Davide Castelvecchi at Nature has the story this morning of a press conference held earlier today at Kyoto University to announce the publication by Publications of the Research Institute for Mathematical Sciences (RIMS) of Mochizuki’s purported proof of the abc conjecture.
This is very odd. As the Nature subheadline explains, “some experts say author Shinichi Mochizuki failed to fix fatal flaw”. It’s completely unheard of for a major journal to publish a proof of an important result when experts have publicly stated that the proof is flawed and are standing behind that statement. That Mochizuki is the chief editor of the journal and that the announcement was made by two of his RIMS colleagues doesn’t help at all with the situation.
For background on the problem with the proof, see an earlier blog entry here. In the Nature article Peter Scholze states:
My judgment has not changed in any way since I wrote that manuscript with Jakob Stix.
and there’s
“I think it is safe to say that there has not been much change in the community opinion since 2018,” says Kiran Kedlaya, a number theorist at the University of California, San Diego, who was among the experts who put considerable effort over several years trying to verify the proof.
I asked around this morning and no one I know who is wellinformed about this has heard of any reason to change their opinion that Mochizuki does not have a proof.
Ivan Fesenko today has a long article entitled On Pioneering Mathematical Research, On the Occasion of Announcement of Forthcoming Publication of the IUT Papers by Shinichi Mochizuki. Much like earlier articles from him (I’d missed this one), it’s full of denunciations of anyone (including Scholze) who has expressed skepticism about the proof as an incompetent. There’s a lot about how Mochizuki’s work on the purported proof is an inspiration to the world, ending with:
In the UK, the recent new additional funding of mathematics, work on which was inspired by the pioneering research of Sh. Mochizuki, will address some of these issues.
which refers to the British government decision discussed here.
There is a really good inspirational story in recent years about successful pioneering mathematical research, but it’s the one about Scholze’s work, not the proof of abc that experts don’t believe, even if it gets published.
Update: See the comment posted here from Peter Scholze further explaining the underlying problem with the Mochizuki proof.
Last Updated on
Publication of RIMS does not really qualify as a major journal. So everything is going to be alright…
In Fesenko’s article one finds the following remarkable line:
“Misinformation and disinformation in science has become a very serious issue, not only for mathematics.”
I think this is in fact a very relevant point – but, perhaps, just not in the way that Fesenko intends it.
Serious question.
I have looked through some of Mochizuki’s rebuttals. I was amazed by the almost “internet insultsasdefensive” language thrown in there. As well as his pedagogical choice of describing things with layers upon layers of analogy, that could be said more directly. Especially when some of the analogies seem to be chosen almost as if to insult someone if they question it. It feels like what a physics theorist I knew used to joke as some authors trying to “prove by intimidation”. But that’s in physics, where a bit of hand waving is allowed (no point in waiting for a millenial prize problem to be solved to posit an effective field theory). It’s confusing how this manner of dialog aids any purpose in mathematics.

Anyway, I got a bit off tract. My question is:
Is it possible the style / pedagogical choices are actually causing real problems? Would it be helpful at this point if someone that states they understand the theory (but unlike Fesenko, will explain in a more useful manner than Mochizuki), if they rewrite the proof in their own words? Or at least the more difficult portions?

Or is the claim that it is pedagogically impossible to present the more difficult portions in any different manner?
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Castelvecchi characterizes the style of Mochizuki’s papers as impenetrable and idiosyncratic, with which I wholeheartedly agree. However, I wonder why in this discussion Go Yamashita’s writeup (http://www.kurims.kyotou.ac.jp/~gokun/DOCUMENTS/abc_ver6.pdf) has apparently received so little attention. While I unfortunately can’t follow the mathematics, I find the style to be very clear and traditional in the best sense. In particular, he gives a proof of the infamous IUTchIII Corollary 3.12 (Corollary 13.13 in his paper), which I find to lend itself better to a discussion.
Peng/IWonder,
For a long time the style, length, organization and idiosyncrasies of the Mochizuki papers seemed to be the main problem, keeping experts from being able to fully understand and thus check the proof, and Go Yamashita’s version promised to improve the situation. But once Scholze and Stix identified a specific issue, spent a lot of time discussing it with Mochizuki, and ended up convinced this was a gap in his proof, that completely changed the situation. Few people are going to devote a lot of time to studying a very complicated proof that at a crucial point has a gap. What’s needed is for Mochizuki or someone else to put forward a convincing response to the issue raised by Scholze/Stix. This doesn’t seem to have happened, and behavior like attributing the problem to Scholze being an incompetent not only doesn’t help, but just convinces others that engaging with Mochizuki and those around him to better understand the issue is a waste of time. The announcement that the journal will publish anyway also doesn’t help the situation at all.
Fesenko’s article seems like a rehash of something he wrote a few years ago, “Remarks on aspects of modern pioneering mathematical research” (https://www.maths.nottingham.ac.uk/plp/pmzibf/rapm.pdf)
Talor Dupuy says
https://twitter.com/DupuyTaylor/status/1246142127538614272?s=19
Scholz may be wrong
Hi. I do read both Japanese and English, and I feel like I’m obligated to give you more information on this matter which may only be provided in Japanese so far. First of all, I am not an expert nor trying to convince you whether IUT holds to be true or not; I am only providing additional information.
–
In 2018, Scholze and Stix released the document to show the famous fatal flow, so did Mochizuki to show the validity of his work. In April, 2019, Fumihiro Kato, a Japanese mathematician who verified IUT, has published a book about IUT for nonexpert Japanese audience. A few months later, I checked Amazon Japan to view readers’ reviews, which most people gave 5 or 4 stars. Then I found one negative review briefly saying, “Who’s gonna read this book? Math community now knows there is fatal flow in IUT.” Kato responded to this review. He posted the same response on his twitter account, so we know this response is really came from the author. He, in a very humble manner, stated that IUT is still in the middle of proofreading (at that time), but IUT experts are now confident to say that Scholze and Stix were simply misunderstanding lines of argument.
–
In January 2020, Mochizuki updated his private blog only available in Japanese. For while, people thought this blog is not Mochizuki’s but is made up by some geek. Surprisingly, Kato has told us in his book that the author of the blog is indeed the Mochizuki himself. Anyway, the updated article indicated the Mochizuki’s frustration for his papers not being published for such a long time, and he rebuked the publisher in quite strong language. He didn’t mention the name of the journal, but readers were aware that was RIMS since the article was available only in Japanese. Which implies that he was not a chief editor of RIMS for the IUT matter. When RIMS had a press conference yesterday, they indeed explained that Mochizuki was completely excluded from refereeing the papers.
