not see why would this all be equivalent to forming the associated derivator of derived categories for all diagrams, in characteristic zero.
I don’t know much about dg- or $A_\infty$ derived categories, but I’m pretty sure that Lurie’s stable $(\infty,1)$-category is (or can be) obtained in the usual way that $(\infty,1)$-categories are obtained from a model structure on chain complexes, and derivators are also obtained from model structures. I don’t know to what extent this has been written down formally, but I would be shocked if starting from a model category, producing an $(\infty,1)$-category, and then passing to a derivator would not give the same result as passing directly to the derivator–that’s just saying that limits and colimits in the $(\infty,1)$-categorical sense are modeled by homotopy limits and colimits in the usual model-categorical sense.
Ok, this is helpful, but we are talking also about the inverse in this stable aspect: is passing to the derivator in this standard way exactly containing the same information as passing to stable infinity. You said if I understood that stable (?) (infinity,1)-category has at least that much information as the derivator. Is it the same other way around, that nothing is lost from stable infinity to the derivator.
I don’t think it’s quite right to say that derivators characterize families of homotopy categories of diagram categories – I don’t know any reason why every derivator must be of the form $Ho(C)$, although I also don’t know of any other examples.
I see. Hm.
Is it the same other way around, that nothing is lost from stable infinity to the derivator.
I would be surprised if it were. Although I’ve been surprised before. I asked Goncalo Tabuada about that, but he didn’t know. I think from a derivator you can reconstruct a category enriched over the homotopy category of spaces, but I don’t see any way to reconstruct anything more refined than that.
It is true that if you have a map of $(\infty,1)$-categories which induces an equivalence on $Ho(Top)$-categories, then it was already an equivalence of $(\infty,1)$-categories. So if the $Ho(Top)$-enrichment is reconstructible from the derivator, as I believe I’ve been told, then it should be the case that passing from $(\infty,1)$-categories to derivators is at least conservative (equivalence-reflecting). But it might still be the case that two inequivalent $(\infty,1$)-categories could have equivalent derivators, if the equivalence is not induced by a map of $(\infty,1)$-categories.
Can you point out some constructions that would show that the definition of derivator is useful? Can one do anything with it that is not, or not as easily done with genuine models of oo,1-categories?
I feel pretty sure there is nothing you can do with them that you couldn’t do with $(\infty,1)$-categories. Some people certainly seem to feel that they are easier to work with. I’m still learning how to use both derivators and $(\infty,1)$-categories myself, and I’m not sure yet what I think.
It’s possible that Kate Ponto and I will have a good example in the near future. One of the reasons I want to learn about derivators is to see whether they can help us simplify something we’re trying to prove – that traces in triangulated bicategories are additive along distinguished triangles. This is a horrible mess to do at the level of triangulated categories, unsurprisingly, but it’s also a fairly horrible mess at the level of model categories, and in order to do it with $(\infty,1)$-categories we’d have to pick a model of $(\infty,2)$-categories, extend Kate’s notion of “shadow” to the $\infty$-context, and then do all of our work with a suitable kind of $\infty$-limits and colimits. On the other hand, it’s quite easy to say what a locally-derivator bicategory should be, since derivators form a 2-category, and I suspect that that may be enough structure to prove the theorem in the easiest way, as a convenient halfway point between model categories or $(\infty,1)$-categories and homotopy/triangulated categories.
Created pointed derivator and stable derivator, with links to+from stable (infinity,1)-category and triangulated category.
Added some discussion of how to compute homotopy Kan extensions to derivator.
Thanks, Mike. Very useful, indeed. I added briefly a link from “homotopy Kan extension” to “derivator”.
I’ve been adding some more material to pointed derivator, drawing from Heller’s work, but his proofs are often quite sketchy and I’m having trouble filling in the details.
Created basic localizer, and explained at derivator how the fact that nerve equivalences are $D$-equivalences reduces to Cisinski’s theorem that nerve equivalences are the smallest basic localizer.
Created homotopy exact square.
I streamlined the proof of the reflectivity of relative diagram categories at pointed derivator, and derived from it a construction of the extraordinary inverse image functors. I had written a claim that the suspension and loop functors could be defined in terms of the extraordinary inverse images, which I read somewhere (I don’t remember where), but now I don’t see how to do that, so I took it out for now.
