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Atrium > Mathematics, Physics & Philosophy: A cartesian model category of (oo,oo)-categories

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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 26th 2011
    • (edited Jul 26th 2011)

    So, I’ve been working on some things for the past few months, and I was wondering if anybody here might find this interesting: Using Joyal and Cisinski’s proposed generating set for the model structure on (oo,oo)-categories, I have shown that this model structure is cartesian and has many of the properties one might want (every (oo,oo)-category is enriched in (oo,oo)-categories, a well-behaved “suspension” functor with an adjoint, a comparison with Rezk’s (oo,n)-categories, and a few other things) . I haven’t written it up yet, but I have proved all of the important results in my notebook. I’m not sure what the current state of the art is regarding things like higher Grothendieck constructions etc, especially in this framework, but I think that this area of research is wide open.

    Not to sound silly, but I was wondering if I could discuss it a little more in private with someone on the n-team and perhaps coauthor something, if that sounds anodyne to one of you.

    A big problem I’m having is dealing with higher limits/colimits because there does not seem to be a good enough analogue for the join at this altitude. I’m also not totally comfortable with lax stuff, but a lot of this stuff should be easily generalizable from the strict case.

    Cheers,

    Harry

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2011

    This sounds most interesting. But I am not up to speed with the developments here. Can you point me to some more details on that Joyal/Cisinski proposal that you mention?

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 26th 2011
    • (edited Jul 26th 2011)

    In Joyal’s original notes on quasicategories (it’s called hc2.pdf, I think), he conjectures that the correct model category of (oo,oo)-categories is (by his and Cisinski’s reckoning) the model category of presheaves on Θ\Theta where the cofibrations are exactly the monomorphisms, and the weak equivalences are given by the localizer W(S)W(S) where SS is the set of spine inclusions.

    Dimitri Ara’s thesis has a precise description of what we mean by a spine inclusion (namely the inclusion of the globular sum of presheaves of globes into the presheaf represented by the globular sum in Θ\Theta).

    A few areas where I’m looking to extend this research are:

    1.) Showing that this model category Θ^\widehat{\Theta} is Quillen equivalent to the model category of Θ^\widehat{\Theta}-enriched categories (using Lurie’s construction of such a model category in appendix 3 of HTT). The main impediment here is finding an analogue of the homotopy-coherent nerve and in particular its left adjoint (since the cobar construction used in the (∞,1)(\infty,1) case does not appear to generalize). This would, among other things, show that Θ^\widehat{\Theta} is indeed itself an (∞,∞)(\infty,\infty) category.

    2.) Proving a Grothendieck construction (if this takes a form similar to that of Lurie’s straightening/unstraightening constructions, this would imply 1.) a fortiori).

    3.) Generalizing the usual universal constructions (limits, colimits, limits/colimits, kan extensions). Weighted (co)limits would have to wait until a useful form of 1.) or 2.) is proven

    4.) Given a sequence f:N→{0,1}f:\mathbf{N}\to \{0,1\} of directions, defining what we mean by an ff-lax (co)limit, perhaps using a formulation of 3.)

    Using some very nice technology generalizing some of Rezk’s technology, I have a definition of a generalized inner horn, but I haven’t really thought hard about any way to get a “coviariant” or “contravariant” model structure going on. I’m hoping that a generalized cartesian/cocartesian model structure might be formulated without relying on stratificiation, but it is only a dim hope, and any way one slices it, it seems like that might be a necessary component.

    A major advantage of this model structure is that it doesn’t have to deal with annoying things like completeness conditions (this is an annoyance that crops up in every model structure using simplicial presheaves and is largely an artifact of something silly)

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJul 27th 2011

    That’s exciting! I look forward to reading it. In principle I’d love to talk more about it too, but right now I really don’t have the time.

    A major advantage of this model structure is that it doesn’t have to deal with annoying things like completeness conditions

    It took me a while to figure out what you meant here, so let me say it in case anyone else is confused: I think you mean “completeness” in the sense it is used in complete Segal space. I agree that it’s nice to get rid of that, and of the “extra level” of homotopy-ness that simplicial presheaves introduce, although it’s not clear to me that it’s an “artifact of something silly”.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2011

    Does anyone (say Joyal or Cisinski or you) have a hunch as to if or how weak complicial sets might sit in this conjectured model for (∞,∞)(\infty,\infty)-categories?