Dupuy should have any lecture in Princeton for Diophantine geometers.
It may be a good possible way to realize certain understandings and consensus in mathematical community out of Kyoto.
sleep/K.N.
For more of the twitter exchange with Dupuy, see here:
https://twitter.com/meu_gato/status/1246220210891190272
where he seems to state that he is not able to provide a convincing proof of the problematic Corollary 3.12 (and agrees that Mochizuki publishing this in the journal he is chief editor of is a bad idea, no matter what procedures were followed).
The problem here is simple: neither Mochizuki nor anyone else has written up a convincing response to ScholzeStix. Fesenko has devoted pages and pages to ad hominem attacks on them, nothing to a technical refutation of their argument. He’s trying to make instead the opposite of an argument from authority, arguing that Peter Scholze is an incompetent. No one is going to buy this. Mochizuki apparently believes there is no reason to make significant changes to his article to address their concerns. He’s entitled to his viewpoint that he is correct and they just don’t understand, but any author who wants to convince the math community that he has a proof needs to be willing to work to explain to others what they are not understanding.
So, sure, ScholzeStix may be wrong, maybe Mochizuki’s methods do give a proof. But an explanation of how this works convincing to experts needs to be provided and it hasn’t been yet.
KS,
I don’t think what’s needed is more lectures about IUT. What’s needed is a more convincing argument for Corollary 3.12, and my reading of Dupuy’s twitter commentary is that he says he can’t provide it.
In my opinion, the corollary may be related to core theoretic framework of IUT , especially as how scheme theoretic universes are constructed and unified.
Mochizuki have been emphasized in his terminology monoanalytic generalized distinction of scheme theoretic objects , but which aren’t mere ordinary scheme theoretic ones.
True problem may be such difficulty of theory , rather than so called too simplification by ScholzeStix.
I hope that Dupuy or some other experts will lecture Diophantine geometers on core ideas and strategies of IUT in for instance Princeton possibly.
Can a referee go public after publication is made, or is that poor practise? Or could at least the referee reports be themselves published? They would help form better judgement on the situation: the experts would either say “nothing conclusive here”, or “ha, I had missed that, got it”.
@Peng,
My understanding is that there exists a paper (1) written by Mochizuki (and others), (2) using the language of IUT theory, (3) that is correct, (4) which could be made much clearer by removing the IUT language and writing it in more typical mathematical style. Unfortunately the paper for which I understand this is not the papers on ABC, but rather a paper he wrote after, purely about fundamental groups / anabelian geometry. (My understanding comes from a friend who has read it.)
So one can certainly say that there exists a paper where the style is the main problem. It just may not be true for the main IUT paper.
.
@Peter I don’t think you can fully separate the ScholzeStix criticism from problems with style. My understanding is that the ScholzeStix criticism comes from (what they see as) changing the style of the argument so that it is more clear, and then observing that the modified argument is wrong, but Mochizuki says that these changes are substantive. If the argument specifically of Corollary 3.12 were written in a different style, there would be no issue.
.
@IWonder I know Scholze has said at some point that other expositions, including Go Yamashita’s, are not clearer on the crucial 3.12. I don’t know if this is true, as I haven’t read it.
.
@Peter I am actually optimistic about Taylor Dupuy’s work, specifically because I believe that if, in the future, he says he understands the proof of Corollary 3.12, then he will be able to write a comprehensible proof of it. If this is true then there would be two possible resolutions of his project and this would be a good thing.
I know that mathematicians are weird – but this is completely off the scale. Peter, could you please explain if there is any mathematical reason why, in eight years, Mochizuki cannot find an alternative approach to his proof? Your statement that “He’s entitled to his viewpoint that he is correct” is too kind. To me, as a nonmathematical scientist, it seems crazy.
David Leavitt,
Yes, this is completely off the scale. Having a serious math journal hold a press conference to announce that they’re publishing a proof of a huge longstanding conjecture even though the consensus of experts in the field is that the proof is flawed is extremely weird.
It’s not surprising Mochizuki hasn’t come up with an alternative proof. That could be hard and he thinks this one is correct. The question is why, if he and others are convinced the proof is correct, they can’t come up with arguments that will convince others of this.
justcurious1,
The tradition is to keep referee reports confidential, and in this case it’s not clear if breaking the tradition would help. Much of the work of the referees was surely on parts of the proof other than the controversial Corollary 3.12. On the controversial part of the proof, they’re siding with Mochizuki that what he has written is correct and complete. If they have an argument for this that would convince ScholeStix and others, it would be helpful if that were made public. This all comes down to the same problem of what is missing here that is needed to justify acceptance of the proof.
I am a Japanese science journalist. I’ve been watching your discussions on this site with interest.
In Japan, Dr. Mochizuki’s paper, published in PRIMS, has the media buzzing that the ABC forecast has been proven. However, I disagree. I have read your arguments and understand that some researchers have claimed that Dr. Mochizuki’s theory is correct and has proven a weak ABC theory, but have not convinced many experts.
Is there any chance that this situation could change?
I think Dr. Mochizuki will have to explain and convince himself. What do you think, Dr.Woit?
Shinichi Aoki,
For there to really be a proof, it needs to be convincing to a consensus of experts in the field, and that has not happened here (the great majority of experts are not convinced). Scholze and Stix accurately entitled what they wrote about this as “Why abc is still a conjecture”
http://www.kurims.kyotou.ac.jp/~motizuki/SS201808.pdf
Unfortunately there does not seem to be much change since that was written. The publication of the article does not help at all, even hurts, since it removes some of the motivation for Mochizuki himself to write an improved version of his proof that might convince others.
I have asked this before in the comment section here, but I will repeat myself in the hope that someone knowledgeable is reading it:
It is well known that an effective proof of abc would imply an effective Roth’s theorem (and consequently effective versions of many other things, like Falting’s theorem), Mochizuki however has (to my knowledge) never stated any explicit bounds as a result of his work. It seems that he initially thought his proof of abc was noneffective due to a specific reason, this noneffective step was later removed by someone else (I have forgotten his name). Therefore his claimed and now published proof appears to have the feature that no explicit bounds follow from it, yet no one seems able to point to the source of this noneffectivity. To me that is a very strong indication that the proof is not complete.
