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David Michael Roberts (Apr 08 2025 at 03:16):Theorems about this higher Gray tensor product were stated by Gaitsgory and Rozenblyum in (volume 1 of) their book published in 2017, but which is more or less the 'foundational' document setting out a bunch of ground theory for the recent proof of the geometric Langlands conjecture(s). However they openly stated they don't give proofs, and they don't know of any in the literature. I'd be grateful to know the current status of these statements and pointers to places where they have been proved, if at all:
https://mathoverflow.net/q/490685/4177
John Baez (Apr 08 2025 at 21:25):Oh, that's interesting. Maybe there's work left to be done in the current proof of the geometric Langlands conjecture, just as there was in Lurie's proof of the cobordism hypothesis. (Lurie, of course, called it a "proof sketch".)
David Michael Roberts (Apr 08 2025 at 22:06):A commenter at MathOverflow points out:
I haven't gone through the results with a fine-toothed comb to check how everything lines up, but it's certainly progress.
Kevin Carlson (Apr 08 2025 at 22:13):The commenter was me! I am finding myself a tad annoyed at how hard it is to get a clear sense of the status of these statements. I think Abellán in particular seems to be writing a bit defensively or something, like he's nervous about making clear claims about whether he's resolving something by such eminences as G&R. I wish we had a nice clear Schelling point somewhere to simply clear this up, and I guess your post will be it now, David, so thanks for thinking to do this! It's been intermittently bugging me for 8 years.
David Michael Roberts (Apr 08 2025 at 22:19):@Kevin Carlson yes, sorry! I haven't even had breakfast yet, I forgot to say.
I'm very much reminded of what Clark Barwick said about the state of homotopy theory as a field, whereby people claim things and the proofs don't appear (or appear in print) and so young people who could make a reputation by resolving an open-stated conjecture by a well-known mathematician cannot get credit, for merely writing up a proof by someone else. (This is a subtly different problem to the folklore disease in category theory, I think, but not unrelated)
David Michael Roberts (Apr 08 2025 at 22:23):I certainly pulled my punches at the MO question, and the disclaimer is to ward off irrelevant arguments, but I honestly don't think people should claim big proofs without specifying clearly there are open problems. I haven't read the big papers on GLC, but if they had a warning in the introduction specifying that the authors left to others the establishment of certain technical facts that aren't expected to be false, then I would be impressed by the honesty. Clearly building a theory of the Gray tensor for -categories is not the same kind of thing as the bulk of the proof, but it's non-zero.
On Mathstodon I was a little more explicit and laid out both sides of the argument one could have, with precedents: https://mathstodon.xyz/@highergeometer/114300309037773091
Kevin Carlson (Apr 08 2025 at 22:24):Yes, I could tell you were gritting your teeth a bit there, as was I!
Nathanael Arkor (Apr 09 2025 at 05:56):I find it entirely misleading (even deceitful) to state something in the form of a proposition or theorem when it doesn't have a proof, and I find it frustrating that referees permit it. It would be easy for a reader to cite such a "theorem" without even realising it hasn't been proven elsewhere.
Nathanael Arkor (Apr 09 2025 at 06:01):However, down that road lies claims like those made by the team that spelled out hastily-sketched technical details in Perelman's proof of the Poincaré conjecture, and then kinda claimed they 'finished' the proof.
I believe the mathematical community would discourage such claims. (I don't see why those that "finish" a proof should be credited more highly than those who have contributed most to the proof.) Certainly this appears to have happened in the case you mention (as you point out later in the thread).
Nathanael Arkor (Apr 09 2025 at 06:02):(But it's entirely possible these considerations are the reason why Gaitsgory–Rozenblyum chose to present the statements the way they did.)
David Michael Roberts (Apr 09 2025 at 06:04):For those that haven't seen it:
We do not have a good culture of problems and conjectures. The people at the top of our field do not, as a rule, issue problems or programs of conjectures that shape our subject for years to come. In fact, in many cases, they simply announce results with only an outline of proof – and never generate a complete proof. Then, when others work to develop proofs, they are not said to have solved a problem of So-and-So; rather, they have completed the write-up of So-and-So’s proof or given a new proof of So-and-So’s theorem.
https://www.maths.ed.ac.uk/~cbarwick/papers/future.pdf
David Michael Roberts (Apr 09 2025 at 06:06):Other subjects have high-status visionaries^[7] who are no sketchier in details than those in homotopy theory, but whose unproved insights are nevertheless known as conjectures, problems, and programs.
[7] Examples. Kontsevich, Deligne, Langlands
Amar Hadzihasanovic (Apr 09 2025 at 06:43):@Clémence Chanavat has an upcoming paper on the Gray tensor product of -categories; Clémence, is there something we can say in relation to this problem? Like, at the level of "if the G&R def is equivalent to ours in the n=2 case, then we have the properties that are needed"?
