Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added to inter-universal Teichmüller theory a pointer to the recent note
(Though after reading I am not sure if that note helps so much.)
Linked to Initial Θ-data
Scholze’s refutation is out: https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/
Thanks! The situation is very sad. I am not in the slightest able to judge, but I am suspicious of a comment that Scholze and Stix make about identifying a load of things for simplicity. I mean, if category theory tells us one thing, it is that one can lose information in this way. What I find sad is the sociological aspect that every number theorist and their dog will jump on this, whilst it is absolutely clear that Mochizuki is trying to do something very original, and whether or not he has succeeded, I would be astonished if there are not deep ideas of value in the IUTT papers.
All the documents are hosted by Mochizuki here
I wonder what, technically, is meant by “poly-isomorphism”.
I don’t feel the situation is so sad. There’s a lot of general noise, to be sure, but to me it sounds as if the air is beginning to clear at the highest levels of discussion (as they currently are), which has been such a long time in coming. Maybe in a year or two the general takeaway from Mochizuki’s work will become appreciated.
“Full poly-isomorphism” appears to refer to the set of all isomorphisms between a pair of objects. Otherwise it’s some set of isomorphisms between a fixed pair of objects (sometimes, apparently, chosen to be an orbit of some action on the full hom-set).
To elaborate a bit on #4, it seems to me that Scholze and Stix’s argument goes something like the following.
a) Identify a load of things with simpler ones.
b) Make some arguments involving the simpler ones.
c) Deduce that Mochizuki’s argument in a key step is invalid because he is not careful enough in his identifications.
Would this be convincing if it were not for the sociological context?!
@Richard
not really, since it possible that ’identify a load of things’ used the wrong isomorphism. If we’ve learned anything from Univalent Foundations it’s that it is fine to talk of isomorphic things as equal, just that we have to remember which element of the identity type we mean. Scholze–Stix are using refl in many places, some of which may be ok, and potentially some may not.
EDIT: I mean, it would be ok if Scholze–Stix managed to avoid the problem I mention. It’s fine to say that you have different copies of the real numbers (or, more appropriately, one-dimensional vector spaces over the reals), just that anytime they are related somehow, one doesn’t just assume that it’s via the identity map. Mochizuki claims that they assume a bunch of maps are linear, when they’re not, and this is the best bit of information in his three (so far) documents. Everything else that I can understand seems to be window dressing, or a failure to have fully internalised the principle of equivalence (I got the feeling of this latter option when I read the appendix of IUTT4 on universes).
Mochizuki claims that they assume a bunch of maps are linear, when they’re not, and this is the best bit of information in his three (so far) documents.
Could you point me to where you saw that?
when I read the appendix of IUTT4 on universes
The first part of that looks quite approachable, I should try to read it in detail!
I’ve begun reading the appendix, and what he writes looks good and interesting to me. Which bits were you thinking of with regard to the principal of equivalence?
Richard #11 see this MO comment for your first question.
Regards #12, the fact M worries about categories and sets that make up those categories accidentally participating in circular $\in$ cycles, which is of course forbidden by Foundation in ZFC. Since mathematics is largely foundation-agnostic, one could presumably formalise everything he is doing in CCAF, or at least in AST and no one would be the wiser. Not to mention that appendix is never actually referred to or used!
His insistence on different labels for specific instances of isomorphic objects but being vague on actual structural/functorial reasoning, and talking about ’alien copies’ when a simple category-theoretic explation would do makes me feel (and I may be wrong) that his thinking is rather material, but he’s trying to be structural. I would think that Scholze is much more structural in his thinking, and seems to have a reasonable grasp of when identifying isomorphic objects is ok, but there could be some subtlety that is missed in all the camouflage of M’s flowery writing.
Thanks! I will look into it further. One point that strikes me is that it seems that, if we draw an analogy with simple algebraic objects like a group, Mochizuki seems to work, in the appendix at least, with something akin to presentations of the objects rather than the objects themselves. One can certainly define categories of presentations (of groups, say), and these are much more rigid than the categories of algebraic objects themselves. In particular, there are numerous isomorphisms on the level of the algebraic objects which do not lift to isomorphisms of presentations. The difficulty is of course to do anything useful at the more rigid level; if this picture is correct, and if Mochizuki has indeed managed to do some kind of anabelian geometry at the level of ’presentations of elliptic curves’ rather than elliptic curves themselves, that would be extremely interesting in itself.
I have actually tried something like this myself in knot theory, with quandles, etc. In order to be able to get something done, one seems invariably to end up needing to endow the presentations with quite a lot of structure; maybe that would explain why the definitions in Mochizuki’s work are so intricate.
I am intrigued to learn here that the critical steps in the proof are not some hard number theory-arguments, but, if your all comments are anything to go by, subtleties of passing to equivalence classes, hence of de-categorification.
