20/04/30 23:10:17.44 uKv4ftJu.net
ショルツ先生の発言、どんどん後退しているな（＾＾；
URLﾘﾝｸ(www.math.columbia.edu)
Not Even Wrong
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Posted on April 3, 2020 by woit
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Peter Scholze says:
April 30, 2020 at 3:32 am
Dear Taylor,
I certainly understood your point there ? you might also take the ring Z[√-1].
There is of course a big difference between the ring Z and the “theory” it defines, i.e. roughly the class of all subsets of all finite powers Z^n that are definable by polynomial equations.
The latter is indeed a highly nontrivial category (where morphisms are given by definable graphs); it is of course not equivalent to a category with one object and two morphisms.
If a category like this is in place in Mochizuki’s work, I’m happy to hear about it!
Reading the IUT papers, however, you are presented with some extremely difficult notion of a Hodge theater, together with a highly non-obvious notion of isomorphisms of such: Isomorphisms do not preserve nearly as much structure as you would expect them to, and this is by design as Mochizuki points out.
So I find it very hard to “guess” what something like a surrounding “theory” might be.
For all I can see, Hodge theaters fit neither into the framework of “structures” as used in the wikipedia entry URLﾘﾝｸ(en.wikipedia.org)(model_theory) you linked to, nor the topos-theoretic framework of Caramello.
(Regarding the first one: A “structure” in the sense of model theory has first of all an underlying set.
I find it hard to take a Hodge theater and produce some interesting set that is functorial in isomorphisms of Hodge theaters, the problem being the very lax notion of isomorphisms of Hodge theaters.)
However, these long discussions are all about interpretations.
Regarding the mathematics proper: I stand by the claim made in our manuscript, and have indicated the proof above.