Clémence Chanavat (Apr 09 2025 at 07:36):Since the Gray tensor product of diagrammatic sets is a Day convolution, we get for free indeed the preservation of colimits and associativity, and it's also a left Quillen bifunctor, so I suppose that counts as having good properties. I don't know about this functor, which seems to be, at first glance, a way to represent the Gray product by the cartesian product, but please correct me if I'm wrong.
Clémence Chanavat (Apr 09 2025 at 07:36):The problem is again gonna be "if the G&R def is equivalent to the diagrammatic def"; the Gray tensor product of any variant of marked simplicial sets is "easily" proved to satisfy the good required properties, but it seems that it takes another 100pages from Abellán to make it coincide with the one of G&R...
Clémence Chanavat (Apr 09 2025 at 07:57):As a side question it makes me think of: it seems to me that the current uses of -categories, (for something else than than just developing the theory) consistently happen in the setting of iterated Segal spaces: the two examples I'll use to back up my claim are this G&R geometric Langland conjecture, and say the cobordism hypothesis by Ayala-Francis. Maybe this should be moved to another thread, but I wonder why this is the case:
I also don't think this is too much the case for , so why is that that people are not working with, say, variations on complicial sets, which are just fancy quasicategories with some marking anyway?
Amar Hadzihasanovic (Apr 09 2025 at 08:47):Just a general consideration, but it seems to me that, forgetting for now the part, there are two kinds of results, constructions etc. about -categories:
For those of the second sort, presumably it is quite convenient to work with Segal models which are founded on iterated weak enrichment, whereas for the first sort shaped models (and in particularly the diagrammatic one for being the most "uniform") would have an edge.
The Gray tensor product, on the other hand, is paradigmatic of the first sort, even in the strict setting, so it is not surprising that Segal models struggle with it.
JR (Apr 09 2025 at 14:01):David Michael Roberts said:
For those that haven't seen it:
We do not have a good culture of problems and conjectures. The people at the top of our field do not, as a rule, issue problems or programs of conjectures that shape our subject for years to come. In fact, in many cases, they simply announce results with only an outline of proof – and never generate a complete proof. Then, when others work to develop proofs, they are not said to have solved a problem of So-and-So; rather, they have completed the write-up of So-and-So’s proof or given a new proof of So-and-So’s theorem.
https://doi.org/10.1007/s11229-018-01981-1
We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation.
David Michael Roberts (Apr 09 2025 at 20:53):Open Access link https://philsci-archive.pitt.edu/id/eprint/15133
Fernando Abellán (Apr 11 2025 at 09:06):I didn't mean to write defensive in the paper though... I guess it is important to make clear that it was not a theorem as stated in GR book.
In any case, I'm soon posting a new version of the paper where the theory is built from scratch. I prove directly that the Gray tensor product satisfies all the desired properties without resorting to models or comparisons (colimit preservation, associativity and that it defines a monoidal structure on the (infinity )2-category of 2-categories and icons )
I will also show that all versions (that I know of) in the literature of the Gray tensor product (of 2-cats) agree.
Happy to answer any questions! :)
Kevin Carlson (Apr 11 2025 at 16:43):Glad to hear it! So it sounds like your new paper will fully close everything left open in Gaitsgory-Rozenblyum, if all goes according to plan?
Kevin Carlson (Apr 11 2025 at 18:57):@Fernando Abellán You might consider clarifying the situation on David's MO post as well. It's gotten many upvotes but no expert clarification and it seems like you understand what's going on here better than anyone.
Fernando Abellán (Apr 11 2025 at 19:43):From the things I mentioned the only new thing with respect to the previous version of the paper is, comparing the GHL tensor product with the one done with Gray cubes via Day convolution and the monoidal structure.
The important thing was to make the paper model-independent and not have so many terrible proofs. I apologize for the previous version, is too dense :sweat_smile:
Joe Moeller (Apr 11 2025 at 20:47):I'll mention this relevant paper by Tim Campion, which I don't think was mentioned yet:
https://arxiv.org/abs/2311.00205
David Michael Roberts (Apr 13 2025 at 05:33):@Fernando Abellán the main question is: does your model-independent version cover the GR definition and resolve their unproved claims?
Fernando Abellán (Apr 13 2025 at 07:22):@David Michael Roberts , I think those claims are already resolved in the current version of my preprint. The main claims I am referring to are: The Gray tensor product is associative and commutes with colimits separately in each variable. (There are some other unproven formulas for the Gray tensor product in terms of the square functor that I didn't work out last time but I will include this time.)
In the current version this is proven as follows: We know these properties hold for the Gagna-Harpaz-Lanari version in terms of scaled simplicial sets, so my approach is to compare this version with the GR approach.
