Mochizuki's notion of a *Frobenioid* introduced in [The geometry of Frobenioids I](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Geometry%20of%20Frobenioids%20I.pdf) 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?
* ...



Are fully general Frobenioids necessary?
From [the horse's mouth](http://www.kurims.kyoto-u.ac.jp/~motizuki/Responses%20to%20questions%20on%20Frobenioids.pdf):



> 
> 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 $\mathbb C$].
> 
> 
> 


and



> 
> 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](https://www.maths.nottingham.ac.uk/plp/pmzibf/files/Frobenioids1.pdf) [slides](https://www.maths.nottingham.ac.uk/plp/pmzibf/files/Frobenioids2.pdf) 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.


