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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2010
    • (edited Mar 9th 2010)

    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.

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 9th 2010
    • (edited Mar 9th 2010)

    @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?

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 9th 2010

    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.

    • CommentRowNumber4.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 9th 2010

    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?

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 9th 2010
    • (edited Mar 9th 2010)

    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')

    • CommentRowNumber6.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 9th 2010

    Ah, that makes sense. I'm still wondering if Urs meant something more profound with that statement though.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2010

    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.

    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeMar 9th 2010

    @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.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 10th 2010
    @Urs,

    of course, my mistake.

    @Harry

    Note that (pre)derivators aren't about weak equivalences at all, but are more like (pre)sheaves of homotopy theories of diagrams. I found an article in JPAA a little while back that shows the 2-category of combinatorial model categories, localised (in the sense of Pronk) at the Quillen equivalences, is equivalent to a full sub-2-category of that of derivators. This full sub-2-category consisted of the derivators with a 'presentation' by the image of a small category under a Yoneda-like embedding, Cat -> Der. I'll track down the link and add it to the page of derivators.
    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeApr 28th 2010

    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.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2010
    • (edited Apr 28th 2010)

    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.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeApr 29th 2010

    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.

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeApr 30th 2010

    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.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeApr 30th 2010

    I don’t know much about dg- or A A_\infty derived categories, but I’m pretty sure that Lurie’s stable (,1)(\infty,1)-category is (or can be) obtained in the usual way that (,1)(\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 (,1)(\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 (,1)(\infty,1)-categorical sense are modeled by homotopy limits and colimits in the usual model-categorical sense.

    • CommentRowNumber15.
    • CommentAuthorzskoda
    • CommentTimeApr 30th 2010

    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.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2010

    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)Ho(C), although I also don’t know of any other examples.

    I see. Hm.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeApr 30th 2010

    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 (,1)(\infty,1)-categories which induces an equivalence on Ho(Top)Ho(Top)-categories, then it was already an equivalence of (,1)(\infty,1)-categories. So if the Ho(Top)Ho(Top)-enrichment is reconstructible from the derivator, as I believe I’ve been told, then it should be the case that passing from (,1)(\infty,1)-categories to derivators is at least conservative (equivalence-reflecting). But it might still be the case that two inequivalent (,1(\infty,1)-categories could have equivalent derivators, if the equivalence is not induced by a map of (,1)(\infty,1)-categories.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2010

    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?

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeMay 1st 2010

    I feel pretty sure there is nothing you can do with them that you couldn’t do with (,1)(\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 (,1)(\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 (,1)(\infty,1)-categories we’d have to pick a model of (,2)(\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 (,1)(\infty,1)-categories and homotopy/triangulated categories.

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeMay 1st 2010
    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeMay 1st 2010

    Added some discussion of how to compute homotopy Kan extensions to derivator.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2010

    Thanks, Mike. Very useful, indeed. I added briefly a link from “homotopy Kan extension” to “derivator”.

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeMay 27th 2010

    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.

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2010

    Created basic localizer, and explained at derivator how the fact that nerve equivalences are DD-equivalences reduces to Cisinski’s theorem that nerve equivalences are the smallest basic localizer.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeMay 30th 2010
    • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeJun 5th 2010

    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.

    • CommentRowNumber27.
    • CommentAuthorMike Shulman
    • CommentTimeJun 6th 2010

    Ok, I worked it out: Σ=t ?s *\Sigma = t^? s_* and Ω=s !t !\Omega = s^! t_!, where ss and tt 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.

    • CommentRowNumber28.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 10th 2010

    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)?

    • CommentRowNumber29.
    • CommentAuthorMike Shulman
    • CommentTimeJun 10th 2010

    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.

    • CommentRowNumber30.
    • CommentAuthorzskoda
    • CommentTimeJun 10th 2010
    • (edited Jun 10th 2010)

    What if instead of sSet we take some other test category in the sense of Grothendieck ?

    • CommentRowNumber31.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 10th 2010
    • (edited Jun 10th 2010)

    @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.

    • CommentRowNumber32.
    • CommentAuthorzskoda
    • CommentTimeJun 10th 2010

    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.

    • CommentRowNumber33.
    • CommentAuthorMike Shulman
    • CommentTimeJun 10th 2010

    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.

    • CommentRowNumber34.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 10th 2010
    • (edited Jun 10th 2010)

    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

    • CommentRowNumber35.
    • CommentAuthorMike Shulman
    • CommentTimeJun 11th 2010

    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(Gpd)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.

    • CommentRowNumber36.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 11th 2010

    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.

    • CommentRowNumber37.
    • CommentAuthorMike Shulman
    • CommentTimeJun 11th 2010

    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?

    • CommentRowNumber38.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 11th 2010

    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.

    • CommentRowNumber39.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 11th 2010

    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!