If you could nicely extract and highlight just that point in, say, a brief blog post, that might be something really valuable for a larger part of the otherwise perplexed community, to take away from the whole event.
Now I am beginning to think that those people who, on MO and elsewhere, suggested that proof checkers might help here, and whose suggestions I saw being shot down for allegedly being way too naive about the actual problem with the proof, might really have had the right idea.
If, from what Scholze seems to be saying, the bulk of the IUT articles is actually easy going arguments (“nothing much is happening here”), with all the subtlety then concentrated in one single lemma sorting through nothing but a dizzying web of isomorphisms, then that might actually be just the right hing to try to look at with a proof checker.
Kiran Kedlaya, in the section “Mochizuki’s approach to diophantine approximation” here, wonders whether HoTT could be helpful.
But there is more than enough hard number theory for me at least to be unable to say something that would have a chance of not being dismissed by number theorists. But I remain very interested to work through Mochizuki’s papers, and am still blocked by the lack of explicit examples of the basic phenomena (when I mentioned this to Mochizuki, he wrote to say that he would encourage his students, etc, to consider including such examples in future expository work). I must say that I was disappointed by the MathOverflow post that I made a while ago asking for an explicit example; there were several number theorists who left comments there who have expressed skepticism about Mochizuki’s work, yet I do not see how one can be qualified to do so if one cannot easily give an example of the first significant definition…!
skepticism about Mochizuki’s work, yet I do not see how one can be qualified to do so if one cannot easily give an example of the first significant definition…!
Interesting. The same argument would seem to apply to non-skepticism, too. Hopefully this is not all just about the empty set, then. :-)
@Urs #14 was that directed at me?
Yes, at you and Richard.
Ah, I see. I’ll can put something together. I was going to do so on my own blog, but the n-café would be more visible.
Why not your own blog? Make it your thing. You seem to have genuine and useful insight here, make yourself heard.
Just a quick further note to #13: comment (LbEx1) in Mochizuki’s report (page 25), for instance, seems to reflect more or less exactly what I was getting at. In particular, I do not think I would share David’s suggestion that Mochizuki may be misunderstanding structural mathematics; whilst the terminology is unconventional, I think he may simply be trying to make the same kind of point as I made in #13, in which case his point is certainly a valid one, as one certainly cannot ’simplify’ presentations to the algebraic objects they present and expect everything to work out.
Ah, thanks, I’ll check that out. I skimmed the long report very briefly, but couldn’t face reading it in-depth at the time.
A quote of Poincaré I just read
Mathematics is the art of giving the same name to different things (Science and Method, 1908)
I’ve now looked at the report that RIchard mentioned in his comment above.
I’m not so convinced by (LbEx2), since the colimit he discusses could be calculated using $\mathbb{N}$ as the diagram shape, which takes care of worrying about $A_0$, and since in fact $f\cdot$ is an endomorphism, really this is a diagram of shape $\mathbf{B}\mathbb{N}$ [Edit this is not correct, though the diagram does factor through $\mathbf{B}\mathbb{N}$, so its image is of that shape - see comment #42 below ]. There is no reason whatsoever to use different labels for ’different copies’ of $A$.
I don’t claim to understand perfections of rings, but I’m definitely suspicious of (LbEx3), claiming that
Here, we observe that, if one forgets the labels $n\in \mathbb{Z}$, i.e., and just considers the inductive system given by a single copy of $A$ and the transition morphism $\phi\colon A\to A$, then one computes easily that the resulting inductive limit is isomorphic to a direct sum…
other sources I’ve looked at define the perfection by using again a diagram of shape $\mathbf{B}\mathbb{N}$ [Edit again, the image of the diagram is of this shape]. The direct sum is most definitely not the colimit of such a diagram.
The argument about line segments and loops (which is just an observation about graphs) in (LbEx1) is not relevant here, since colimits of diagrams of endomorphisms are what they are. If one is working in ind-objects then it’s a bit more subtle, but the specific diagram shape isn’t necessarily uniquely specified. SM writing snide things about Scholze and Stix not understanding the theory of heights is one thing, but confusing basic category theory like this is super weird.
It gets stranger: (LbEx4) is an observation about field extensions which makes a fallacious argument akin to the hole problem in relativity. Any decent abstract treatment of Galois theory would be able to resolve this ’issue’.