    • CommentRowNumber6.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 27th 2011
    • (edited Jul 27th 2011)

    @Mike: I called the completeness condition an “artifact of something silly” in this case because it arises from taking the simplicial completion of a model structure and therefore only really contains the information of the trivial fibrations. It isn’t always an artifact (see Lurie’s definition of a the model category of complete Segal objects with respect to a distributor in his paper on the Goodwillie Calculus or Rezk’s original definition of complete Segal spaces (complete Segal objects in the simplicial model category of spaces). There should be a Quillen equivalence between complete segal objects in the category of cellular sets (Psh(Θ)Psh(\Theta)) and the model category of cellular sets itself, but instead of thinking of CSS(Psh(Θ)CSS(Psh(\Theta) as simplicial presheaves on Θ\Theta, I think it’s more suggestive to think of them as cellular presheaves on Δ\Delta (we’re thinking of them as categories enriched in cellular sets, not vice-versa).

    @Urs: I haven’t heard anything from either of them about a comparison with weak complicial sets, but it seems like a comparison might be pretty nasty, given how complicated weak complicial sets are. I mean, weak complicial sets don’t even form a cartesian model category, do they?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2011

    it seems like a comparison might be pretty nasty,

    That’s probably so. But it would be very useful to understand it. It is a bit of a strange situation that Verity has this concrete proposal for a model structure for (∞,∞)(\infty,\infty)-categories for such a long time already and nobody seems to know what to do with it.

    • CommentRowNumber8.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 27th 2011

    @Urs: I think a major impediment to working with them is that they don’t form a topos. Also, I haven’t met many people who are very comfortable even with strict ∞\infty-categories, and I’ve met even fewer who are comfortable working with their ω-nerves.

    However, I think Dimitri Ara and/or Cisinski may have at least thought about it, so you could probably e-mail one of them.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJul 29th 2011

    I would say, let’s not be too parochial in insisting that our model categories must be cartesian or a topos. Isn’t monoidal and combinatorial good enough for lots of things?

    • CommentRowNumber10.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 29th 2011
    • (edited Jul 29th 2011)

    Sure, but those are definitely extremely nice properties to have when modeling category-like objects. Also, since most algebraic objects can be modeled as particular kinds of objects internal to a topos, we get a lot of interesting model structures automatically. Also as a base for enrichment, even in Lurie’s appendix 3, cartesian model categories are by far the best to enrich over, simply because we can say so much more.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJul 29th 2011

    Automatically? Really?

    • CommentRowNumber12.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 29th 2011
    • (edited Jul 29th 2011)

    It’s a theorem of Quillen that under such circumstances, we automatically get a model structure on the category of algebras for certain kinds of monads, so I believe so, yes (for presheaf toposes, at least).

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJul 29th 2011

    Quillen proved a theorem specifically about toposes?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2011
    • (edited Jul 29th 2011)

    I am aware of Quillen’s statement about model structures on simplicial algebras over a Lawvere theory as described at model structure on simplicial algebras.

    Generally, concerning the discussion here: of course its useful to have model structures with as many nice properties as possible. But most importantly is to have as many different model structures as possible, with as many Quillen adjunctions/equivalences between them as possible.

    Verity’s model structure may be lacking a bunch of nice properties, but it has other advantages. (Not the least of them being that it actually has been proven to exist ;-). The main one being that it handles explicitly the filler conditions in one single simplicial set. This may of course also be the source of its main disadvantage: the sprit of nn-fold complete Segal spaces and akin models is in a way to do away with specific fillers and instead have spaces of possible fillers and spaces of such spaces, etc. This pushes more and more combinatorics into more and more homotopy theory. Which is good for some purposes. For others it may be good to have something closer to Verity’s idea.

    The main point I am interested in is: is Verity’s model indeed a model of (∞,∞)(\infty,\infty)-categories? To my mind a major drawback of his theory so far is that he does not (or at least I am not aware of it) give any strong arguments to that end. Given the fact that the nn-fold complete Segal spaces and Θ\Theta-spaces have meanwhile been seen to be the correct model for (∞,n)(\infty,n)-categories, it would seem necessary to see how or if Verity’s model can subsume these.

    That this may be hard to see is true, but besides the point. The harder this is to see, the more important would it be to see it! :-)

    • CommentRowNumber15.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 29th 2011

    Verity’s model structure may be lacking a bunch of nice properties, but it has other advantages. (Not the least of them being that it actually has been proven to exist ;-).

    The model structure I’ve been working with is proven to exist by Cisinski theory (similar to how the model structures for Rezk’s Theta_n-spaces are proven to exist by the general theorems on Bousfield localization). The main issue is whether or not the model structure we generate is in fact cartesian-closed. That the model structure on simplicial presheaves generated by the segal maps is cartesian is in fact the main theorem of Rezk’s Theta_n-space paper!

