Mochizuki's notion of a Frobenioid introduced in The geometry of Frobenioids I is rather elaborate. However, he also introduces a myriad of further properties that a Frobenioid may satisfy, and his main results in that paper are always under considerable additional assumptions. This leads to me to wonder

Question: To what extent is it necessary to think about Frobenioids in their full generality to understand Mochizuki's theory? For applications is it safe to assume that all Frobenioids one will encounter are (for example)

  • of isotropic type?

  • of standard type?

  • obtained as "model Frobenioids" (Thm 5.2)?

  • Frobenius-normalized?

  • ...

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    $\begingroup$ As far as I know, it is not even clear at the moment whether the full Mochizuki's theory (IUT) provides nontrivial results. $\endgroup$ Jan 18, 2019 at 9:52
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    $\begingroup$ @FrancescoPolizzi I want to set that question aside. Let’s assume for the purposes of this question that one is interested (for whatever reason) in reading the IUT papers and needs to understand Frobenioids for that purpose. $\endgroup$
    – Tim Campion
    Jan 18, 2019 at 14:32

1 Answer 1


From the horse's mouth:

Indeed, the only Frobenioids that are used in the IUTeich papers are the following:

(F1) tempered Frobenioids, i.e., a generalization developed in [EtTh], §3, §4, §5, of the geometric example given in [FrdI], Example 6.1, to the case of the tempered coverings that arise in the theory of the theta function;

(F2) Frobenioids associated to number fields as in [FrdI], Example 6.3 (cf., e.g., [IUTchIII], Example 3.6), together with the realifications associated to (certain of) such Frobenioids;

(F3) certain special cases of the p-adic Frobenioids discussed in [FrdII], Example 1.1 (...) together with various related Frobenioids obtained by forming associated realifications or by forming the quotient...

(F4) copies of the archimedean Frobenioid discussed in [FrdII], Example 3.3, (ii) (i.e., the Frobenioid denoted “C”).

Here, we note that (F3) and (F4) are inessential since a Frobenioid as in (F3) essentially amounts to (i.e., may be replaced by) a suitable topological monoid with a continuous action by a topological group, while a Frobenioid as in (F4) essentially amounts to a copy of the topological monoid [that is the punctured closed unit disc of $\mathbb C$].


At any rate, from the point of view of studying IUTeich,

(I1) in [FrdI], one may assume that all Frobenioids are model Frobenioids (cf. [FrdI], Theorem 5.2, (ii)), which implies, in particular, that every object of a Frobenioid is isotropic, and that every morphism of a Frobenioid is co-angular;

(I2) one may in fact ignore [FrdII], §3, §4, §5.

It's worth looking at these slides by Weronika Czerniawska, especially the second set from page 15 on. On page 14 she says "almost all of Frobenioids appearing in IUT are model Frobenioids" and "Those which are not can be easily dealt with without notion of Frobenioid." She then goes on to detail exactly which ones are used.

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    $\begingroup$ Allow me also to point out Taylor Dupuy's explanation of the statement of IUT3 Corollary 3.12 without using Frobenioids, per the comments of Mochizuki and Czerniawska: youtube.com/watch?v=ZstG89SLy2k $\endgroup$ Jan 18, 2019 at 22:32
  • $\begingroup$ Thanks, this is just what I was hoping for! The ideas in the paper related to objects not of "isotropic type" struck me as truly bizarre (especially the definition of "quasi-isotropic type" which goes into the definition of "standard type"), so it's a relief to see that Mochizuki seems to have abandoned whatever plans he might have had for using them. On the other hand, it feels like a cruel let-down to be told that all important Frobenioids are "model Frobenioids" -- these don't really exhibit any of the interesting complexities permissible in a general Frobenioid "of isotropic type". $\endgroup$
    – Tim Campion
    Jan 18, 2019 at 22:44
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    $\begingroup$ @Tim I hope that Frobenioids will become objects of independent study, much like $Sh(X)$ and $QCoh(X)$ are prototypes of interesting categories. $\endgroup$ Jan 19, 2019 at 0:04
  • $\begingroup$ @TimCampion see also the notes from the talk "Introduction to (model) Frobenioids" by Obus here: uvm.edu/~tdupuy/anabelian/VermontNotes_20.pdf $\endgroup$ Jan 22, 2019 at 6:49

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