Here’s batch 2 of my old google+ posts on ‘Inter Universal Teichmuller theory’, or rather on the number theoretic examples of Frobenioids.
June 5th, 2013
Mochizuki’s categorical prime number sieve
And now for the interesting part of Frobenioids1: after replacing a bunch of arithmetic schemes and maps between them by a huge category, we will reconstruct this classical picture by purely categorical means.
Let’s start with the simplest case, that of the ‘baby arithmetic Frobenioid’ dismantling $\mathbf{Spec}(\mathbb{Z}) (that is, the collection of all prime numbers) and replacing it by the category having as its objects
all
and morphisms labeled by triples
Composition of morphisms is well-defined and looks like
The challenge is to recover all prime numbers back from this ‘Frobenioid’. We would like to take an object
We can identify all isomorphisms in the category and check that they are precisely the morphisms labeled
Another class of arrows we can spot categorically are the ‘irreducibles’, which are maps
– those of Frobenius type :
– those of Order type :
We would like to color the froBs Blue and the oRders Red, but there seems to be no way to differentiate between the two classes by purely categorical means, until you spot Mochizuki’s clever little trick.
start with a Red say
and if
On the other hand, if you start with a Blue and compose it with either a Red or a Blue irreducible, the obtained map cannot be factored in more irreducibles.
Thus, we can identify the Order-type morphisms as those irreducibles
Finally we say that two Reds out of
So we do indeed recover all prime numbers from the category.
Similarly, we can see that equivalence classes of Frobs from
There seems to be no categorical way to determine the prime number associated to an equivalence class of Order-morphisms though… Or, am i missing something trivial?
June 7th, 2013
Mochizuki’s Frobenioids for the Working Category Theorist
Many of you, including +David Roberts +Charles Wells +John Baez (and possibly others, i didn’t look at all comments left on all reshares of the past 4 posts in this MinuteMochizuki project) hoped that there might be a more elegant category theoretic description of Frobenioids, the buzz-word apparently being ‘Grothendieck fibration’ …
Hence this attempt to deconstruct Frobenioids. Two caveats though:
– i am not a category theorist (the few who know me IRL are by now ROFL)
– these categories are meant to include all arithmetic information of number fields, which is a messy business, so one should only expect clear cut fibrations in easy situation such as principal ideal domains (think of the integers
Okay, we will try to construct the Frobenioid associated to a number field
The objects will be fractional ideals of K which are just the R-submodules
where the
Next, we define an equivalence relation on this set, calling two fractional ideals
We have a set with an equivalence relation and hence a groupoid where these is a unique isomorphism between any two equivalent objects. This groupoid is precisely the groupoid of isomorphisms of the Frobenioid we’re after.
The number of equivalence classes is finite and these classes correspond to the element of a finite group
Next, we will add the other morphisms. By definition they are all compositions of irreducibles which come in 2 flavours:
– the order-morphisms
– the power-maps
Well, that’s it basically for the layer of the Frobenioid corresponding to the number field
The only extra-type morphisms we still have to include are those between the different layers of the Frobenioid, the green ones which M calls the pull-back morphisms.
They are of the following form: if
a question for category people
Take the simplest situation, that of the integers
My question now is: if for two different primes
June 11th, 2013
my problem with Mochizuki’s Frobenioid1
Let us see how much arithmetic information can be reconstructed from an arithmetic Frobenioids. Recall that for a fixed finite Galois extension
When all objects and morphisms are labeled it is quite easy to reconstruct the Galois field
However, in this reconstruction process we are only allowed t use the category structure, so all objects and morphisms are unlabelled (the situation top left) and we want to reconstruct from it the different layers of the Frobenioid (corresponding to the different subfields) and divide all arrows according to their type (situation bottom left).
First we can look at all isomorphisms. They will divide the category in the dashed regions, some of them will be an entire layer (for example for
Another categorical notion we can use are ‘irreducible morphisms’, that is a morphism
– oRder-maps (Red) : multiplication by a prime ideal
– froBenius or power-maps (Blue) sendingg a fractional ideal
– Galois-maps (Green) extending ideals for a subfield
We would like to determine the colour of these irreducibles purely categorical. The idea is that reds have the property that they can be composed with another irreducible (in fact, of power type) such that the composition can again be decomposed in irreducibles and that there is no a priori bound on the number of these terms (this uses the fact that
The most interesting case is the composition of a Galois map with an order map
but the number
– we can determine all the red maps, which will then give us also the different layers
– we can determine the green maps as they move between different layers
– to the remaining blue ones we can even associate their label
Taking an object in a layer, we get the set of prime ideals of the ring on integers in that field as the set of all red arrows leaving that object unto equivalence (by composing with an isomorphism), so we get the prime spectra
For a ring-extension
Indeed, composing the composition of the Galois map with the order-map
Let
The problem i have is that i do not see a categorical way to label the red arrows in
This suggests that one might use the ‘Arakelov information’ (that is the archimidean valuations) to do this (the bit i left out so far), but i do not see this in the case of
Probably i am missing something so all sorts of enlightenment re welcome!
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