20/05/02 13:47:40 qpZJrq8I.net
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URLﾘﾝｸ(carmonamateo.github.io)
A letter from Mochizuki to Mateo Carmona 13.11.2017
(抜粋)
Dear Carmona,
There is a very substantive mathematical difference between the theory of Galois categories/topoi as developed in SGA1/SGA4 and the theory of anabelioids as developed in my paper "The Geometry of Anabelioids":
Namely, the notion of slimness allows one to work with 1-categories of (slim) anabelioids, whereas the theory of Galois categories/topoi as developed in SGA1/SGA4 gives rise to 2-categories of Galois categories/topoi.
In particular, "Galois groups" (i.e., in the classical sense) arise naturally as groups of 1-morphisms in 1-categories of slim anabelioids, which is a very substantive mathematical difference from the way in
which they arise in 2-categories of Galois categories/topoi, i.e., as groups of 2-morphisms in 2-categories.
This difference between 1- vs. 2-categories or 1- vs. 2-morphisms plays a fundamental role in the theory of anabelioids (as developed both in my paper "The Geometry of Anabelioids", as well as in subsequent papers, e.g., papers on combinatorial anabelian geometry).
Put another way, this difference may be understood as being analogous to the difference between
Algebraic spaces (which form a 1-category)
and
(Deligne-Mumford) algebraic stacks (which form a 2-category).
Of course, algebraic spaces and (Deligne-Mumford) algebraic stacks are closelyrelated, in the sense that both arise by considering gluing operations in the etale topology of schemes.
On the other hand, the substantive difference between 1-and 2-categories gives rise to many substantive mathematical differences in various geometric arguments.
In particular, this substantive difference between 1- and 2-categories is sufficiently significant as to render extremely strange and unnatural any attempt to use the same terminology for both algebraic spaces and algebraic stacks.