(LbEx5) is more geometric in flavour, talking about how different charts on a manifold are secretly different copies of $\mathbb{R}$. SM is talking about how a manifold is a quotient of some $\coprod_I \mathbb{R}^n$, and how someone might get confused if they identify all the copies of $\mathbb{R}^n$ and think that the geometry of manifolds is all about $\mathbb{R}^n$. Structurally, one can talk of the copower $I\odot \mathbb{R}^n$ where $I$ is an indexing set (manifolds are copowered over sets, possibly with cardinality bounds if one insists on second countability). One doesn’t need special labels on the Euclidean space, it’s just that some other construction has happened. And all this is before even talking about the transition functions, which one could be viewed as living in some pseudogroup of local automorphisms of a fixed cartesian space. This paragraph shows a complete absence of structural understanding
If one identifies these distinct copies of Euclidean space that appear in the theory of (topological or differential) manifolds with one another, then it is easy to obtain a “contradiction”. Such a “contradiction” may, of course, be misinterpreted as an “internal contradiction” in the theory of (topological or differential) manifolds. In fact, however, there is no “internal contradiction” in the theory of (topological or differential) manifolds; the apparent “internal contradiction” is nothing more than a superficial consequence of the erroneous operation of omitting the labels used to distinguish the distinct copies of Euclidean space.
SM then goes on to invoke fallibility of human memory to risk forgetting the correct labelling and incurring
the risk that different people will “remember” different labeling appartuses, which result in structurally non-equivalent mathematical structures.
And here I don’t think he means structural in the sense the nLab does, rather “non-equivalent” or “non-isomorphic”. This reads like he’s projecting here…
Could you add a link to the document with those LbEx-s in them?
Hi Urs,
I’ve edited the post, and also now added the link. I don’t claim to understand the rôle of poly-isomorphism, the only thing I can imagine they are good for is if the hom-set is itself an object that is acted on, cf Minyong Kim’s work on nonabelian stuff using ’torsors of paths’ i.e. hom-sets of étale or pro-unipotent fundamental groupoids (see eg this paper, which is perfectly good anabelian geometry). But it’s possible poly-isomorphisms do something nontrivial that I haven’t seen yet.
Thanks. And thanks for your concrete analysis. That’s the way to go. I sense the possibility of basic category theory making a decisive difference in a disputed topic of hard core number theory. If you can bring that out in a little article, much good will have been achieved.
OK, I think have nearly enough to compile now. I will plow through SM’s report some more to see what I can get out of it.
Thanks very much for this, David! I don’t have the opportunity just now, but will read properly through #23 and reflect upon it later.
Actually, (LbEx5) can be answered in a better way: the coproduct $\coprod_I \mathbb{R}^n$ is the colimit in the category of manifolds (or of the free finite coproduct completion) of the constant functor $disc(I) \to Mfld$, $I\ni i \mapsto \mathbb{R}^n$. One doesn’t need ’differently-labelled copies’ of $\mathbb{R}^n$ here.
Re #25: what post?
@Todd my #23 above. It was about half the length originally, and my edits overlapped with Urs’ #24
Oh, I see – you mean #23 is the post? I thought maybe it was a Google+ post or something.
I’m only looking now at Mochizuki’s report. I have to say that I am finding parts shockingly awful, e.g., nos. 2 and 3, as examples of real bitterness and sarcasm. It’s hard for me to imagine substantive discussions taking place this way.
I agree that those comments are very unfortunate, and are not going to help. One can only imagine, though, his frustration. He is also probably aware that someone with Scholze’s reputation suggesting that there is an error is likely to impede many/most from trying to understanding his work, and thus has felt a need to react very defensively. So I think one can have some sympathy and understanding for him in that he felt that he had to write those things, even if one wishes he had not done so.
I suppose you’re right, Richard (and your last comment is yet another example of the kindness and charity that I have seen so often from you).
And I have to agree with David that this diagram labeling business is very odd. One thing though about #23: we do have to distinguish between colimits over diagrams of shape $\mathbf{B} \mathbb{N}$ (a one-object category under my reading of $\mathbf{B}$) and those over diagrams of shape $\bullet \stackrel{f}{\to} \bullet \stackrel{f}{\to} \ldots$. For example, in $Set$ and taking $f: \mathbb{N} \to \mathbb{N}$ to be successor, the colimit of the first would be a one-point set, and the colimit of the second would be $\mathbb{Z}$, as mentioned at integer. Or did I misread you, David?
(Thank you for your generous words, Todd!)
David, I’ve begun reading/reflecting on #23 now. Before I say anything further, let me say that I agree with Urs and that you should definitely go ahead with your blog post, it will certainly be a very useful contribution to the debate!
I still think that I disagree with you, but don’t be put off by that :-).
(LbEx2): I wonder if the problem here is mostly Mochizuki’s ’label’ terminology. He is basically just saying the following as I see it: one cannot take the colimit in non-unital rings and expect to get the same answer as in unital rings. In other words, if one wishes to forget the fact that one has a unit in one’s ring, one cannot just imagine that one’s colimit is going to give the same answer as before. Remember that what he is trying to convince one of is that Scholze and Stix’s simplifications/forgettings are not justifiable.
(LbEx3): I agree this is confusing as written. However, I think the point is that he is looking at the underlying $\mathbb{F}_{p}$-vector spaces, i.e. forgetting the ring structure, after computing the colimit. It is true that as an $\mathbb{F}_{p}$ vector space, $A$ itself is isomorphic to an infinite direct sum of copies of $\mathbb{F}_{p}$.
