We now begin describing the most dramatic part yet of Mochizuki’s epic paper The Geometry of Frobenioids I: General Theory. We will go through section 0: preliminaries and notation.

Recall that the whole paper may be found here:

http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Geometry%20of%20Frobenioids%20I.pdf

Mochizuki starts saying some things that make sense but are not very deep.

Given a partially ordered set E, the category ORD(E) has underlying objects elements of E and morphisms which are inequalities. (You’d almost think we should think about inequality as being related.)

Within number systems we can think about some monoids. The positive integers are a monoid under multiplication and the primes generate it.

You could do this with integers, rationals or reals and look at the positive versions of any of the three. If a monoid is isomorphic to the nonnegative (reals,rationals or integers) as an additive monoid we refer to it as a monoprime monoid. (It is not clear that this helps.)

One can complete monoprime monoids from rational type to real type.

A number field is a finite extension of the rationals.

Monoids are a category with one object the monoid so that elements of the monoid are endomorphisms.

Next we consider the category of all monoids. We can go from monoids to their invertible elements with the +- superscript. (This is important!!! The +- superscript appears throughout Mochizuki’s description of theatres later on.)

You can mod out by invertible elements to get the characteristic and groupify in the normal way to get the group.

A monoid is: Torsion free if no torsion elements. Sharp if no invertible elements. Integral if groupification injective. Saturated if whenever a positiveinteger multiple of an element of the group lives in the monoid so does the element.

One can perfect monoids through an inductive limit under multiplication by all positive integers. This basically makes multiplication bijective (adds fractions.)

Now our attention turns to sharp, saturated, and integral modules and something important is going on. There is a partial order in such a monoid if there is with . An element of such a monoid is prime if it has no factors. (Where’s 1?) Thus we can rebuild the notion of primality within this kind of monoids.

Next time, we’ll go over Mochizuki’s preliminaries on Topological groups and Categories.