And isn’t the above a stronger reason to doubt the correctness of Mochizuki’s work than the appeal to the authority of Scholze? Or har Mochizuki commented somewhere on why the proof remains noneffective?
this is not true. Dupuy and Hilado have derived very explicit estimates for Szpiro’s conjecture in their work taking Cor 3.12 as a black box, and have just released a (prearXiv) preprint in Twitter. As in: “Let A_0 = 84372107405, B_0 = 316495, then the absolute value of the minimal discriminant of an elliptic curve E/F satisfying the standing hypotheses is bounded by … [explicit expression involving these constants and other known quantities]”. This uses work of Serre–Tate is a serious way, and their reworking of Cor 3.12 into nonIUT language (that’s another long paper they are working on). I don’t know how you get more explicit than this.
There’s also older work of Dimitrov (https://arxiv.org/abs/1601.03572) showing that one should get effective bounds (“there exist computable functions such that… [inequality]”), but I don’t think he actually gives them.
@mahmoud I think the issue is that Mochizuki’s work does give explicit formulas, they are just so weak you would have trouble finding an explicit countereaxample. Then any reduction argument to another problem would give you even weaker bounds. Taylor Dupuy and Anton Hilado have been working on giving explicit formulas on the statement https://www.dropbox.com/s/hwdxtpk5ydqhp6g/thm1p10short.pdf?
If you look at Theorem 1.0.5 you can see constants like exp of 84372107405 times ell^4, where I think ell is a prime greater than 19, and thus must be at least 23. That’s before you reduce from a general problem to Mochizuki’s specific case.
I think Mochizuki has more precise estimates in his writeup, but they’re also harder to understand.
Here’s my take on this situation. I know a considerable amount about elliptic curves and Abelian varieties, but the stuff on most highlevel mathematics is way over my head.
It seems to me that Mochizuki’s theory is an attempt to treat some classes of mathematical objects in the same way. That’s not out of this world. The objections by Stix and Scholze boil down to a point where they believe this treatment is flawed. And Mochizuki answers that they do not understand his theory.
In all these years, several mathematicians were in contact with Mochizuki’s group. If the relations he puts forward were so flawed that his proof is beyond repair, then there’s something fundamentally wrong with his theory from the start, and the error would be much greater than a small passage in the controversial corollary.
Instead, what we are seeing is that there are many mathematicians open to the idea and are working on the theory instead of dismissing Mochizuki as a crank.
I find it completely possible that somewhere in the papers Mochizuki made a mistake that’s very hard to fix, just like Wiles did when he proved FLT.
At the same time, I also find it possible a Heegnerstyle scenario, where someone outside the mainstream math community proved a theorem using 19th century math, and people dismissed it on grounds that it cited an incorrect result, which turned out not to be flawed upon closer inspection.
In any case, I see little reason to not publish Mochizuki’s papers. Even if his results are not perfect, many people seem to believe it’s a viable route to attack the problem. That’s enough for me.
Felipe Lopes,
The latest version of Mochizuki’s papers have always been available online, and anyone who wants to work on this has all they need. The only difference with having a published version is that the journal’s editors have put their own reputation and that of their journal behind the claim that the arguments in the papers have been checked and are valid and complete. Given that the consensus of experts is still that this is a flawed proof, I don’t see what the PRIMS editors are accomplishing here other than putting a torch to their journal’s reputation.
Thanks to David Roberts and w for correcting me! I’m not sure how comforting it is that Dupoy & Hilado obviously have worked hard on this and still are unable to make sense of the proof of Mochizuki’s 3.12 though…
(Vesselin Dimitrov was the name I couldn’t remember in my first comment.)
I want put here substantial reference on this issure , namely the corollary 3.12.
Mochizuki updated a his own survey which he recognize in his blog(in Japanese) as a most important summary on the heart of IUT which available at his website , the title as below .
The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Interuniversal Teichmuller Theory.
(crucial arguments are about P142146(example 3.11.4)).
In both above place and his blog , he is explaining that ScholzeStix ‘s misunderstanding is caused by confusing logical relation ∧ with ∨ on the θlink , so there is no problem in the theory originally according to Mochizuki himself.
Especially , there is an emphasis that θlink should be absolutely “∧unification” of Hodge theaters in the system of “logtheta lattice” and its horizonal sequence.
And he said in his blog(again to say , in Japanese but substantially wellargued in an above survey paper) , “logical operator ∧ ” is basically stronger information than ∨ , in the sense of restriction of the relation.
As my “unreliable personal” undersrtanding , this arguement can be seen as just “how scheme theoretic universes are unified and computed systematically” in the context of “arithmetic geometric(but monoanalytic , topological group theoretic) ” construction of each of θdivisors over global number fields.
Here , we should recall that θlink is basically distinctlyunifying device of the Hodge theaters.
Anyway , I think there is any public need of the response to this by ScholzeStix although it’s not certain whether they recognized this matter or not.
All of this is confusing. Let me at least try to clarify something about effectivity and the external reduction.
@David Roberts: One can certainly be more explicit by spelling out – if not bypassing – what exactly those restrictive standing hypotheses mean. This paper by Dupuy and Hilado actually involves all of Mochizuki’s special assumptions (from I Def. 3.1) occurring in his already explicit claim IV 1.10, and even more: [DH] furthermore require the elliptic curve to have a good reduction at all prime ideals of residue characteristic 2. The latter is not assumed by Mochizuki (he rather states an explicit bound on the primeto2 part of the minimal discriminant, under his other assumptions in I Def. 3.1); and the Belyi maps argument does not allow a reduction to the case of good reduction above 2 (it only reduces to a situation of bounded contributions at 2). Besides, the true exponent is a “6” (as in IV 1.10) rather than the lossful “24” in [DH]. My understanding is they rather want to make the statement more conventionally readable.
