The goal of this post, a fool’s errand really, is to try to give a summary of the introduction to Mochizuki’s paper, The Geometry of Frobenioids I: The General Theory, which may be found here:
I would like to add that there is also a wikipedia entry describing these strange objects known as Frobenioids:
I would like to point out that one thing which is odd about this wikipedia entry is that it does not go so far as to define Frobenioids precisely. Instead, it borrows language from the technical summary of the paper. The technical summary is just the first section of the introduction. Roughly, that summary says that there is something called an elementary Frobenioid which we should think of like a semi-direct product of Monoids and a Frobenioid is going to be a category equipped with a functor to elementary Frobenioids with enough properties to guarantee that this functor can be reconstructed category-theoretically. The philosophy is supposed to be that the “base category” from which the monoids are taken is a model for field extensions of a finite field where as the functor assigns divisors.
All of this is contained in section I1, the “technical summary” of the paper. And nothing outside of section I1 is contained in the wikipedia article. If section I1 seems not to make sense, that is because not all technical details in the definitions have been given there. The remainder of this post will consist of a brief summary of the remaining sections of the introduction.
After the technical summary, the paper contains some further intriguing introductory sections. The next of these is I2 which is intriguingly titled “Abstract combinatorialization of Arithmetic geometry.” This section is a kind of philosophical essay about how Galois categories allow one to abstract all info about Field extensions into combinatorial info about the Galois groups. Mochizuki regards this Galois theory as very good for working over nonarchimedean primes, instead one needs divisors for working over the archimedean ones. The idea is that Frobenioids are a kind of compromise that lets you do both. No actual mathematics happens in this section.
Section I3 seems to be difficult to follow if one does not already understand Mochizuki’s rather impenetrable p-adic Teichmuller theory. Mochizuki laments that for some reason the morphism p —> p^n does not extend to all the integers. He says it would make sense if one worked with Monoids instead !!! He relates the idea that the theory of Frobenioids will encode a substantial part of scheme theory, but not, not, not, all of scheme theory.
Section I4 briefly describes a dichotomy between Frobenius and Etale structures. The idea is, apparently, that the Frobenioids will suppress the etale structure of schemes leaving only the other part.
Stay tuned for more posts on later sections of this riveting paper.