I also think that, whilst the notation again has the potential to be confusing, he only intends the diagrams to be taken over the natural numbers, not over $\mathbb{Z}$.
Thus once more, I think the point is simply that one cannot ’forget’ or ’take a quotient’ and expect to get the same answer.
(LbEx4): I don’t see anything wrong with this, though it could be more clearly expressed. I just think he is making the same point once more, in a way quite close in this case to what I wrote in #8.
That is: there is a difference between a) the pushout of two copies of the embedding of $\mathbb{Q}$ in $F_{1}$, and b) the pushout of the embedding of $\mathbb{Q}$ in $F_{1}$ and of the embedding of $\mathbb{Q}$ in $F_{2}$. This is the case even though there is an isomorphism between $F_{1}$ and $F_{2}$: if one ’simplifies’ by identifying $F_{1}$ and $F_{2}$, it does not mean that one can replace the second pushout by the first one and expect everything to work out.
(LbEx5): Actually I think this may be the clearest of all of the 5 examples. He is just making the obvious observation that even though one can work ’locally’ on a manifold as if it was Euclidean space, it does not mean that one can actually simplify things so much as to just take a Euclidean space and work there!
In the setting of IUTT, Scholze and Stix do something very similar to this.
In summary, I may be the only person on the planet outside of those closely acquainted with Mochizuki of whom this true, but if I were forced to ’choose sides’ with regard to guessing which of Scholze/Stix and Mochizuki are the more correct, I would still go with Mochizuki :-)!
@Todd #35
Oh, of course. What I was thinking was that the sequential diagram factors through $\mathbb{N}$, but of course this doesn’t mean you can take the colimit over that. I was wondering why such a rookie mistake, but it was mine :-)
But the thinking is still so far from structural. If we would write down some categories and functors it would be much more clear, and all the bookkeeping would be taken care of.
@Richard #38
That comes down again to diagram shape, not the identity of the fields in question. It’s a pushout diagram, not a coequaliser. Also, the cocone is part of the data of a colimit, you need to know the embeddings of the two extensions of Q (which could be constructed abstractly as quotient rings for all we know) in C to call it a pushout.
Re #41:
Also, the cocone is part of the data of a colimit, you need to know the embeddings of the two extensions of Q in C to call it a pushout.
I agree, and I think this is precisely (part of) the point that I think that Mochizuki is trying to make, and which I tried to explain in #38. I.e. I think he is just saying: we have two different pushout squares, and ’identifying $F_{1}$ and $F_{2}$’ does not magically make those two diagrams the same.
So it seems all he’s trying to say is that given a diagram $X\colon D\to C$ that factors through another diagram, $D\to D' \stackrel{Y}{\to} C$, you cannot (necessarily) compute $colim X$ as $colim Y$, and given a colimit a different choice of cocone gives an isomorphic result, possibly by a non-trivial isomorphism. If that’s the entire content of the disagreement then I’ll be somewhat surprised. This is a perfectly fine structural statement and doesn’t need several pages of blather including statements about human memory being fallible…
Re #43; Yes, I think you are on the right lines here.
If that’s the entire content of the disagreement then I’ll be somewhat surprised
Yes, I do think the substance of the disagreement is simply: one cannot forget or take quotients and expect everything to work out. Of course the precise places in which such simplifications manifest themselves as problematic is a difficult matter, and needs understanding of the details of Mochizuki’s work.
This is a perfectly fine structural statement and doesn’t need several pages of blather including statements about human memory being fallible…
Whilst I would guess that the report is a sincere attempt to clarify things, I do agree that the unconventional terminology is more likely to cause confusion than help, and that the proliferation of justification makes the waters muddier than a simple, direct point would. I imagine that Mochizuki feels that the latter is not possible in a way which captures the nuances of the matter, and that he feels it his duty to try to express those nuances.
[edit the following stuff about rings is a bit off, but the underlying categorical idea is sound]
Regarding #36, it’s again a question of cocones. Assuming the diagram is indexed by a countable, unbounded linear order $L$, then there is no natural basepoint. Take for concreteness the diagram $L\to Ring$ [Edit well only $L\to Ab$] sending every $l\in L$ to $\mathbb{Z}$ and every arrow to the multiplication by 2 map. The colimit of this is the [underlying additive group of the] dyadic rationals , but one needs a choice of cocone to say this, and different cocones with gives scalings of the result, since the colimit is 2-divisible. The isomorphism will be non-unital, as the transition maps are non-unital, as are the maps in the cocone. So a priori one is taking the colimit in the category of non-unital rings, but one could consider this diagram in the category of initial rings with non-unital maps, in which case the vertex of a cocone needs to be a unital ring to start with. A different choice of cocone gives something isomorphic in this category, but still unital.