In my note you linked to, I simply explain that there is nothing inherently ineffective in the external reduction of the full abc conjecture to this explicit but restrictive statement IV 1.10. This is really disjoint from [DH], the point being exactly to bypass the special assumptions. Yes, the increase in the constant will be rather large if fully worked out this way. Anyways the essence of the reduction is exemplified by the following representative statement that had not been known before Mochizuki’s “Arithmetic elliptic curves in general position.” It is one undisputed contribution of Mochizuki’s to the subject of abc that will survive regardless of the ultimate outcome of the IUT saga: Over general number fields F, in the version where an [F:Q] dependence is included, the sharp (6+epsilon exponent) Szpiro discriminantconductor conjecture is already equivalent to the a priori stronger sharp (1+epsilon) upper bound on (twice) the full Faltings height including its Archimedean term; with a change of constants that is traceable to a computable function of epsilon and [F:Q] alone.
@mahmoud, It seems Mochizuki has neither acknowledged nor disputed that his indirect appeal to compactness in “Arithmetic elliptic curves in general position” is straightforwardly turned around into a constructive argument, as I outlined in https://arxiv.org/pdf/1601.03572.pdf . But, along with his collaborators, he has subsequently gone about for effectivity in a different way:
I understand (cf. this announcement: http://www.unigoettingen.de/en/77723.html?cid=20836&date=20200123&fbclid=IwAR1DYidCLs2cwKcBE3TRQmiEwP8WXfSSiYvb5C2SR5rfX9JWKGxF7IYUeGw ) that Mochizuki et. al. have since gone on to expand the claim of IV 1.10 further by also directly including the contribution to the minimal discriminant at 2. That would bypass the external recourse to Belyi maps insofar as the Szpiro conjecture over Q (not strong abc over Q with the “1+epsilon”) is concerned. If they could really also directly add the Archimedean term of the Faltings height (as was claimed at one point in 2018), it would have yielded a full and practical explicit strong abc bound, and the connection to Siegel zeros would have gone through as well.
@Peter Woit “I don’t see what the PRIMS editors are accomplishing here other than putting a torch to their journal’s reputation.” For one thing they are making a lot of publicity for the journal. Whether this is going to harm or boost PRIMS in the long run remains to be seen.
KS,
Mochizuki’s claim “that “ScholzeStix ‘s misunderstanding is caused by confusing logical relation ∧ with ∨” is not plausible and Scholze has in some sense responded to it with his statement to Nature that nothing has changed. This is just a variant of Fesenko’s claim that the explanation for ScholzeStix not accepting this proof is that they are incompetents, and equally hard to take seriously.
I understand that other sciences have highly contentious episodes somewhat comparable to this one, and that of course in mathematics there are some people who maintain for a long time they’ve proved some important result although the community does not accept it (e.g. de Branges and the Riemann hypothesis), but I can’t think of anything in math really similar to this Mochizuki stuff. Does anyone know of a historical parallel?
Sam Hopkins,
People sometimes point to Heegner’s proof of the class number 1 problem, published in 1952, but mistakenly assumed by experts to be incorrect until it was reexamined in the late 1960s. As far as I know though, the supposed problem with the proof was raised after its publication. I’ve never heard of anything like a journal going ahead and publishing a proof over a consensus of experts that it has a gap.
Hi Peter,
I read that SM admits that it might be impossible to derive intermediate results from his purported abc proof, but has anyone else proved any statements about number theory or arithmetic geometry using IUT that are generally accepted (or provide a new proof even of known results)? Basically, has IUT been able to expand beyond its own shores?
DS,
You need someone more expert than me for an informed answer to this. I haven’t though heard of any major results crucially using the IUT stuff. For a related answer, see the comment above
https://www.math.columbia.edu/~woit/wordpress/?p=11709#comment235908
from W.
From The Asahi Shimbun http://www.asahi.com/ajw/articles/13271575
The popular media seem to think it’s a done deal:
Editors of the journal of RIMS asked outside experts to peer review the articles for any problems.
In late 2017, it appeared that the articles would be published, but mathematicians in the West pointed out what they considered inappropriate leaps in logic in a core portion of the articles.
That led the journal editorial board to continue with their assessment. A number of other outside experts were consulted and it was only in February that Mochizuki’s proof was considered to no longer have any problems.
The constant in the inequality is, even in D–H’s version, taking the curve E11a1 for concreteness (they check this satisfies the hypotheses of being “in initial theta data”) something like 10^86858896380650, at the smallest, if we take \ell = 7. So the bound can only get worse? (!!) I don’t know what the appropriate minimal discriminant is that this is supposed to be bounding, but this is not exactly a practical estimate.
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Naive question: Doesn’t “Publications of the Research Institute for Mathematical Sciences” sound like an institute’s own publication? If yes, they are morally entitled to publish their own work, with lower peerreview standards than for globalreach journals, isn’t it so?
tulpoeid,
Different math journals do have different standards, and may have a mission of publishing their own researcher’s work. But “lower standards” means willingness to publish correct but not very interesting work, or maybe even correct but not very well written papers, not publishing incorrect papers.
There was a frontpage article in the Asahi Shinbun (one of the major Japanese newspapers) in the morning edition on April 4. An enormous article that took up about 1/3 the page, with the title in only a slightly smaller font than the lead article.
First, the old news, just for the record:
It states that the four papers were submitted to PRIMS at the same time they were published on Mochizuki’s blog and PRIMS submitted them to multiple outside reviewers at that time. The magazine furthermore asked the outside reviewers to continue their review in the light of the 2017 criticism of the proof, but says that said reviewers have now concluded that the proof is correct.
What I don’t think has been mentioned here is last paragraph of that article.
It states that (a) proofs in pure mathematics of things such as the abc conjecture have in the past led to practical applications and (b) that IUT is expected to be a powerful tool for solving a wide variety of difficult problems in mathematics. Furthermore (c) that a new research center (whose purpose is to promote this theory) led by Mochizuki had been created within the Kyoto University math department in 2019 and that the Ministry of Education, Culture, Sports, Science and Technology has created an annual budget of 4 x 10^7 yen (slightly under 4 x 10^5 USD) for said center.
If the proof weren’t controversial, that last paragraph wouldn’t be notable in the slightest. FWIW, a native speaker nonmathematician friend after reading said article was incredulous that there could be any problem whatsoever with the underlying math.
I have been weighing back and forth commenting again on this matter. However, the news in that last comment by David J. Littleboy convinced me that it might be good, even if futile, to say something again.
