You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
GhaS Shee (Oct 27 2020 at 11:31):I am not a supporter of IUTT because I am, needless to say, far from understanding Anabelian Geometry, p-adic numbers and even understanding ZFC.
However, I recently thought that the idea of changing a universe to another universe with treating an object in the two universes is interesting to some extent, which is like comparing some obejct in two different type systems e.g. between MLTT and Refinement Type System.
( I also think my choice of the term 'universe' might be wrong for some religion. )
When I read the book "Coq Art", I saw a sentence like ;
We say a function is a 'functional' when it is a function that takes functions as its arguments. In the same way, we define 'tactical' as a tactic that takes tactics as its arguments
In category theory, 'functors' seem doing this role. ( or sheaves ? )
This is bottom-up style of defining something.
Here I thought that it might be interesting if we have this kind of bottom-up style changes of universes.
In contrast, in Mochizuki's setting, he seems to put history as a 'graph with some extra data' on the most outside of his category-like objects, i.e. species.
Unlike categories of categories, we could not add some structure to the outside since the most outside is not any more a category-like object but a graph with some extra data.
Thus we might not treat Mochizuki's style as the bottom-up style.
This seems to be top-down style structure-definition since he seemed to know the most outside.
( Or I may not understand well that the generalization species can have histories (i.e. graphs with extra data) as its example, hence that it can be also bottom-up style. But if so, I cannot see why he needs this extra definition of history, so I believe they cannot have histories as its example. )
I have 2 questions;
top-down style and bottom-up style? ( I do not know even the difference between algebra and coalgebra well. Though I might be able to think of them independently, but when I mix them, I could not figure out the whole contents or the relation between them well. Especially I do not know coalgebra well. )
top-down style is? David Michael Roberts (Oct 27 2020 at 13:53):For Mochizuki, as far as I can tell, he demands that a diagram in a category is literally a subcategory, not a functor to the category. This is I think because he wants to work all the time with isomorphism classes of functors, not a great idea!
A "history" seems to be just a diagram, but in this Mochizukian sense, where it is a subcategory (or rather, a subcategory generated by a subgraph, and he only specifies the subgraph).
It's entirely unclear what universes do for Mochizuki. His vague mutterings about needing universes else circular -chains lead to a violation of the axiom of foundation are .... not very enlightening.
GhaS Shee (Oct 27 2020 at 14:27):@David Michael Roberts
Thank you for your answer!
I might see that they do not reveal objects (universes) though they handle them.
Since I do not understand the papers well, and because of your answer, now it might not seem a good way of constructing something with referring the papers ..., but rather I should construct from the bottom without them.
In contrast, Univalence Axiom seems a good way of linking objects in two different levels, which might be a bit different topic from IUTT since it tells the exact 'equality' of two objects.
Now I have read @David Michael Roberts 's report (maybe) again.
https://thehighergeometer.files.wordpress.com/2018/09/mochizuki_final1.pdf
And I am gradually seeing what the report says much more and the situation of the papers seems worse for me now than when I heard the news RIMS accepted them in April.
umm.....
GhaS Shee (Dec 05 2020 at 06:07):I found that Davide Sangiorgi's Bisimulation might answer the almost all my questions;
" What is the difference between algebra and coalgebra ? "
" What are invariances between models ? "
Now I am reading :