I’m less sure it’s poor explaining, even if deep down he’s feeling the correct ideas, than not knowing the relevant terminology to explain the subtleties. I get the feeling that in certain parts of number theory people don’t use categories much.
I haven’t yet come across anything I can understand that has nuances that requires metaphors and ill-explained examples. If there are nuances it may be in the abelian geometry, but others are more equipped to answer that than I. This is not to say Scholze and Stix are correct, they may have overstepped the mark, but SM’s examples are not helping, because they betray a crudeness of appreciation for category theory. Or perhaps he feels people cannot understand abstract category-theoretic reasoning, with reference to diagrams and cocones, and we require all kind of baby analogies to figure it out, but that is an insult to the collective intelligence of mathematicians and a waste of people’s time.
David, just to amplify that to make yor point valid and useful you don’t have to decide who of the two teams is right. Your point would be that, in either case, what makes the discussion lead to disagreement in the first place is the failure of it being formulated in the appropriate modern mathematical language designed to make such trivialities of re-identification be not only trivial but obvious. You would have occasion to quote Freyd’s “purpose of categorical algebra is to show what is trivially trivial” here and you should convey that the writeup and debate of the abc-proof (irrespective of its validity or not) provide a grandiose example of the relevance of this whimsical-sounding dictum.
@Urs
Yes, this is the point I am emphasising in my notes. Also, I just noticed you shared a screenshot that included my incorrect statement in #23 above. Would be nice if something could be done pointing people to the correct argument in #43.
I will share a draft of my notes here when they are closer to being finished.
Sure, have removed it for the moment. Will make an announcement when your notes are more finished.
Re #45 and #46: Thanks for your thoughts, David! I think I’ve written as much as I can usefully say at the moment about that part of the report for now; I think my feeling remains the same, but as before, I agree with Urs that your thoughts will surely be a useful contribution to the debate anyhow. My own suggestion (which you are completely free to ignore!) would be to not put the emphasis on things like ’lack of understanding of structural mathematics’, because Mochizuki is a careful mathematician, and from what I have seen I would be surprised if he has made basic errors; but, as Urs I think suggests, pointing to the fact that careful use of the terminology of structural mathematics could be helpful in trying to resolve the disagreement is a perfectly good and valid point.
The only way to resolve the disagreement in the end, of course, will be to understand Mochizuki’s work in itself. I.e. it may be that Scholze/Stix’s argument can be upgraded to an actual error, or it may not be, but the only way ultimately to tell is to formulate it in Mochizuki’s own terminology and point to a precise statement in his papers that is incorrect.
Thanks, Richard. At the very least, Mochizuki invented his own terminology for existing concepts, and doesn’t link his own formulation to one that everyone else uses, so that it’s unclear what is a metaphor, what is an algorithm, a functor, what can and can’t be done with the system (I mean, mathematically, from the definitions, not what M says can and can’t be done, using his own terminology and metaphors, ’alien copies of scheme theory’ and so on)
Yes, you’ve probably seen this. The newspaper article, originally German, linked to there expresses the terminological difficulties poignantly. (Incidentally I had not seen this article before I wrote #2, even though it has something of the same sentiment.)
The following is not completely clear to me, though; that he has invented a profileration of new terminology is of course true, but whether there is deliberate use of new terminology for existing concepts is not evident to me.
Mochizuki invented his own terminology for existing concepts,
On a different note, I just happened to notice that section A.4, pg.364-365 in Yamashita’s survey seems to address quite directly the point that Scholze and Stix raise as problematic. In particular, it seems that it is clear that there are problems with the argument if one ’carries it out in traditional scheme theory’; that it is really here that Mochizuki’s theory is needed; and that the ’indeterminacies’ are crucial. Scholze and Stix remark that they have not understood the latter. Perhaps it is here that energy needs to be expended.
Re #45: sorry to nitpick, but the chain obtained by the multiplication by 2 map is not a functor to $Ring$ (not even rngs = nonunital rings). Or it looks a little strange to say it this way since the map $\mathbb{Z} \to \mathbb{Z}$ defined by $x \mapsto 2x$ does not preserve multiplication ($(2x)(2y) \neq 2(x y)$). I surmise that what is envisaged is that the ideal inclusion $2 \mathbb{Z} \hookrightarrow \mathbb{Z}$ is a rng morphism. But that’s a different statement since neither abelian group isomorphism $\mathbb{Z} \to 2\mathbb{Z}$ is a rng map.
Gah… Not nitpicking at all, I just didn’t think it through. My better write-up does not include this example in any case.
@Richard
If I understood what ’indeterminacies’ were I might feel better. It seems with the polymorphism stuff M may want to work in some quotient category, where the objects are the same but the hom-sets have been quotiented. This is the case for the guiding examples he gives in the Report.
Either that or he’s resisting using 2-categories, which he explicitly states in the case of Cat, preferring instead to work with isomorphism classes of functors. I find this super weird, given his insistence of the importance of labelling: the value of a functor at a single object is then specified up to mere isomorphism, in the technical sense, or is rather a clique, or anaobject.