Ok, I worked it out: $\Sigma = t^? s_*$ and $\Omega = s^! t_!$, where $s$ and $t$ are the two inclusions of the source and target into the interval category. It’s kind of annoying to have to give names to all the functors that appear in these calculations.
Hey Mike, I was reading the article derivator, and I have a (hopefully) quick question. In what sense is SSet the terminal object (I guess of the 2-category of prederivators)?
The universal property of the derivator Ho(sSet) that I know of is that it’s the reflection of the terminal prederivator into the 2-category of derivators and cocontinuous maps (“the free cocompletion of a point”). I wouldn’t be surprised if it were also the terminal object in the 2-category of “derivator toposes” and “geometric morphisms”, but AFAIK those words haven’t even been defined yet.
What if instead of sSet we take some other test category in the sense of Grothendieck ?
@Zoran: I’m reasonably certain that the answer you seek is in the preface of Astérisque 308 by Cisinski. Since it is not written in the definition-theorem-proof style, I’m having a rather hard time reading it, but I’ll put a translation of it up on the lab later (the French in the preface is a bit more complicated than in the general exposition). In particular, one of the main results (if I read this correctly) of the book is that every test category has a minimal model structure on its presheaf category equivalent to the Quillen model structure on SSet.
My question was does if that makes it satisfy the universal property mentioned by Mike above ? I surely expect yes, but would like Mike's expert opinion.
Any presentation of the same homotopy theory will give rise to the same derivator with the same universal property. I am by no means an expert on test categories, but my understanding is that, as Harry says, the whole point of them is that they present the same homotopy theory as sSet, and hence the same derivator.
In particular, it seems that we can derive everything from the axioms for a test category rather than a specific model (simplicial sets, cubical sets, etc.), which makes everything totally canonical. I haven’t gotten deep enough into it yet to say, but I suspect that there is a canonical description of the Joyal model structure for a general test category giving a homotopy theory equivalent to quasicategories. Note that this doesn’t have so much to do with derivators. However, it’s my hope that this framework gives a more canonical definition of the Joyal model structure in the generality of test categories rather than sSets
In the world of derivators, we don’t even need to say anything about test categories. We just know what it means to have colimits, in the derivator sense, and therefore to be a free cocompletion of the point, and that characterizes the derivator $Ho(\infty Gpd)$.
I haven’t yet been convinced that the generality of “test categories” is interesting, but then again, I didn’t used to care about derivators or basic localizers either, and they seem to be part of the same circle of ideas; so probably I’m just missing it so far.
You may find this kind of surprising since you’ve been a topologist for so long, but the use of simplices is really unmotivated at the beginning of AT.
Only if you have a teacher who doesn’t motivate them! (-:
Actually, I don’t think I’ve ever found the use of simplices to be “well-motivated” in any a priori way. The main justification I know for using simplices is that they work so well. But given that they do work so well, why should we put a lot of effort into developing a general theory of which other presheaf categories we could use that probably wouldn’t work so well? How does that make the use of simplices any more motivated?
Certainly there is interest in particular other presheaf categories, like cubical sets, which have different good properties. But under what circumstances would I care about an arbitrary test category, which probably has none of the good properties of simplicial or cubical sets? Or does the theory of general test categories somehow simplify the study of special cases like simplicial and cubical sets?
Or does the theory of general test categories somehow simplify the study of special cases like simplicial and cubical sets?
This is what I aim to find out by reading Astérisque 308. The introduction leads me to believe that we can develop homotopy theory in a canonical way making no choice of a test category, but at the moment, I’m only on page 60 and have not yet been introduced to test categories.
Perhaps a better way to say it is that it is independent, up to equivalence of (oo,1)-categories, of the choice of test category? I’m interested in the problem of when a functor between test categories induces such an equivalence (clearly not all do: the constant functor at an object doesn’t) - I tried asking on MO, but the question was a little confused. Cisinski gave me a nice answer, but not to the question I thought I was asking!