    • CommentRowNumber40.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 11th 2010
    • (edited Jun 11th 2010)
    @David: The following is based on information I've gleaned from the introduction to Astérisque 308 (I call it this because the real title is much longer) and Jardine's paper "Categorical Homotopy Theory" as well as a few of my own inferences that seem to follow a fortiori. I don't claim to know any of this in any depth:

    The theory of test categories is more fundamental than the theory of quasicategories. This is not in the context of derivator theory. In particular, it says that there exists a model of (oo,1)-categories (to contrast with quasicategories, which are simplicial models) on the category PSh(A) for any test category A (this is an easy corollary of the fact that any model structure on SSet has a Quillen equivalent model structure on PSh(A) for any test category A, and that the theory of quasicategories is given on SSet by the Joyal model structure). What this means, apparently, is that we can switch between a huge variety of models for (oo,1)-categories by switching the underlying test category A.

    Further, the fact that Quillen equivalences are presentations of (oo,1) equivalences should be independent of our choice of test category. So in a sense, the theory of test categories gives the underlying theory of (oo,1) as a special case and makes it presentation independent as far as we're giving a presheaf category model of (oo,1) categories.
    • CommentRowNumber41.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 11th 2010

    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.

    • CommentRowNumber42.
    • CommentAuthorMike Shulman
    • CommentTimeJun 11th 2010

    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 (,1)(\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?

    • CommentRowNumber43.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 11th 2010
    • (edited Jun 11th 2010)

    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 (,1)(\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!

    • CommentRowNumber44.
    • CommentAuthorMike Shulman
    • CommentTimeJun 11th 2010

    Well, maybe you can find out the answers and tell the rest of us! (-:

    • CommentRowNumber45.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 12th 2010

    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.

    • CommentRowNumber46.
    • CommentAuthorMike Shulman
    • CommentTimeJun 12th 2010

    In other news more related to the original topic of this thread (-: I created pullback in a derivator and monomorphism in a derivator.

    • CommentRowNumber47.
    • CommentAuthorMike Shulman
    • CommentTimeJun 12th 2010

    By the way, the identification of these derivator notions with the (,1)(\infty,1)-categorical ones depends on using (,1)(\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.

    • CommentRowNumber48.
    • CommentAuthorMike Shulman
    • CommentTimeJun 13th 2010

    I started playing around at enriched derivator.

    • CommentRowNumber49.
    • CommentAuthorzskoda
    • CommentTimeJun 14th 2010

    Harry, ask Urs about the personal lab if you need it.

    • CommentRowNumber50.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 14th 2010

    I e-mailed Andrew, and he said that he’s submitting it to the steering committee.

    • CommentRowNumber51.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2010
    • (edited May 11th 2011)

    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 1S^1-Equivariant simplicial algebras and de Rham theory (arXiv)

    all the homotopy theory is done in terms of derivators

    • CommentRowNumber52.
    • CommentAuthorMike Shulman
    • CommentTimeMay 11th 2011

    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?

    • CommentRowNumber53.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2011

    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.

    • CommentRowNumber54.
    • CommentAuthorMike Shulman
    • CommentTimeMay 12th 2011

    That looks good, thanks!

    • CommentRowNumber55.
    • CommentAuthorMike Shulman
    • CommentTimeMay 14th 2011

    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.

    • CommentRowNumber56.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2011

    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?

    • CommentRowNumber57.
    • CommentAuthorMike Shulman
    • CommentTimeMay 17th 2011

    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 (,1)(\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.

    • CommentRowNumber58.
    • CommentAuthorMike Shulman
    • CommentTimeMay 18th 2011

    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.

    • CommentRowNumber59.
    • CommentAuthorMike Shulman
    • CommentTimeMay 26th 2011

    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).

    • CommentRowNumber60.
    • CommentAuthorMike Shulman
    • CommentTimeJun 8th 2011

    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.

    • CommentRowNumber61.
    • CommentAuthorMike Shulman
    • CommentTimeJun 9th 2011

    Added some historical remarks about the axioms to derivator.

    • CommentRowNumber62.
    • CommentAuthorZhen Lin
    • CommentTimeApr 30th 2013

    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?

    • CommentRowNumber63.
    • CommentAuthorTim_Porter
    • CommentTimeApr 30th 2013
    • (edited Apr 30th 2013)

    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.)

    • CommentRowNumber64.
    • CommentAuthorZhen Lin
    • CommentTimeApr 30th 2013

    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”!

    • CommentRowNumber65.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 30th 2013

    But compare “right exactness” vs. “left exactness”, where the first refers to colimits and the second to limits?

    • CommentRowNumber66.
    • CommentAuthorMike Shulman
    • CommentTimeMay 1st 2013

    Handedness is just a mess.

