An exercise in colimits, contra Mochizuki

I’m not sure why I do this, in a vain hope that someone in Kyoto will listen to what I write about basic category theory. I got an email from the organising committee for IUT workshops (actually direct from Mochizuki) that seems to be rather widely sent, inviting people to register to see the (quasi-secret) recordings of recent IUT workshops. The email itself if not secret, you can see it here. It further complains about what is dubbed the ‘RCS’, or “Redundant Copies School”, basically Scholze and Stix and those who agree on their approach to try to understand the IUT papers. The email says

Unfortunately, it has come to my attention that certain misunderstandings concerning IUT continue to persist in certain parts of the world. Perhaps the most famous misunderstanding concerns an asserted identification of “redundant copies”. This misunderstanding involves well-known, essentially elementary mathematics at the beginning graduate level concerning the general nonsense surrounding “gluings”. For instance, if one “applies” this misunderstanding to the well-known gluing construction of the projective line, then one concludes that the two copies of the affine line that appear in this gluing are “redundant’’, hence may be identified. This identification leads immediately to a contradiction, i.e., to a “proof” that the projective line cannot exist! More details may be found in the Introduction to [EssLgc] and the references given there.

referencing this document, which is, to say the least, not a mathematical justification of any part of IUT.

Anyway, in case there are people who think Mochizuki can do no wrong, I wrote out a complete and elementary construction, with proof, of the projective line, using no “redundant” copies of the affine line. It uses the standard definition of pushout (=a gluing construction) from Mac Lane’s classic textbook Categories for the Working Mathematician, and the usual definition of the projective line. It is not terribly interesting, but someone had to do it.

I maintain, with good reason, that the type of reasoning in the note is what the so-called ‘RCS’ is doing. It is standard category theory and standard mathematics. There is no linguistic trickery or confusion or deeply detrimental disruptions here. Mochizuki is using non-standard definitions of standard terminology, and then complaining that other people’s definitions (which are the standard ones) lead to contradictions. They really don’t, if one doesn’t insist on trying to ignore the differences, and I don’t understand why he persists in it.

This says nothing about IUT itself, but given the claim of a

remarkably close structural resemblance between the gluing that appears in the standard construction of the projective line and the gluing constituted by the Θ-link of inter-universal Teichmüller theory,

not to mention

from the point of view of arithmetic geometry, the discussion of

the projective line as a gluing of ring schemes along a multiplicative group scheme

given in Example 2.4.7 yields a remarkably elementary qualitative model/analogue of the essential logical structure surrounding the gluing given by the Θ-link in inter-universal Teichmüller theory.

and the fact the claims of a contradiction in the construction of the projective line à la ‘RCS’ are erroneous, I think that one must take a really hard look at the analogous claims around the contradiction arising from the Θ-link, and adjust one’s priors accordingly.

7 thoughts on “An exercise in colimits, contra Mochizuki

    1. From a non-expert point of view? It looks like normal mathematics, and very interesting. From what I understand, it’s trying to take the intuitive same idea of trying to do some kind of arithmetic deformation theory that Mochizuki says IUT is supposed to do (which, recall, is not apparently not possible with existing tools).


    1. We’ve had some correspondence 🙂 I have some more I need to write, but I’m not quite sure exactly what just yet, because I need to email some people and talk to them a bit more.


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