Inter-universal geometry と ABC予想 (応援スレ) 49

[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 49 (1002ﾚｽ)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

554: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/18(日) 12:55:01.21 ID:ZLSkSSTT

>>553

つづき

Ｐ50
(i) Types of mathematical objects: In the following discussion, we shall often
speak of “types of mathematical objects”, i.e., such as groups, rings, topological
spaces equipped with some additional structure, schemes, etc. This notion of a “type
of mathematical object” is formalized in [IUTchIV], §3, by introducing the notion of
a “species”. On the other hand, the details of this formalization are not so important
for the following discussion of the notion of multiradiality. A “type of mathematical
object” determines an associated category consisting of mathematical objects of this
type ? i.e., in a given universe, or model of set theory ? and morphisms between such
mathematical objects. On the other hand, in general, the structure of this associated
category [i.e., as an abstract category!] contains considerably less information than the
information that determines the “type of mathematical object” that one started with.
For instance, if p is a prime number, then the “type of mathematical object” given by
rings isomorphic to Z/pZ [and ring homomorphisms] yields a category whose equivalence
class as an abstract category is manifestly independent of the prime number p.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1600350445/554

555: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/18(日) 12:55:19.30 ID:ZLSkSSTT

>>554
つづき

Ｐ147
[cf. the discussion preceding [Pano], Theorem 4.1].
(ii) Explicit examples of connections to classical theories: Next, we review
various explicit examples of connections between inter-universal Teichm¨uller theory, as
exposed thus far in the present paper, and various classical theories:
(1cls) Recall from the discussion of §2.10 that the notion of a “universe”, as well as
the use of multiple universes within the discussion of a single set-up in arithmetic
geometry, already occurs in the mathematics of the 1960’s, i.e., in the mathematics
of Galois categories and ´etale topoi associated to schemes [cf. [SGA1], [SGA4]].

(2cls) One important aspect of the appearance of universes in the theory of Galois
categories is the inner automorphism indeterminacies that occur when one
relates Galois categories associated to distinct schemes via a morphism between such
schemes [cf. [SGA1], Expos´e V, §5, §6, §7]. These indeterminacies may be regarded
as distant ancestors, or prototypes, of the more drastic indeterminacies ? cf.,
e.g., the indeterminacies (Ind1), (Ind2), (Ind3) discussed in §3.7, (i) ? that occur
in inter-universal Teichm¨uller theory.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1600350445/555

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.040s