Yes, it remains a formidable task to understand the logic of the papers. I think quite often: I’ll try to just sketch the logic of the approach, taking as black boxes any ingredients from anabelian geometry. But even this looks extremely difficult.
The thing is that it could be that there is a revolutionary approach to algebraic geometry over $\mathbb{F}_{1}$ that has been developed, it’s just a question of finding some comprehensible way to describe what is being done. Maybe the best way would be to take some simple example and try to ’carry out IUTT’ for it, irrespective of the question of the abc-conjecture. For instance, one might think of trying to do something with $\mathsf{Spec}(\mathbb{Z})$. But the étale fundamental group of the latter is trivial, so I would guess that one cannot do anything much with this example directly, since one does not have the anabelian geometry ingredients available.
I cannot, by the way, quite understand Scholze and Stix’s objection that there is not much anabelian geometry involved in IUTT. The role of anabelian geometry looks absolutely crucial to me, it looks like it provides the non-formal input to the IUTT formalism.
My biggest question now that I would like answered is whether the so-called log-Theta lattice is meant to be an object in a diagram (2-)category, and so considered some kind of formal colimit, or whether one should be actually taking its colimit in some (2-)category of Hodge theatres. M talks of gluing, but I haven’t seen, in my limited reading, any statement that confirms this. Knowing this, together with the trivial observation above about colimit shapes, could be extremely useful at getting around the communication impasse in the dispute over labels.
I was even thinking of emailing Mochizuki…
OK, it turns out temperoids and anabelioids are just pretoposes given by continuous actions of certain prodiscrete/profinite topological groups on countable/finite sets respectively. Morphisms are just inverse image functors (cocontinuous and lex in the reverse direction).
A trivial note: ’capsules’ seem to be just objects in the free finite coproduct completion, and ’capsule morphisms’ the usual maps.
A number of times the definition of a Hodge theater requires ’a category equivalent to such-and-such a given category’, with no explicit equivalence. I guess this is like asking for a generic Grothendieck topos without giving a site of definition? But in fact it’s not ’equivalent to a category of such-and-such form’, but one specific category. I’d want to prove a theorem of the kind: Every Hodge theater is equivalent to one where the categories that make it up are that specific category.
OK, so I have most of a draft here. It tapers off a bit, because I feel like I might be starting to repeat myself, but I do want to add a reasonable discussion about #9, and also something about isomorphism of diagrams, since that is the main tactic that Scholze–Stix seem to be using. I think also perhaps something about M’s use of isomorphism classes of functors, which I can’t get used to, and which may be hiding some infelicities.
Richard, please let me know if there are unjustified criticisms that need toning down.
Hi David,
just to alert you that the document breaks off in the middle of a sentence on p. 7.
At least on my system. The last it says is literally
…are known to be isomorphic (without such specified isomorphisms being given), then one knows that
and then it ends.
Yes. :-/ But I wasted a lot of time today trying to read the actual IUTT papers and also Hoshi’s document Richard linked to above, to see if I could make any real comment on M’s use of diagrams. So I will finish that section tomorrow.
Mostly, I want to know: is it too long? I can’t imagine having more than another page or so of text. Is the example with fundamental groups too wordy/out of place? Reader feedback would be appreciated as to what’s there so far.
I find the strong point is where you directly compare to IUTT constructions/arguments.
If the discussion of the fundamental group is just an illustration of the general point without specific reference to a construction/argument that appear in IUTT, then let’s just make sure that it doesn’t come across as you adding just yet another analogy to the discussion. It may be very useful to readers, of course, but should not get in the way of your more concrete points further down. Maybe make it an appendix? Or maybe at least add a sentence telling readers that they may want to skip over this if they don’t think they need to see this example (again).
Yes, it did feel like more analogies, which is not that helpful if you don’t know it, and obvious if you do. I’ve removed it from my local copy for now. If Richard, or anyone else, feels it’s relevant, I’ll put it in an appendix.
I haven’t finished, but so far I’m liking what I’m reading. It speaks eloquently of the (very basic, elementary) role of category theory in maintaining basic hygiene and orderly thinking. It’s amazing to me how often this lesson is ignored; really it’s hard for me to imagine being a mathematician and not having such basic skills.
