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What is the simplest (at the lowest level feasible) explanation of the approach of “forgetting the history” of a mathematical object, as used in Inter-universal Teichmüller Theory (IUTT)? Please explain:

  • What are these operations and their history? (Without that we get nowhere!)
  • How can an exposition or proof forbid the reader to use previous known information about an object? (This just sounds unreasonable to me.)
  • What is a really simple example of a history with re-initialisation playing a useful rôle? (This would help one to understand and accept the method.)
  • Is this method broadly accepted? (Scholze and Stix do not mention it in their reports², though it is evidently very important to Mochizuki.)

N.B.

  • I am not asking what is the status of the purported proof of the abc conjecture, though that is obviously relevant.
  • I admit to not having so much as started to try to study the IUTT papers³, but do not think that amounts to an unreasonable lack of effort to solve my problem myself, given that I understand that many professional mathematicians have shied away from them. I am hoping that someone else has at least got far enough to answer.

Background

Shinichi Mochizuki, initiator of IUTT, says in §5 of his report¹ of discussions with critics of IUTT that:

¡My précis!

The logical origin of the differences in viewpoint and so of misunderstandings by critics might be the different approaches to histories of operations on mathematical objects (e.g. structures):

Conventional: Conventionally, one regards all operations as parts of a single history, wherein they are all accessible.

IUTT: IUTT frequently uses re-initialisation operations — i.e. one “forgets” the previous history of an object, regarding it as inaccessible in subsequent discussions. Re-initialisation necessitates labels for “before and after versions” of an object and explicit specification of the types (in IUTT: “species”) of objects, particularly before and after re-initialisation (e.g. “automorphism groups of fields” / “abstract topological groups”).

References

¹ http://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf – Report by Shinichi Mochizuki of discussions in March 2018 between himself and Yuichiro Hoshi (expounding IUTT) and Peter Scholze and Jakob Stix (questioning its methods).

² http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf (August 2018), critique by Scholze and Stix, one of the other documents referred to in ¹

³ S. Mochizuki, Inter-universal Teichmüller Theory (August 2012):

I: Construction of Hodge Theaters, RIMS Preprint 1756
II: Hodge-Arakelov-theoretic Evaluation, RIMS Preprint 1757
III: Canonical Splittings of the Log-theta-lattice, RIMS Preprint 1758
IV: Log-volume Computations and Set-theoretic Foundations, RIMS Preprint 1759

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    $\begingroup$ I've tried to read into this many times. I fail everytime. I don't even claim to understand a word that man speaks unfortunately. $\endgroup$ – Don Thousand Oct 10 '18 at 13:27
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    $\begingroup$ You may be interested in this blog post and (specially) the accompanying note. $\endgroup$ – Andrés E. Caicedo Oct 10 '18 at 13:32
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    $\begingroup$ @AndrésE.Caicedo: Thanks; the page does not add all that much, but his PDF about Mochizukis comments looks interesting. He says One can ... wonder what ‘histories of operations’ is supposed to mean, or ‘re-initialization operations’. If the diagrams shown there are supposed to represent diagram shapes over which one is taking colimits, then it is a category-theoretic triviality that one gets different colimits (recall the quote of Freyd above!). I need some effort to understand that. $\endgroup$ – PJTraill Oct 10 '18 at 13:41
  • $\begingroup$ Yes, it is a brief note but very much in the direction of your question. $\endgroup$ – Andrés E. Caicedo Oct 10 '18 at 13:44
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    $\begingroup$ Whatever about Mochizuki's approach, the real question is, what is this "conventional approach" he refers to? In what sense do we conventionally "regard all operations as part of a single history"? What does that mean? You can't understand how he intends to diverge from the traditional approach if you have no idea what it is he's diverging from. $\endgroup$ – Jack M Oct 10 '18 at 13:56
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(I am the author of the notes Andrés linked to in the comments)

As far as I can understand it, Mochizuki uses a highly non-standard definition of the phrase 'diagram in a category'. For a category theorist (as I am) and practically anyone else, a diagram in a category $C$ has two parts: a small (think: finite, countable, etc) category $D$, called the diagram shape, and a functor $d: D \to C$, the diagram proper. Example diagram shapes are: $$ \cdots \to \bullet \to \bullet \to \bullet \to \cdots $$ (one object per integer), or $$ \bullet \to \bullet \to \bullet \to \cdots $$ (one object per natural number), or $$ \bullet \to \bullet \to \cdots \qquad \bullet \to \bullet \to\cdots $$ (one object per natural number, but arrows only between even and even, and odd and odd), and so on. The functor $d$ may be anything, even non-injective—even sending every object of $D$ to the same object in $C$.

Mochizuki, on the other hand, says in the IUT papers that he works with categories and isomorphism classes of functors (which is really, really weird). So a diagram would then only be defined up to natural isomorphism, and this would be antithetical to his habit of specifying objects very precisely. I think that to get around this, he has adopted the habit of defining a diagram in $C$ to be a subcategory of $C$. This is equivalent to demanding that all diagrams are specified by injective functors. This is why he insists on creating new copies of things. Otherwise a non-injective diagram functor $d:D\to C$ collapses to its image. Thus a diagram of shape $$ \cdots \to \bullet \to \bullet \to \bullet \to \cdots $$ that maps all objects to the same object $x$, and all the arrows connecting successive objects to the same endomorphism $f: x\to x$, for Mochizuki is the same thing as the diagram with one object and a family of endomorphisms given by powers of $f$.

This is not standard, and I've seen it nowhere else.

In ordinary category theory these are completely different diagrams, and one cannot make this replacement. In the third example above, a person following standard category theory could map all the objects in the diagram shape to the same object, and there would be no complaints. But for Mochizuki, one would have to make a new copy of the object that was used in the first node of the first connected component of the diagram to use as the first node of the second connected component. This is what I think he means by forgetting the previous history of an object.

Additionally, there are issues with leaving forgetful functors implicit, so that it is not enough to talk about the underlying multiplicative monoid of a field: one must cook up some scheme whereby one has forgotten that the monoid came from a field. Given a field $k\in \mathbf{Fields}$, there is a monoid $U(k) = (k,\times)\in \mathbf{Monoids}$. For everyone else, $U(k)$ is just an object of the category of monoids. But for Mochizuki it seems one needs to make this extra operation by passing to an arbitrary monoid that is anonymously isomorphic to $U(k)$ (in the sense that there isn't a supplied isomorphism witnessing this fact).

Combining these two non-standard approaches to category-theoretic machinery, Mochizuki seems saddled with an approach that is confusing and misleading, given that he partly uses standard terminology in a non-standard way and also uses idiosyncratic psychological terminology for what should be a commonplace procedure.

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    $\begingroup$ Thank-you very much for your explanation, which I could immediately mark up, as it is evidently helpful and presumably involved appreciable effort. I am not sure whether I am going to be able to accept this (or any) answer, but that has more to do with my limited knowledge; I hope to make the effort, but it may take me some time! $\endgroup$ – PJTraill Mar 3 at 12:11
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    $\begingroup$ P.S. I suppose I would also accept this answer if sufficiently many comments by experienced contributors supported it. $\endgroup$ – PJTraill Mar 4 at 13:21
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    $\begingroup$ Thanks, fixed ! $\endgroup$ – David Roberts Mar 8 at 0:18

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