That gleaning is what I pretty much thought (and perhaps didn’t say very well): the homotopy theory you get (the whole lot, not just the model category) from a test category is independent of the choice of test category. I don’t know what you get making no choice of test category at all, which scans in my brain into ’don’t use a test category’ :P If I get what you mean, then you are saying one can work with an arbitrary test category A, but not pick one out to work with. In this case I agree again.
The theory of test categories is more fundamental than the theory of quasicategories.
I would draw a distinction between “more general” and “more fundamental.” The notion of essentially algebraic theory is more general than the theory of categories, but when we want to study categories, we study categories, not essentially algebraic theories. I realize this is a different situation, because all test categories present the same homotopy theory, but I couldn’t come up with a better analogy right now; my general point is that more generality is not necessarily useful.
In particular, I expect that quasicategories are a much better behaved and more useful model for $(\infty,1)$-categories than are presheaves on most test categories. For instance, they are a cartesian closed model structure, they admit a fully faithful nerve functor from 1-categories with a product-preserving left adjoint, and they are also the target of the homotopy coherent nerve from simplicially enriched categories. I wouldn’t expect most test categories to have all the properties like that. So what’s the advantage of working in the more general context of an arbitrary test category?
Hmm.. Perhaps a more correct statement would be that the theory of test categories allows us to build an (oo,1)-category theory that gives us a presentation of each (oo,1)-category (these are fibrant objects in the model structure corresponding to the Joyal model structure on a general test category) for each test category. We see that in this framework, quasicategories are a nice presentation of the underlying concept.
In particular, I expect that quasicategories are a much better behaved and more useful model for $(\infty,1)$-categories than are presheaves on most test categories. For instance, they are a cartesian closed model structure, they admit a fully faithful nerve functor from 1-categories with a product-preserving left adjoint, and they are also the target of the homotopy coherent nerve from simplicially enriched categories. I wouldn’t expect most test categories to have all the properties like that. So what’s the advantage of working in the more general context of an arbitrary test category?
I’d be very interested to see answers to those questions!
Well, maybe you can find out the answers and tell the rest of us! (-:
Any chance I could have a personal web to write up notes? I want to translate the definitions and statements of important theorems, as well as write up some of the preliminary material not included in the book. The exposition is pretty terse, and I think I’ll also write up explanations. I’d write them straight to the lab, but I’d rather not leave incomplete pages up like cylinder on a presheaf.
In other news more related to the original topic of this thread (-: I created pullback in a derivator and monomorphism in a derivator.
By the way, the identification of these derivator notions with the $(\infty,1)$-categorical ones depends on using $(\infty,1)$-categorical limits and colimits to construct a homotopy derivator. This is “obvious,” but I haven’t seen a proof of it anywhere, based for instance on the more explicit quasicategorical definitions of limit and colimit.
I started playing around at enriched derivator.
Harry, ask Urs about the personal lab if you need it.
I e-mailed Andrew, and he said that he’s submitting it to the steering committee.
just for the record, in case anyone finds this of interest (because there do not seem to be too many examples like this in the literature):
in the article
B. Toën, G. Vezzosi, $S^1$-Equivariant simplicial algebras and de Rham theory (arXiv)
all the homotopy theory is done in terms of derivators
Returning to this subject after a long hiatus, I have added to pointed derivator a characterization of some of the squares of “categories with specified zero objects” which are “homotopy exact” in the appropriate sense for pointed derivators. I haven’t seen a theorem of this sort in the parts of the literature that I’ve read, but it could be that I just haven’t gotten to it yet – has anyone else?
Being new to the entry, I got stuck for a moment at the point where you said “follows by (1) and (2)”. Because first I started scanning the entry for equations numbered (1) and (2).
So I have numbered the lemmas now and expanded that phrase to “follows by the first assumptiuon and by the second assumption and lemma 2”.
But check that I didn’t screw it up. I have to rush off now.
That looks good, thanks!
I added a refined version of the previous theorem, in which the second condition is made relative to a collection of zero-objects in the target whose zero-ness is automatic.
Kate Ponto and I have been forced to come up with these characterizations in order to prove the additivity of traces in stable bicategorical derivators. We’re not especially pleased with the way they look at the moment, but at least they’re algorithmically verifiable (modulo being able to check that a category has a contractible nerve, which is of course hard in general, but in practice the tiny categories that arise in concrete applications tend to have initial or terminal objects, which is easy to notice). And in fact I’ve written some code that verifies them, which is both quicker and less error-prone than doing it by hand. However, we’d still love to hear if anyone knows of better conditions, or of a better way to deal with derivator calculus in the pointed case.