Just a few quick reactions:
page 4, “specific representations of objects matter and make,
in the same paragraph, “Mochizuki’s claim”,
page 5, first paragraph: the second sentence made sense to me (it is sufficient that the subposet inclusion $R \hookrightarrow L$ be a final functor), but the third did not: I did not see the relevance of the condition $l \lt l' \lt s(l)$ for no $l'$. Don’t you only need $s$ to be inflationary, i.e., $l \leq s(l)$, for finality to obtain?
page 6, first paragraph: I have to say that I found myself getting confused here in the discussion of the example. $\mathbb{Q}$ is initial in the category of fields of characteristic $0$, so if a pushout of this form exists, it will be the coproduct of $K$ and $K'$. I didn’t think the category of characteristic $0$ fields actually had coproducts. My instinctive memory is that the coproduct, in the category of commutative rings anyway, is $K \otimes K'$, which breaks up as a product of $\dim_\mathbb{Q}(K)$ copies of $K'$ if $K, K'$ are isomorphic Galois extensions: not a field. Even in the category of fields, it feels very unlikely that you can construct the coproduct of $\mathbb{Q}(\sqrt{2})$ with itself: how can you embed two copies of this field “disjointly” (meaning, roughly, independently) within a number field? Also, the coequalizer diagram for this example looks odd because those two maps $\mathbb{Q} \rightrightarrows K$ are the same, by initiality.
Typo on page 5: “ingoring” instead of “ignoring”.
Thanks, Todd, certainly didn’t think that one through. I think the example with the fields can be be discussed independently of pushouts, namely which category the isomorphism lives in: $Fields$ vs $SubFields(\mathbb{C})$.
Version 0.1.1 is now here.
And now there is a finished draft, version 0.2, here. I will write a short blog post and then give a link to the final version after comments here.
p. 6 “known to isomorphic”
p. 3 “strutural”
P. 4, the paragraph starting with “An even more striking example” may need stating its punchline: Is it “and this simple state of affairs seems to be what LbEx5 is all about” or is it “in contrast to what LbEx5 seems to be saying”?
On p. 2 it would seem in order to add a side note with pointer to a reference text on category theory, given that the point is that parts of the intended audience may have no inkling. I’d suggest to point to MacLane and to Borceux.
Thanks for the corrections.
Re #73: Why not Leinster and/or Riehl, both of which are free? Or do you mean that I should point to serious authorities on the matter? (off-topic, I seem to have lost my hardcopy of Tom’s book :-/ … )
Sure, or Riehl. What I meant was: not just MacLane.
Re #72 I have finished off the paragraph thus:
Now of course one has the factorisation $\disc(I) \to \ast \xrightarrow{A} C$, and the colimit over a trivial diagram $\ast \xrightarrow{A} C$ is just the object $A$ again, so keeping the information of the diagram shape is crucial. To join up with (LbEx5), every manifold is the quotient of some coproduct $\coprod_I \mathbb{R}^n$ in the category of manifolds, by an equivalence relation that is itself some other coproduct $\coprod_J \mathbb{R}^n$. Nothing here reduces to the triviality that (LbEx5) seems to claim, despite all diagrams only ever using a single copy of $\mathbb{R}^n$.
Available here (version 0.3)
Ah, interesting. I would have expected (not having looked at the report) that the conclusion was going to be that Mochizuki is doing it right, if only informally and verbosely.
Now if one could identify such a problematic step not just in the “report”, but in the IUTT articles themselves?!
It could be that Mochizuki is doing something right, but he claims that collapsing things will lead to trivialities. My argument is: not necessarily, unless one is doing it incoherently (for instance, replacing diagrams by other diagrams that are only object-wise isomorphic, not naturally isomorphic.
Maybe this means that it’s easy to overestimate the relevant background of some of the intended audience. Maybe the useful reference to point to is something more elementary, such as Lawvere-Schanuel & Lawvere-Rosebrough.
Yes, now that I think about it, please add that. Maybe also think of the audience you will have on NEW. These readers won’t be able to get anything out of Mike’s erudite discussion of structuralism, they will need to be taken by the hand.
OK, I added Lawvere–Schanuel and Spivak’s Category theory for the sciences.
Hi David, I’ve skim-read the latest version, and I think it is shaping up to be a nice contribution to the debate! There are a few points where I probably do not fully concur, but I do not have time unfortunately just now to go into them. Anyhow, I don’t think my opinion is especially important! As long as you have the kernel of a valid point from a slightly different angle, that is a very useful thing. I would suggest to just send it to Mochizuki and to Scholze/Stix for comments. I have never been touch with the latter two authors, but I expect that Mochizuki will reply.
Thanks, Richard. I think I will merely make some blog posting, perhaps make a small amount of noise about it (I gather Urs will likewise advertise it) and see what happens. I’d be nervous about emailing all these big shots (including a newly-minted Fields medallist) and claiming they don’t understand elementary category theory.
Well, you know, they’re just ordinary people. I’d probably try not to write anything that you wouldn’t ask if you were in discussion with them in person, i.e. try to phrase things with the same degree of politeness, balance, and constructiveness (which is not to say that you are not doing so already, just that it’s for you to make that judgement I’d say :-)).