The new approach is not so different, the key difference is that everything is built up from scratch without ever mentioning model categories. So I construct a Gray tensor product (equivalent to that of GR almost by definition), show that it has the expected properties (hence GR's version does) and further compare this tensor product with the other existing approaches.
I hope this helps.
Kevin Carlson (Apr 14 2025 at 16:36):I’m a bit surprised you say you *think * the claims are already resolved. Those claims were left open in a book that’s been the foundation of a program to resolve geometric Langlands that hundreds of people have been working on. If I were you working on this, I’d be sure to have an extremely crisp section right in my introductions listing exactly where and how to use my work to resolve each open statement in R-G, because I’d be assuming finding out about the open questions from R-G was a major source of readers.
Fernando Abellán (Apr 14 2025 at 16:57):I guess I was intending to say " I am not sure if I am forgetting about some claims" I probably rushed my answer. I think in the current version it is clearly stated the conjectures I solve. See Theorem 2, Corollary 1 and Corollary 3 in the introduction.
Kevin Carlson (Apr 14 2025 at 16:58):Thanks! Do you mind just dropping the arxiv link you're looking at again? Lots to scroll through above here.
Fernando Abellán (Apr 14 2025 at 17:03):@Kevin Carlson sure no problem https://arxiv.org/pdf/2311.12746.
John Baez (Apr 14 2025 at 21:15):If I were you working on this, I’d be sure to have an extremely crisp section right in my introductions listing exactly where and how to use my work to resolve each open statement in R-G, because I’d be assuming finding out about the open questions from R-G was a major source of readers.
Yes, that would make the paper much more useful - and also more popular, which will be good if Fernando wants to get a permanent job. (Just some unsolicited advice from an old mathematician who has been on many hiring committees.)
David Michael Roberts (Apr 15 2025 at 01:09):For reference, here's the passage from G&R where they spell out what they say are the questions left open:
0.4. Status of the assertions.
0.4.1. Unfortunately, the existing literature on (∞, 2)-categories does not contain
the proofs of all the statements that we need. We decided to leave some of the
statements unproved, and supply the corresponding proofs elsewhere (including
the proofs here would have altered the order of the exposition, and would have
come at the expense of clarity).
0.4.2. Here is the list of the unproved statements:
- Proposition 3.2.6 says that the formation of Gray product commutes with colimits in each variable.
- Proposition 3.2.9 asserts the associativity of the Gray product.
- Theorems 4.1.3, Theorem 4.3.5, Theorem 4.6.3 and Theorem 5.2.3 are all generalizations of the assertion that the functor $\mathrm{Seq}^{\mathrm{ext}}_{\bullet}$ (or, rather, its variant $\mathrm{Sq}^{\mathrm{Pair}}_{\bullet,\bullet}$)
is fully faithful with specified essential image.- Proposition 4.5.4 gives an explicit description of the Gray product in terms of
the functor $\mathrm{Sq}^{\mathrm{Pair}}_{\bullet,\bullet}$.It is quite possible that references for (some of) the above statements do exist, and we would be grateful if the reader could point them out to us.
I can't say definitively that this is in fact the entire list of things they don't prove, but I would suggest, @Fernando Abellán that you refer to this 10.0.4.2 (i.e. chapter 10, section 0.4.2 above) and be extremely explicit what you have completed from this list.
On page 3 of the linked document, you write in item 6.:
"...thus answering another question in [GR17]."
which is underselling the result. You mention earlier in item 3
We show in Corollary 1 that the Gray tensor product of complete Segal objects (or 2-
fold complete Segal spaces) in (∞, 1)-categories given in [GR17] defines a closed monoidal
structure.
which doesn't say that this was an open problem in GR17 and a property (or corollaries of this) that they crucially rely on.
Even in the appendix, the statement
"As a corollary, we derive that the Gray tensor product of (∞, 2)-categories as defined by Gaitsgory-Rozenblyum is equivalent to that of Gagna-Harpaz-Lanari."
absolutely buries the lede that you are filling unproved steps from the foundations of Gaitsgory's program.
I myself would say in the abstract that you resolve all (or some) of the claims in GR17 about (oo,2)-categories, because most people would have no idea that what you claim does this, or its importance.
John Baez (Apr 15 2025 at 01:27):I agree with what David Roberts said, especially the last sentence! If your abstract doesn't say why your paper is interesting to the many fans of geometric Langlands, they won't read it and you'll be missing a great opportunity.
Kevin Carlson (Apr 15 2025 at 02:00):And also—they won’t all read it anyway, so you want them to find out why they ought to mention you anytime this topic comes up as quickly as possible!
Félix Loubaton (Apr 15 2025 at 08:50):Hello!