And in fact I’ve written some code that verifies them, which is both quicker and less error-prone than doing it by hand.
I find this amazing. How are you writing this code? Is this a spin-off of your work on homotopy type theory?
I did originally try writing it in Agda, one of the proof assistants that uses dependent type theory and which we also use for HoTT, although the statements in question here are not themselves homotopical at all (just about sets). That would have had the advantage of actually producing a formal proof of the desired statements. But my lack of Agda-fu was slowing me down, as was the relative slowness of Agda itself (or perhaps that was also just my lack of fu), so I switched to a different language for now (namely, D, my new favorite “real-world” programming language). I may go back to reimplement it in Agda or Coq later, once things have settled down.
I am kind of excited about it, this being the first time I’ve actually written a computer program to help me prove something (other than HoTT things which are all about using the computer). It wasn’t really at all difficult; it’s just that this is the first time I’ve run into something where I thought it would be helpful. I may write some more about the experience, when this paper is all finished and I post about it at the Café.
The algorithmic nature of the calculus of homotopy exact squares demonstrated thereby also strengthens my belief that derivators are a useful tool in the arsenal of the $(\infty,1)$-category theorist. At one point we started comparing some proofs in derivator-language versus quasicategory-language, but I think we only got to the pullback lemma. Maybe we can go back to that sometime.
Perhaps I should also mention that my code only deals with the case of finite posets, which is much easier than general (even finite) categories. But it’s also sufficient for many purposes (since any “commutative diagram” one can actually fit on a finite sheet of paper must be a finite poset)… although insufficient for some others. Once the code is a little cleaner, I’ll post it somewhere.
I have added to derivator, pointed derivator, and stable derivator a reference to Jens Franke’s (never-published?) preprint Uniqueness theorems for certain triangulated categories with an Adams spectral sequence, which contains a development of stable derivators from the point of view of enrichment over pointed sets, apparently originally done independently of both Grothendieck and Heller. I have also expanded somewhat at pointed derivator the sketch of the construction of such an enrichment out of the other definition of pointed derivator (as a derivator with a zero object).
I’ve added to homotopy exact square another example, due to Groth. I’ve also added to pullback in a derivator a proof of the pasting lemma, and also a “detection lemma” due to Franke. All have straightforward proofs in the language of homotopy exact squares.
Added some historical remarks about the axioms to derivator.
In Cisinski’s 2003 paper, he talks about « dérivateurs faible à gauche », which seem to be the same thing as what the nLab article might call a complete semiderivator. What does « faible à gauche » mean here?
Just a point : the faible and the ’à gauche’ are probably not directly linked. In TAC he also talks about ones ‘à droite’. The contrast of the two definitions should answer your query. (Hopefully the language is relatively easy to decipher but ask if something needs translation.)
Yes, there are left/right versions, but the handedness is the opposite of what I expected, which was to have complete semiderivators to be “right” and cocomplete ones to be “left”!
But compare “right exactness” vs. “left exactness”, where the first refers to colimits and the second to limits?
Handedness is just a mess.
I edited the statement of Cisinski’s theorem on model categories to be more precise. If the model categories in question have functorial factorisations, then one can use the methods of DHKS [Homotopy limit functors etc.] to prove the theorem. (My impression is that Cisinski’s methods are essentially the same, but I have not read the paper.)
Edited the idea section of derivator to read as
An (∞,1)-category $C$ can be flattened into a 1-category $ho(C)$, called its homotopy category, by forgetting higher morphisms. A derivator is a refinement of $ho(C)$, in the sense that it retains enough information about $C$ for many purposes, like computing homotopy colimits and homotopy limits. Roughly speaking, the idea is to retain the data of the homotopy categories of all categories of diagrams in $C$, together with the induced functors between them. Since derivators can be studied using only ordinary 2-category theory, they are often practical in situations when one requires more information than the homotopy category retains, but not the whole (∞,1)-category $C$. […]
As usual, please correct if necessary. Also, a question: in this thread, Mike writes
I believe you can compute the mapping space between two objects [in a derivator] (as an object of the homotopy category of spaces)
This surprises me. Wouldn’t it mean that one doesn’t lose any information by passing from a cocomplete and complete infinity-category to the associated derivator?