OK here is version 0.4 and what I intend to write as a blog post (shamelessly stolen from the intro)
In March 2018 Peter Scholze and Jacob Stix travelled to Japan to visit Shinichi Mochizuki to discuss with him his claimed proof of the $abc$ conjecture. In documents released in September 2018, Scholze–Stix claimed the key Lemma~3.12 of Mochizuki’s third Inter-Universal Teichm"uller Theory (IUTT) paper reduced to a trivial inequality under certain harmless simplifications, invalidating the claimed proof. Scholze apparently had concerns about the proof of Lemma 3.12 for some time; it has been reported that a number of other arithmetic geometers independently arrived at the same conclusion. Mochizuki agreed with the conclusion that under the given simplifications the result became trivial, but not that the simplifications were harmless. However, Scholze and Stix were not convinced by the arguments as to why their simplifications drastically altered the theory, and we stand at an impasse.
The documents released by both sides include two versions of a report by Scholze–Stix, titled Why $abc$ is still a conjecture, each with an accomanying reply by Mochizuki, as well as a 41-page article, Report on discussions, held during the period March 15 – 20, 2018, concerning Inter-Universal Teichm"uller Theory (IUTCH). This latter document is written in a style consistent with Mochizuki’s IUTT papers, and his other documents concerning IUTT. As such, it can be difficult (at least for me) to extract concrete and precisely-defined mathematical results that aren’t mere analogies or metaphors. Rather than analogies, one should strive to express the necessary ideas or objections in as precise terms as possible, and I argue that one should use category theory to clean up all the parts of the arguments that are not actual number theory or arithmetic geometry.
I made some more detailed notes about this here.
I’d be happy to stop thinking about this :-), progress becomes very, very slow when I do.
Maybe just one quick suggestion: “clean up all the parts” sounds to me a little naïve. Maybe just drop the “all”?
Also I reacted a bit to the final sentence in #76: I don’t think Mochizuki is claiming that anything is trivial, I think he is simply saying that a colimit of a diagram involving many copies of $\mathbb{R}^{n}$ cannot be replaced by a diagram whose source is the final category, sent to a single copy of $\mathbb{R}^{n}$.
Richard (#86), I strongly suspect this is what he is meaning about the labels, but it isn’t obvious: he is talking about making new copies of the objects themselves, not about indexing diagrams. Undergraduate linear algebra depends on the fact that all n-dimensional vector spaces over the reals are isomorphic to one single such, and while it brings conceptual clarity to use the full category of vector spaces, it doesn’t affect any calculations if you do things correctly.
Also: removed the ’all’ from #85, thanks.
Well, it’s going up. I want to relax over this coming weekend (and I’m going away, so won’t be available for discussion after tonight until Monday afternoon).
The post is here
Re #87: hmm, I’d say I’m 99% sure that what Mochizuki has in mind is what I wrote in #86! But no problem, this point is no doubt minor in the overall scheme of things. Well done on getting the note out! Hopefully it will lead to some interesting discussions!
On a different note, my own (non-expert and subjective) feeling at present is that Mochizuki is doing something magical with ’indeterminacies’ that is highly interesting, but unfortunately, due to unfamiliarity with anabelian geometry, I just cannot get an overview of the logic of how he does it.
Here is one concrete idea that definitely appears in Mochizuki’s papers. It is ’obvious’, but still I had not really thought about it before. It is that addition and multiplication (more generally, ’arithmetic’, as Mochizuki puts it) in $\mathbb{F}_{p}$ are ’essentially the same as’ in $\mathbb{Z}$ when $p$ is large (i.e. as long as whatever integers one is working with are sufficiently small). So if one considers the additive and multiplicative monoids of $\mathbb{F}_{p}$ for all $p$, one should have something that roughly resembles $\mathbb{Z}$ (or at least $\mathbb{N}$). It seems that (on a much more sophisticated) level, a Hodge theatre is a way of organising such a structure for an elliptic curve. But one could experiment with trying to do something with this idea ’from scratch’ instead.
Now left a comment at NEW pointing to my post.
The post’s link to the PDF returns 404 for me ….
Sorry, fixed that now.
Thanks! Also just noticed that you’re missing a ’z’ in Mochizuki’s name in the post title. :-)
:-/ Fixed. Getting tired…
Typo: “appartuses” instead of “apparatuses”.
@Dmitri thanks. I think that might be in the original, as I copy/pasted, but I can fix it.
I have an updated version of my notes here. As always, comments welcome. A big remaining issue for me, and not one I’m equipped to tackle, is the claim that Scholze-Stix’s hexagon diagram is wrong because the maps involved aren’t linear, as they presume. I’ve started an email discussion with Mochizuki, and that is something I want to get to eventually.
Great! For me, the strangest part of Scholze and Stix’s manuscript concerns what I wrote in the last paragraph of #57, and which Ivan Fesenko also remarked upon: unless I misunderstand what they mean, Scholze and Stix suggest that the anabelian geometry is not significant, whereas to me it looks absolutely crucial; I think it is what actually is the heart of Mochizuki’s work.
Great that you emailed Mochizuki!