I may have not expressed this clearly enough in my manuscript with Stix, but there is just no way that anything like what Mochizuki does can work. (I would not make this claim as strong as I am making it if I had not discussed this for with Mochizuki in Kyoto for a whole week; the following point is extremely basic, and Mochizuki could not convince me that one dot of it is misguided, during that whole week.) It strikes deep into my heart to think that in the name of pure mathematics, an institute could be founded for research on such questions, and I sincerely hope that this will not come back to haunt pure mathematics.
The reason it cannot work is a theorem of Mochizuki himself. This states that a hyperbolic curve $X$ over a $p$adic field $K$ (maybe with some assumptions, all of which are always satisfied in all cases relevant to IUT) is determined up to isomorphism by its fundamental group $\pi_1(X)$, and in fact automorphisms of $X$ are bijective with outer automorphisms of $\pi_1(X)$. Thus, the data of $X$ is completely equivalent to the data of $\pi_1(X)$ as a profinite group up to conjugation. In IUT, Mochizuki always considers the latter type of data, but of course up to equivalence of groupoids this makes no difference. (The passage back and forth is even constructive, by another result of Mochizuki.)
Mochizuki claims that by replacing $X$ by $\pi_1(X)$, things can happen that cannot otherwise happen. Examples are given concerning the action of $\pi_1(X)$ on certain associated monoids. We discussed this at very great length in Kyoto, but none of these examples carried any actual content. Note that any potential noncommutativity of some diagram that results from identifying $\pi_1(X)$’s via isomorphisms of $X$’s could not possibly be resolved by using some other isomorphism of $\pi_1(X)$’s — all of them come from isomorphisms of $X$’s! Mochizuki considers infinitely many distinct isomorphic copies of $\pi_1(X)$’s, but could not tell us what goes wrong if we simply identify all of them with one another, and with $\pi_1(X_0)$ for some fixed $X_0$ — there is no diagram that commutes in his situation but does not commute under this further identification. (In my manuscript with Stix, we simply went through Mochizuki’s argument with this further identification, pinpointing what goes wrong. If this further identification causes problems, just tell us which diagram it is whose commutativity is rescued by not explicitly identifying $\pi_1(X)$’s.)
However, what I really want to do with this comment is to point out that there seems to be significant confusion over just the above point on $X$’s vs $\pi_1(X)$’s. Recently, arXiv:2003.01890v1 appeared, in which the author (Kirti Joshi) gives some survey on results related to Mochizuki’s work. In the introduction, on page 7, he explicitly claims that one could find nonisomorphic $X$’s giving rise to the same $\pi_1(X)$, and even more, in Remark 2.1 on page 14 he explains that my reading of the situation is a common misunderstanding. Even more, in Corollary 21.2 on page 47, he states something “wellknown to everyone at RIMS” giving an explicit example of this phenomenon of nonisomorphic $X$’s giving rise to the same $\pi_1(X)$.
With this appearing on arXiv, I was indeed quite confused — did I in fact misunderstand this basic point all this time? If the above claims would have been true, I would see how Mochizuki’s strategy might have a nonzero chance of succeeding. But I was quite sure that in our discussions in Kyoto, Mochizuki agreed with me on that basic point; and the proof of Theorem 21.1 in that survey (of which Corollary 21.2 is indeed a corollary) was wrong. In any case, I emailed Joshi indicating my confusion, and he has since checked back with Mochizuki and retracted all of these claims (he told me a new version will be on arXiv soon). In particular, the fact “wellknown to everyone at RIMS” is wrong, and in contradiction to this earlier correct anabelian theorem of Mochizuki.
I’m really frustrated with the current situation. What EricB reports from the Asahi Shinbun also sounds deeply troubling, effectively arguing along national lines; again, this strikes deep into my heart. I’m really quite surprised by the strong backing that Mochizuki gets from the many eminent people (who I highly respect) at RIMS.
If I can in any way help to mitigate the situation, I’d be most happy to.
Hi Peter!
First, hope your pandemic is going well. Mine is going ok. Hard to get things done without daycare.
Second, let me say that for hyperbolic curves over a padic field K (with no extra hypotheses like strictly Belyi type or Belyi type or canonical lift) that pi_1(Z) “determines” Z is open. Also, I personally would advocate against using words like “determines”, “reconstructs”, etc that have been causing sooo many problems in discussing all of these things. For the uninitiated, let me say that what Peter claims here is that outer isomorphisms of fundamental groups as topological groups are in bijection with isomorphisms of curves over a field K. The difference between what Mochizuki did in his relative case together with his interpretation of G_K and the absolute Grothendieck conjecture I claim is subtle (maybe I am missing something though, I’m a little intimidated saying this so publicly to be honest).
–to see this is a nontrivial topic consider for example the introduction to this manuscript here: http://www.kurims.kyotou.ac.jp/preprint/file/RIMS1892.pdf
Mochizuki’s theorem states that for Z and W hyperbolic curves over a padic field K, outer isomorphisms pi_1(Z) \to pi_1(W) *over* G_K — meaning they morphisms in an overcategory where pi_1(Z) \to G_K and \pi_1(W) \to G_K — are in bijection with isomorphisms between Z and W.
To prove this, it suffices for example to show pi_1(Z) interprets the field K. I have written down an unreviewed proof which I don’t think it is so difficult which I can share if you want.
Anyway, I think what you are thinking about that that pi_1(Z) admits and interpretation GG(pi_1(Z)) naturally isomorphic to G_K and since we have both the fundamental group and the augmentation map we should be in the setting of Mochizuki’s theorem and are done. This is not quite correct. What is at issue is that given f:pi_1(Z) \to pi_1(W) one does not necessarily know that GG(f) is inner (which would puts you into the hypotheses of the Mochizuki’s proof of the *relative* Grothendieck conjecture).
Also, Mochizuki has conjectured in print it is absolute Grothendieck over padic is not true—See Remark 1.3.5.1 of this paper: http://www.kurims.kyotou.ac.jp/~motizuki/Absolute%20Anabelian%20Geometry.pdf but seems to be nonspecific nowadays. I don’t know. We would need to ask him.