It’s not quite true as stated. More precisely, if there is a mapping space, then its homotopy type is determined: Cisinski showed that for any derivator $\mathcal{D}$, the category $\mathcal{D}^1$ is a (say, left) $Ho(sSet)$-module. This shouldn’t be so surprising: just as one can compute $Set$-copowers using coproducts, one can compute $Ho(sSet)$-tensors using homotopy colimits. At the same time, I wouldn’t say that nothing is lost: an $(\infty, 1)$-category is not just a $Ho(sSet)$-enriched category, after all. But perhaps no “information” is lost, just “structure”.
Thanks, that makes sense.
Yes, what I meant was that if it exists (as it always does if you started with a model category or $(\infty,1)$-category), then you can compute it. (In general, it always exists as a presheaf on $Ho(Top)$, but it may not be representable — although I don’t know any examples where it isn’t.)
FWIW, I believe Heller constructed the $Ho(Top)$-module structure before Cisinski. And I would certainly consider “structure” to be a kind of “information”!
Can anyone give some specific references to constructions of such module structures?
@Mike #71
Ah yes, that makes sense. I still haven’t gotten around to looking through Heller’s monograph. As for “structure” vs “information”, what I meant was the faithfulness (or perhaps lack thereof) of the functor that sends a quasicategory to its prederivator.
@Karol
If you’re asking about the $Ho(sSet)$-action on $Ho(\mathcal{M})$ for a (complete and cocomplete) model category $\mathcal{M}$ (with functorial factorisations), see Theorem 5.5.3 in [Hovey, Model categories]. For derivators, see Théorème 5.22 in [Cisinski, Propriétés universelles et extensions de Kan dérivées].
@Karol It is worth looking also at Heller’s original form.
A. Heller, Homotopy in Functor Categories, Trans. Amer. Math. Soc., 272,, (1982), 185 –202.
and A. Heller, 1988, Homotopy Theories, number 383 in Memoirs Amer. Math. Soc., Amer. Math. Soc.
@Zhen it’s tricky to say what you mean by ’faithful’ when talking about $\infty$-categories, but I don’t think that functor is faithful for anything you might mean by it. It is conservative when restricted to locally presentable objects, though.
That seems odd. Perhaps I should clarify.
Let $\mathbf{Qcat}$ be the quasicategory-enriched category of quasicategories. Then there is a Yoneda-type functor $\mathcal{D} : \mathbf{Qcat} \to [\mathbf{Cat}^{op}, \mathbf{Cat}]$ defined by $Y \mapsto \tau_1 [-, Y]$, where $\tau_1 : \mathbf{Qcat} \to \mathbf{Cat}$ is the left adjoint of inclusion. This $\mathcal{D} (Y)$ is what I mean by the prederivator of $Y$. $\mathcal{D} : \mathbf{Qcat} \to [\mathbf{Cat}^{op}, \mathbf{Cat}]$ is definitely faithful as a functor of ordinary categories, because one can extract from $\mathcal{D} (Y)$ by “evil” means the original quasicategory $Y$. $\mathcal{D}$ is also homotopical in the sense of sending Joyal equivalences to componentwise equivalences. I think $\mathcal{D}$ is also homotopically faithful in the sense that $\operatorname{Ho} \mathbf{Qcat} \to \operatorname{Ho} [\mathbf{Cat}^{op}, \mathbf{Cat}]$ is faithful in the ordinary sense.
I’d like to believe that $\mathcal{D}$ is even homotopically conservative – but my homotopy-fu is too feeble to prove this.
Offhand, I don’t see any reason to believe it to be either “homotopically faithful” or conservative.
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.
Added reference
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. )
What exactly is missing? An identification of the homotopy 2-category of combinatorial model categories with that of locally presentable $(\infty,1)$-categories?
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?
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…
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.
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?
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.
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.
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.
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.
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.
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?
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.
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 #96: Okay, I started a draft, let’s see where it goes.
Excellent, thanks! Looking forward to it.
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