I wanted to give a bit of an overview of the progress on the unproven claims of the Gaitsgory-Rozenblyum book. To summarize, it seems to me that they are all proven except for one (but which should be soon).
In what follows, I will refer to works that use a priori other definitions of the Gray tensor product that the one of Gaitsgory-Rozenblyum. However, as was mentioned earlier, Fernando proved that the Gray product in the Gaitsgory-Rozenblyum version is the same as in the Gagna-Harpaz-Lanari version. Moreover, even though I don’t think it’s clearly written in the literature (which is unfortunate, I admit), the Gagna-Harpaz-Lanari product is equivalent to all the other definitions I know.
There were 8 claims for which the proofs were missing.
• Propositions 3.2.6 and 3.3.9 concern the Gray tensor product (commutation with colimits and associativity). They should therefore follow from the comparison between the definition by Gaitsgory-Rozenblyum and the one by Gagna-Harpaz-Lanari, since Gagna-Harpaz-Lanari demonstrate that their version satisfies these properties (https://arxiv.org/abs/2006.14495)
• Proposition 3.3.5 discusses the fact that the iterated Gray product of simplices is an (strict) 2-category. To my knowledge, this result is not stated as such in the literature. However, I think one can deduce it quite directly from the results of Campion and Maehara presented in their article https://arxiv.org/abs/2304.05965
• Then there are Theorems 4.1.3, 4.3.5, 4.3.8, and 5.2.3 which discuss the square functor and its variants. The first was proven by Fernando (https://arxiv.org/abs/2311.12746), and the last three in my latest preprint (https://arxiv.org/abs/2503.19242)
• Finally, there is Proposition 4.5.4 which gives yet another definition of the Gray tensor product.
It should appear in a forthcoming article written with Jaco Ruit, but also in the new version of Fernando’s paper.
It is very possible that I forgot to cite some works, and I apologize in advance. Let me know if this is the case !
David Michael Roberts (Apr 15 2025 at 09:17):@Tim Campion @Yuki Maehara can you speak to the claim about Prop 3.3.5 in the previous post?
Félix Loubaton (Apr 15 2025 at 09:59):After looking more closely, I think that once we admit that all these Gray tensor products coincide, Corollary 7.11 of Maehara’s article https://arxiv.org/pdf/2003.11757 implies Proposition 3.3.5 of Gaitsgory-Rozenblyum.
Yuki Maehara (Apr 15 2025 at 11:52):Short answer: Yes, I think Proposition 3.3.5 can be proved using the 2-quasi-categorical Gray tensor product assuming Fernando's result.
Details: I don't know exactly what Fernando proves, so I'll start with the minimal assumption: the 2-quasi-categorical Gray tensor product computes the "correct" object at least in the binary case. (It is a left Quillen bifunctor with respect to Ara's model structure, and the induced oo-bifunctor agrees with the one defined by Day extension in https://arxiv.org/pdf/2304.05965; please see Remark 3.4 of that paper.) Although this tensor product is not associative up to isomorphism, it is associative up to homotopy in the best possible sense I can think of. (This is Corollary 7.11 of https://arxiv.org/pdf/2003.11757 that Félix mentioned; "cell(...)" there is a subclass of the trivial cofibrations.) In particular, [n_1,...,n_k] in Proposition 3.3.5 of Gaitsgory-Rozenblyum can be computed by simply applying the k-ary Gray tensor product (Definition 3.1 of https://arxiv.org/pdf/2003.11757) to the representable Theta_2-sets corresponding to [n_i]'s, but this is defined to be the nerve of the corresponding 2-categorical Gray tensor product.
Fernando Abellán (Apr 15 2025 at 11:57):Just to be clear. In the current version, there is no comparison with the 2-quasi-categorical Gray tensor.
This is not hard to prove and will be included in the new version.
Thank you Félix and Yuki for further clarifying the status of the conjectures.
David Michael Roberts (Apr 15 2025 at 12:45):Thank you to everyone who chimed in! Can someone post an answer over on MathOverflow that explains the above status of the various claims? Of course, it can wait until after the new version of the paper by Fernando is out, and/or the Loubaton–Ruit paper lands. There's no real rush, I just want to get this all on the public record.
Also, I strongly suggest someone include in their paper a proof of Proposition 3.3.5 using all the tech mentioned by Félix and Yuki, rather than it being left for experts to know because they read the technical literature and can see how the pieces make it fit together. What we have here is a chance to set a good example and stamp out the emergence of a new piece of folklore.
Félix Loubaton (Apr 15 2025 at 13:08):Of course! I will do it. I think this is indeed a good idea to do it once the new version of Fernando’s paper or my project with Ruit is online. That way, we’ll really be able to say that everything is proven!
David Michael Roberts (Apr 16 2025 at 02:00):Excellent, thanks everyone.