[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 44 (1002ﾚｽ)
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27(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/12(日)14:00 ID:hAg37Ryy(25/38) AAS
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The Scholze?Stix simplifications
So given all this discussion of peculiarities on Mochizuki’s side, what can be said about the approach of Scholze?Stix?
Many times they say they are identifying certain objects of interest that are known to be isomorphic/equivalent. Mochizuki objects to this,
but it is not a priori clear that identifying objects is destructive: in the examples above of colimits, one did not need to ensure that different objects were the values of different nodes in the diagram shape.
The book-keeping is taking place at the diagram level, not at the specific identity of the objects.
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28(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/12(日)14:00 ID:hAg37Ryy(26/38) AAS
>>27

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If one then has some other isomorphism, then it can be turned into an automorphism of X’ (say).
Consider the case one has some diagram12 X: D → C of objects where all the objects X(d) in the image of the diagram are known to be isomorphic to a fixed object Xo.
Then given an isomorphic diagram X’ : D → C, via some given natural isomorphism a : X 〜→ X’,
and where X’(d) = X’ for all d ∈ D, there is a canonical isomorphism colim X '〜＝ colim X’.
There is no guarantee13 that the arrows of D are sent to identity maps by X’; in fact if the arrows in the image of X are not invertible, then neither will the arrows in the image of X’. What is going on is that even though one might assume for simplicity that all the objects of the diagram are sent to the same object,
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