I personally believe if it is true then there is sort of a Zilber trichotemy/diochotemy thing going on. That is “absolute Grothendieck over padic” iff pi_1(Z) interprets a field. Maybe I am using Zilber trichotemy/dichotemy wrong, but I stand by the statement. The reason for this is because in all the special cases where absolute Grothendieck over a padic field holds, (canonical lifts, beyli type, strictly belyi type) there is an interpretations of fields. Maybe this is wrong though and a reader can point out an example where we know the absolute Grothendieck conjecture over padic fields without some special hypothesis being imposed.
Third, your counterpoint is a heuristic and not a disproof. For example, Dieudonne modules are equivalent to finite group schemes and we don’t call them worthless. We could go blue in the face coming up with examples of two equivalent objects one of which is useful and the other which is not. Maybe I am missing something, but I’m not sure how productive these meta discussions are. I think your point is that he needs to “use something”/”do something”. I agree, but it is not a disproof.
Alternatively, we can argue who has the burden of proof here… which I think is a more compelling argument for rejecting the whole thing. I think we can all agree the notation in his manuscripts are a big dumpster fire.
Fourth, regarding the deformation theory, his entire theory is up to a generic isomorphism (=polyisomorphism) the base G. That is the fundamental objects he considers are (like) a fundamental group Pi with a map Pi \to G_K where G_K is considered up to automorphism. In fact, any time he does one of these polyisomorphisms (which I think we should be calling generic isomorphisms), we should probably be thinking of some sort of poor man’s universal family.
Next we can ask “does this *DO* anything”? Well, for one, it certainly provides a formalism for talking about how things change under automorphism. There are a lot of interesting representations of automorphisms of fundamental groups acting on various interpretations which seem to me to be vedry nontrivial… etale theta being a principal example. Here is another thing I haven’t been able to puzzle out but maybe you can help: Let Z be a hyperbolic curve over a padic field. Fix the augmentatio map f:pi_1(Z) > G_K, let g:G_K > G_K be and outer isomorphism. What is the base change of f by g?
pi_1(X) \times_{G_K, g} G_K = ???
It is like some sort of Frobenius twist looking thing—except not twisted by the Frobenius but by some outer morphisms of the fundamental group of the base. Is this the fundamental group of a curve? Anyway, it seems to me you can apply many of his interpretations to this object and that constructions are uniform in this sort of thing. This I am not 100% on and wish I didn’t have to talk about these things I don’t understand so well publicly. Maybe the readers can tell us what breaks here.
So, that’s all I have to say about that for now.
Best,
Taylor
P.S. Can someone tell me how to add new lines? Hopefully I can edit my comment later. Oooh, looks like spaces are included. They just didn’t show up in the preview. Niccce.
@Taylor Dupuy
Your clarification is very helpful for arbitrary curves, but as you say at the beginning, is relevant when there are no extra hypotheses like strictly Belyi type. Since IUT focuses on etale fundamental groups of oncepunctured elliptic curves, and these are of strictly Belyi type, I don’t see why they are helpful.
In case anyone following is confused about what theorem of Mochizuki is meant, it is Corollary 1.10 of his paper “Absolute Anabelian Cuspidalizations of Proper
Hyperbolic Curves”. (Not his earlier theorem about arbitrary curves.)
Even if this “stricly Belyi type” condition were somehow avoided, then the existence of extra isomorphisms of the abstract fundamental group would still be an open problem. It seems hard to imagine how these isomorphisms could ever be used in a proof of some concrete inequality between two real numbers without proving that at least one exists.
It is probably possible to make rigorous a lot of what Peter is saying, that any proof of XYZ form cannot possibly work, or maybe more precisely that Mochizukki’s proof is equivalent to the wrong proof sketched by Scholze and Stix.
But one also should include notcompletelyrigorous evidence when deciding on how much burden of proof to assign to the author of a paper. The fact that every plausible use of a particular mathematical construction would not help in a particular argument, but the construction adds unnecessary complexity to the argument, and thus would make it harder to see any mistake, and two smart mathematicians have tried to remove the construction and found an incorrect argument, is strong evidence against the paper, even though it is possible that (1) there is a different way to remove the construction which leads to a correct argument or (2) there exists a clear explanation of how the construction is used to solve a problem which is not possible without it.
> Fix the augmentation map f:pi_1(Z) > G_K, let g:G_K > G_K be and outer isomorphism. What is the base change of f by g?…. Is this the fundamental group of a curve?
Isn’t the content of Mochizuki’s theorem that, if Z of strictly Belyi type, then this is not the fundamental group of a curve? Of course this may be a curve in some other case.
W writes:
>My understanding is that there exists a paper (1) written by Mochizuki (and others), (2) using the language of IUT theory, (3) that is correct, (4) which could be made much clearer by removing the IUT language and writing it in more typical mathematical style. Unfortunately the paper for which I understand this is not the papers on ABC, but rather a paper he wrote after, purely about fundamental groups / anabelian geometry. (My understanding comes from a friend who has read it.)
I’m not sure if I’m the friend to whom W is referring here, but I have had this experience and wanted to add a brief comment about it. I’ve gone through this paper:
https://projecteuclid.org/download/pdf_1/euclid.aspm/1540417834
and this paper:
http://www.kurims.kyotou.ac.jp/~yuichiro/rims1870revised.pdf
with some amount of care, and concluded that despite the nonstandard language, they are both essentially correct. As W writes, either could be made much clearer (and shorter!) by rewriting them in standard language. What’s worth noting here is that I was able to understand what was written in these papers and convince myself that they were correct, despite the nonstandard exposition. On the other hand, I was unable to do this with crucial parts of the IUT papers, even after putting substantial time into doing so.
@W, just to fix your precise reference for the convenience of others following along: You mean either Corollary 2.3 on page 500 of that paper (Absolute anabelian cuspidalization of proper hyperbolic curves); or Corollary 1.10, part (iii) on page 43 of the subsequent paper (Topics in absolute anabelian geometry III).
Those disprove Corollary 21.2 of Joshi’s preprint.
Taylor, thanks a lot for your answer!
Regarding your first point: Yes, I’m doing fine in these strange times; I hope you are too. Fortunately we are still allowed to enjoy the incoming spring.
Regarding your second point: As W observes, your objection only seems relevant when the curves are not of strictly Belyi type, but all the relevant ones for IUT are. So this objection is a red herring. (Meanwhile, it is clearly interesting to sort out whether strictly Belyi type is necessary, and I wish you luck in improving Mochizuki’s anabelian results!)
