[FOM] Shinichi Mochizuki on set-theoretical/foundational issues

MartDowd at aol.com MartDowd at aol.com
Sat Jun 1 10:37:30 EDT 2013

I've been looking further into the question of why Mochizuki wants  to
extend the standard methods of category theory that have been in use  by
mathematicians since the 1950's, and exactly what he is proposing.   I'll
use the same bibliographic citations as in Mochizuki's  "Inter-universal"
series of papers.
In IUTchIV Mochizuki states that:
On the other hand, if one  restricts one's attention to such a category,
then one must keep in  mind the fact that the structure of the category -
i.e., which  consists only of a collection of objects and morphisms
satisfying  certain properties! - does not include any mention of the
various sets  and conditions satisfied by those sets that give rise to
the "type of  mathematical object" under consideration.  For instance, the
data  consisting of the underlying set of a group, the group multiplication
law on the group, and the properties satisfied by this group  multiplication
law cannot be recovered [at least in an a priori  sense!] from the structure
of the "category of groups".
This much at least is subject to debate.  A group is a tuple  <G,1,x> where
1 \in G and x is a binary function on G, satisfying  certain axioms.  One
then defines a group homomorphism, and the category  Grp whose objects are
the groups and whose morphisms from A to B are the  pairs <f,B> where f is a
homomorphism with domain A and codomain  contained in B.  Showing that the
axioms of a category are satisfied  makes use of the set-theoretic
definitions; proving properties of the  category, such as completeness and
co-completeness, does also.  In  general, the "concrete" structure of Grp
may be referred to if  necessary.
I'm going to leave off here for now; but I think that the question should  
looked in to, whether Mochizuki's constructions can be carried out  using
standard category-theoretic methods.  In this case, the  foundational issues
are as usual.
- Martin Dowd

In a message dated 5/31/2013 4:38:49 P.M. Pacific Daylight Time,  
MartDowd at aol.com writes:

For any  x\in V, P^-1(x) is a fiber, which is isomorphic to V as a 
structure for the  lnguage of set theory.
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