    • CommentRowNumber67.
    • CommentAuthorZhen Lin
    • CommentTimeMay 5th 2013

    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.)

    • CommentRowNumber68.
    • CommentAuthoradeelkh
    • CommentTimeAug 25th 2014

    Edited the idea section of derivator to read as

    An (∞,1)-category CC can be flattened into a 1-category ho(C)ho(C), called its homotopy category, by forgetting higher morphisms. A derivator is a refinement of ho(C)ho(C), in the sense that it retains enough information about CC 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 CC, 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 CC. […]

    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?

    • CommentRowNumber69.
    • CommentAuthorZhen Lin
    • CommentTimeAug 25th 2014
    • (edited Aug 25th 2014)

    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 𝒟\mathcal{D}, the category 𝒟 1\mathcal{D}^1 is a (say, left) Ho(sSet)Ho(sSet)-module. This shouldn’t be so surprising: just as one can compute SetSet-copowers using coproducts, one can compute Ho(sSet)Ho(sSet)-tensors using homotopy colimits. At the same time, I wouldn’t say that nothing is lost: an (,1)(\infty, 1)-category is not just a Ho(sSet)Ho(sSet)-enriched category, after all. But perhaps no “information” is lost, just “structure”.

    • CommentRowNumber70.
    • CommentAuthoradeelkh
    • CommentTimeAug 25th 2014

    Thanks, that makes sense.

    • CommentRowNumber71.
    • CommentAuthorMike Shulman
    • CommentTimeAug 25th 2014

    Yes, what I meant was that if it exists (as it always does if you started with a model category or (,1)(\infty,1)-category), then you can compute it. (In general, it always exists as a presheaf on Ho(Top)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)Ho(Top)-module structure before Cisinski. And I would certainly consider “structure” to be a kind of “information”!

    • CommentRowNumber72.
    • CommentAuthorKarol Szumiło
    • CommentTimeAug 26th 2014

    Can anyone give some specific references to constructions of such module structures?

    • CommentRowNumber73.
    • CommentAuthorZhen Lin
    • CommentTimeAug 26th 2014

    @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)Ho(sSet)-action on Ho()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].

    • CommentRowNumber74.
    • CommentAuthorTim_Porter
    • CommentTimeAug 26th 2014

    @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.

    • CommentRowNumber75.
    • CommentAuthorMike Shulman
    • CommentTimeAug 26th 2014

    @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.

    • CommentRowNumber76.
    • CommentAuthorZhen Lin
    • CommentTimeAug 26th 2014
    • (edited Aug 26th 2014)

    That seems odd. Perhaps I should clarify.

    Let Qcat\mathbf{Qcat} be the quasicategory-enriched category of quasicategories. Then there is a Yoneda-type functor 𝒟:Qcat[Cat op,Cat]\mathcal{D} : \mathbf{Qcat} \to [\mathbf{Cat}^{op}, \mathbf{Cat}] defined by Yτ 1[,Y]Y \mapsto \tau_1 [-, Y], where τ 1:QcatCat\tau_1 : \mathbf{Qcat} \to \mathbf{Cat} is the left adjoint of inclusion. This 𝒟(Y)\mathcal{D} (Y) is what I mean by the prederivator of YY. 𝒟:Qcat[Cat op,Cat]\mathcal{D} : \mathbf{Qcat} \to [\mathbf{Cat}^{op}, \mathbf{Cat}] is definitely faithful as a functor of ordinary categories, because one can extract from 𝒟(Y)\mathcal{D} (Y) by “evil” means the original quasicategory YY. 𝒟\mathcal{D} is also homotopical in the sense of sending Joyal equivalences to componentwise equivalences. I think 𝒟\mathcal{D} is also homotopically faithful in the sense that HoQcatHo[Cat op,Cat]\operatorname{Ho} \mathbf{Qcat} \to \operatorname{Ho} [\mathbf{Cat}^{op}, \mathbf{Cat}] is faithful in the ordinary sense.

    I’d like to believe that 𝒟\mathcal{D} is even homotopically conservative – but my homotopy-fu is too feeble to prove this.

    • CommentRowNumber77.
    • CommentAuthorMike Shulman
    • CommentTimeAug 26th 2014

    Offhand, I don’t see any reason to believe it to be either “homotopically faithful” or conservative.

    • CommentRowNumber78.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2018

    Added a citation for non-strong derivators not arising from homotopy theory or higher categories.

    diff, v60, current

    • CommentRowNumber79.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2018

    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.

    diff, v61, current

    • CommentRowNumber80.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2018

    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)Ho(Pr(,1)Cat). Ho(PrDer) \longrightarrow Ho(Pr(\infty,1)Cat) \,.

    diff, v62, current

    • CommentRowNumber81.
    • CommentAuthorMike Shulman
    • CommentTimeOct 26th 2020

    Added reference

    diff, v64, current

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