Regarding your third point: Of course I am very well aware of the power of category equivalences. But there must be something you can do on the other side that you can’t do on the first. As I said, during one week Mochizuki was not able to give a single relevant example. So to me it seems like a category equivalence that simply obfuscates things.
Regarding your fourth point: I am at a complete loss what one wants to do with full polyisomorphisms. For the convenience of readers following along: A full polyisomorphism between two isomorphic objects $A$ and $B$ of a category $C$ is the set of all isomorphisms between $A$ and $B$. Mochizuki often says that he identifies two objects $A$ and $B$ along the full polyisomorphism. To be clear, this is (up to equivalence of categories) no data at all, there is a unique full polyisomorphism, so you can’t “pick one” (or rather, you can always pick one, and only one). On the other hand, to identify to objects in a category, you need to pick a specific isomorphism! Of course, you can just pick any one of them, but you can’t pick all of them! If $A$ and $B$ are sets and $a\in A$ is an element, then what is the image of $a$ in $B$ under the full polyisomorphism? It makes no sense. Mochizuki has repeatedly told us in Kyoto that because some diagram does not commute when $A$ and $B$ are identified via the obvious isomorphism (that usually exists in his situation), he has to identify them only along the full polyisomorphism. But (in his situation) there is not a single isomorphism between $A$ and $B$ that makes the diagram commute! So how does the full polyisomorphism help? You can’t make the logarithm map into a map of fields by pre and postcomposing with field automorphisms!
(To be clear, I am willing to accept that there is a nonzero chance that some of these things might make sense under certain circumstances. Again, let me stress that we discussed these very matters for one week in Kyoto.)
From what I understand, the objection to my manuscript with Stix is that we did some identifications that are not allowed. This is just the identification I was talking about: Mochizuki considers infinitely many distinct copies of $\pi_1(X)$ and is only allowing himself to identify them along the full polyisomorphism. We do not see any diagram that commutes with this choice, but does not commute when we identify them along the identity map, once taking all $\pi_1(X)$ to be equal to the actual $\pi_1(X_0)$ for our fixed curve $X_0$. In any case, the passage from $X$’s to $\pi_1(X)$’s is giving you no extra flexibility, as discussed in my previous comment.
Best wishes,
Peter
Hopefully this is productive…
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second point (and W’s comment on the base change thing):
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I’m not sure how much to engage here since this isn’t strictly logically necessary for a disproof. Since I think this is an important point in terms of our understanding of what is going on I’m going to say a few words. First, for the readers, Peter and W are focusing on the bad nonarchimedeans cases, I believe. In this situation, I’m not 100% sure that all the curves in what I would call the “zoo of covers” are SBT but certainly most of them are as they are analytifications of base changes of hyperbolic curves from number fields. This is Belyi’s Theorem. There are a number of other points that we should consider: there are log structures too which should make things even more rigid (my understanding is that you need this information for reconstruction of the special fiber), you have these punctured tempered universal covers (these are certainly not algebraic), there are groups involved in monotheta environments going on, and stacks fundamental groups. My point is that it is still conceivable to me that there is some sort of deformation theory going on and I don’t want to confirm or deny this. But yeah, a lot of them are. The base fields certainly are not.
In terms of uniformity of constructions in $G$ or “strange base changes”; Mochizuki performs many constructions with respect to $(\Pi,G)$ where for a pair we consider a generic isomorphism $\mathbf{G}(\Pi) \to G$ — here $\mathbf{G}$ is the interpretation of a structure isomorphic to $G_K$ in $\Pi$. This makes $G$ “up to automorphism”. You need Lemma 1.1.4.ii of “Absolute Anabelian Geometry of Hyperbolic Curves” for the interpretation $\mathbf{G}$.
W: “Isn’t the content of Mochizuki’s theorem that, if Z of strictly Belyi type, then this is not the fundamental group of a curve?”
I would say no, this is not the content of his theorem. But I don’t know what you mean here. Mochizuki’s theorem is a statement about morphisms not objects. The begining of the claim is
“given that $\Pi \to G$ and $\Pi’\to G$ are augmented fundamental groups of hyperbolic curves…”
There may be statements that allow you to classify when a group is the fundamental group of something but you will have to ask Daniel Litt or Emmanuel Lepage or Jakob Stix about this. They certainly will be much more knowledgable than me. By the way, this is what Mochizuki means when he calls things “bianabelian”.
Let $\Pi = \pi_1(Z) \times_{G_K,g} G_K$. I think you can run the $\mathbf{G}(\Pi)$ interpretation on this thing. I haven’t worked out what this does… it is unclear if $\operatorname{pr}_2:\Pi\to G_K$ is naturally the same as to $p_{\mathbf{G}}: \Pi \to \mathbf{G}(\Pi)$.
Also, Peter, don’t hold your breath waiting for new results in this direction. What I said is pretty much the extent of what I “know” about absolute grothendieck conjecture over padic fields.
Also, I think we can all agree whether something can change or not due to Mochizuki’s consideration of potentially lossy functors is one of the more interesting parts of this story and it is important that we as a community clarify the situation. I think Kirti’s emphasis on this is a good idea.
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fourth point:
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https://imgflip.com/i/3vr2vv
Full polyisomorphisms should be considered as “generic isomorphisms”. I completely agree with this viewpoint. There are some instances where you have polyisomorphisms which not full and these keep track of finite indeterminacy rather than an arbitrary choice.
For better or for worse Mochizuki decided this is the language he wanted to use to describe these things. When things are omitted without justification the statements made are no longer Mochizuki’s. We can get mad at him all we want for over questionable style choices but it doesn’t change his assertions. This, I believe, is grounds for rejecting a paper, but it doesn’t disqualify the proof.
I want to give a couple comments on polyisomorphisms for the readers.
Comment A) First, I want to point out a really really really bad style problem that come with this choice of “polyisomorphism language”: Many commutative diagrams involving full polyisomorphisms are tautologically commutative which makes many claims vacuous.
example: Theorem 3.11.iii.a, the $\vdash \times \mu$ prime strip commutativity statements.
These are everywhere.
This style choice forces the readers to search unnecessarily for nonfull polyisomorphisms which, frankly, is a big pain (an example of a nonfull polyisomorphism can be found in the definitions of the bridges of the Hodge Theaters for example). This doesn’t make him wrong though. Just not the best expositor.