    Unless I missed a joke there!

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 30th 2011

    The model structure I’ve been working with is proven to exist by Cisinski theory

    Oh, I see. Right. My fault.

    • CommentRowNumber17.
    • CommentAuthorzskoda
    • CommentTimeJul 30th 2011

    I can not access the nnLab now, but is there a page dedicated to model structures for (infinity, infinity)-categories ? To put some iudeas, conjectures, and links to references like Ara’s thesis and Joyal’s work ?

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJul 30th 2011
    • (edited Jul 30th 2011)

    but is there a page dedicated to model structures for (infinity, infinity)-categories ?

    Not yet. You could create one.

    What there is currently is

    • a broken link to the model structure on weak complicital sets here

    • and then various pages on model structures for (∞,n)(\infty,n)-categories, of which the one on Theta-space vaguely claims that this works for n=∞n = \infty

    • CommentRowNumber19.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 30th 2011
    • (edited Jul 30th 2011)

    I think it’s worth noting: Ara’s thesis is noticeably silent on the issue of the model structure in question. However, its framework of globular extensions (and in particular, with Θ\Theta as the initial categorical globular extension) and description of strict ∞\infinity-categories is precisely the motivation for Joyal and Cisinski’s choice of generators of the class of weak equivalences for Psh(Θ)Psh(\Theta).

    The best currently-published paper on this model structure is Rezk’s paper for finite nn. However, the iterative approach used therein does not at all help in the infinite case (or at least, one cannot simply read things off from it without doing a fair bit of re-engineering). It does not treat the n=∞n=\infty case, but it contains a lot of important ideas that do extend to that case.

    I’m currently working on showing that a certain analogue of the homotopy-coherent realization (left adjoint of the homotopy coherent nerve) is left-quillen. This is a fair bit harder than it sounds, given that the usual proof that the homotopy-coherent realization is indeed left-quillen makes use of a lot of the explicit combinatorics of simplicial sets. I’d actually like to show that it’s a quillen equivalence, since such a result together with the cartesianness result would certainly be substantial enough to publish, however, I expect this to be a nontrivial undertaking (whence my solicitation for a coauthor).

    Such a result could conceivably be extended to a full Grothendieck construction, but I feel that would require a better understanding of (grothendieck) fibrations in the strict case.

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeJul 30th 2011

    Do we really need to say “(∞,∞)(\infty,\infty)-category”?

    Has “∞\infty-category” been so thoroughly hijacked to mean “(∞,1)(\infty,1)-category” that we can’t use it with its original meaning any more? If so, maybe we could instead say something like “ω\omega-category” which has the same meaning but hasn’t (as far as I know) been similarly hijacked?

    • CommentRowNumber21.
    • CommentAuthorzskoda
    • CommentTimeJul 30th 2011

    It has. Many (and Urs and I did at some point) by ω\omega-categories mean strict ∞\infty-categories. I have no problem with calling ∞\infty-category all the variants, and by (∞,∞)(\infty,\infty) specifically “the weakest” version.

    • CommentRowNumber22.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 30th 2011

    The only reason why I don’t use the term “ω\omega-categories” is that it was my understanding that the original meaning of that definition included “infinite dimensional cells” or something.

    However, I concur that we should just use “ω\omega-category” if nobody is opposed to it and there is no risk of confusion.

    • CommentRowNumber23.
    • CommentAuthorzskoda
    • CommentTimeJul 30th 2011
    • (edited Jul 30th 2011)

    (∞,∞)(\infty,\infty)-has no risk for confusion. If the context is known just ∞\infty is fine. Few letters of ink are less a sacrifice than additional requirements on the baggage of the reader.

    • CommentRowNumber24.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 31st 2011

    Just a note in the introduction, or after defining (∞,∞)(\infty,\infty)-categories, that “∞\infty-category” means this, and that Lurie does it differently, caveat lector, etc.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeJul 31st 2011

    I find “(∞,∞)(\infty,\infty)-category” to be far too many syllables for regular use; even “∞\infty-category” is pushing it. Somebody might once have defined ω\omega-categories to include infinite-dimensional cells, but I think no one thinks it means that any more. And I can’t see any good reason to restrict that word to the strict case.

    • CommentRowNumber26.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 31st 2011

    Alright, I’m writing up the paper now. Who knew that the most annoying part of writing it is restating other people’s results?

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