Comment B) If $A$ and $B$ are isomorphic objects then $\operatorname{Isom}(A,B) = f \operatorname{Aut}(A) = \operatorname{Aut}(B) f$ for any fixed isomorphism $f:A \to B$. In my head I always “push” the generic isomorphism into an arbitary automorphisms of one object. This is pretty tautological and Peter already does this but I just wanted to say that applying this systematically allows you to reduce a lot of things and perform computations.
Comment C) I have spent a bit of time literally identifying objects if they had a polyisomorphisms between them. Although you can actually get some pretty interesting “global objects” this is not what Mochizuki had in mind. One example comes from the socalled full polyisomorphism monotheta environments and the loglinked fields. If you do this you end up identifying a bunch of roots of unity in a bunch of different fields — this is well defined because the fields are all loglinked. Anyway, my point here is that this is not what Mochizuki had in mind so you maybe don’t want to do this.
(I can talk at length about other style issues. Another example is invocation of interpretations “in Hodge Theaters”. This is one I feel that just sends readers on a wild goose chase reading page after page of definitions. Things are typically defined by much less and in an optimal presentation one shouldn’t consider superfluous structure when a reduct will do.)
Ok, so what do generic isomorphisms do besides confuse readers?
First, they are intended to keeps track of automorphisms. Mochizuki’s theory is really an investigation of the behavior of interpretations under automorphisms and permutations of the interpreting structure. I say permutations because sometimes they don’t respect the category and are maps of sets. The stupidest example I can think of: Let $G\to G’$ be a map of groups with kernel $N$. Set theoretic permutations of $G$ which fix cosets $gN$ setwise induce the identity on $G’$.
Here are some example questions Mochizuki addresses:
–How does the kummer class of (an $l$th root of a pullback of) the Jacobi theta change when you take an automorphims of $\pi_1^{temp}(\underline{\underline{X}}^+)$?
–What about the evaluation points (conj classes of decomposition groups)?
–What types of automorphisms/actions stabilize what construction? (I am thinking about monotheta environments here and the purpose they serve)
–What happens to a the measure space we construct when we take automorphisms of the interpreting $G$? (this is what Ind2 is)
Second, generic automorphisms can serve as a sort of “poor person’s deformation space”. I think Kirti discusses this well in his updated manuscripts so I’m not going to talk about this so much. I think, as Kirti has suggested, we should be asking ourselves which of these generic isomorphisms are actually representable. Emmanuel Lepage, if he is reading, might be able to say more about the usage of full and essentially surjective functors from the perspective of Gerbes. He told me something about this once.
Third, regarding your “what is the image of an element under a full polyisomorphism”. This goes back to what the point of all these things are. We are really investigating interpretations under automorphisms. At the end of the day we are looking to construct a “multiradial representation of the theta pilot object” which is a region obtained by a procedure involving certain orbits. This is supposed to be 1) independent of any choices and 2) relatable back to our original $P_q$. We partially address (1) in my first manuscript with Anton which should be available soon. You can find some stuff on my vlog about indeterminacy diagrams. Anyway, the point of these comments is that the polyisomorphisms are Mochizuki’s way of dealing with choices, it is also safe to make a choice but you need to make things independent of this choice.
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fifth point:
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–Regarding your manuscript with Jakob, the construction as you have stated imposes two normalizations that are alleged to be simultaneously enforced which lead to a trivial contradiction of the form A=B and A!=B. I think we all agree on this. Mochizuki says this is a straw man. I can’t find these assertions in the manuscript.
I do however think that statements of the form “All proofs that use X must have property Y” could be very useful provided 1) we can make X completely rigorous and 2) Mochizuki’s language can be pinned down in a way that makes X verifiable.
Aside: Actually, this was part of my motivation for looking at these interpretations. If you can show that two objects are equivalent (e.g. Frobenioids and pairs $(\Pi,M)$) then if there exists a proof using Frobenioids then there exists a proof using pairs $(\Pi,M)$. These sort of reductions allow you to make assumptions about the structures that are used in the proof. This is what safely allows us to get rid of unappealing constructions.
I personally find a lot of the language very hard to falsify/parse.
–Regarding the sentences “This is just the identification I was talking about: Mochizuki considers infinitely many distinct copies of $\pi_1(X_0)$ and is only allowing himself to identify them along the full polyisomorphism. We do not see any diagram that commutes with this choice, but does not commute when we identify them along the identity map, once taking all $\pi_1(X)$ to be equal to the actual $\pi_1(X_0)$ for our fixed curve $X_0$.”
First, I don’t think this is a faithful presentation of Mochizuki’s setup but one thing I can say that might make you feel better: in this particular example of bad nonarch primes setting $X_0 = \underline{\underline{X}}_{\underline{v}}$ (this should be a double underline) then automorphisms of the fundamental group do all sorts of things to the zoo of covers. All of those dihedral symmetries act, all of the $\underline{\mathbb{Z}}$ symmetries act etc. Also, all of the stuff that is not interpreted from a fundamental group stays fixed. This seems particularly relevant in the context of “monotheta cyclotomic synchronization”, but again, if I could finish the proof we wouldn’t be having this discussion at all.
Sorry if I made any mistakes anywhere…
Taylor — you’ve written quite a lot, but frankly I don’t really see why anyone should continue discussing this until someone can point to a specific spot in the IUT papers which defeats Peter’s objection. For example, if I understand correctly, you suggest some juice might be obtained by looking at maps of pi_1 which are not over G_K. If so, where are such maps used in IUT? And Peter gives a specific challenge — point to some diagram whose commutativity is rescued by not making the identifications Peter makes. If there is such a diagram, where is it?
I’m convinced there is no such diagram, because that is not how category theory works. The whole issue with identifications (as in: demanding objects are distinct copies vs having them be the same object) is a red herring. Mochizuki doesn’t want to do it for spurious technical reasons that one can ignore (or alternatively, humour him and agree to play along). I’m more suspicious that people could be thinking about objects that are living in different categories. But what do I know?
David — indeed there are objects in different categories being identified here (eg a curve and its fundamental group). That’s because we’re speaking English, not trying to make formal mathematical statements. Formally, one might (for example) look for a diagram which is not commutative but whose image under some functor becomes commutative, say. That is something that can happen in “category theory.” In any case, I agree it’s likely there’s no such diagram.