[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む63 (1002ﾚｽ)
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191(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/04/05(金) 13:56:49.10 ID:VayTWyHw(9/15) AAS
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ind-object：インド象？ｗ（＾＾
https://ncatlab.org/nlab/show/ind-object
nLab
ind-object Last revised on April 12, 2018
Contents
1. Idea
2. Definition
As diagrams
As filtered colimits of representable presheaves
3. Examples
4. Properties
The category of ind-objects
Recognition of Ind-objects
Functoriality
The case that C already admits filtered colimits
5. Applications
6. In higher category theory
In (∞,1)-categories
7. Related concepts
8. References
(抜粋)
1. Idea
An ind-object of a category C is a formal filtered colimit of objects of C. Here “formal” means that the colimit is taken in the category of presheaves of C (the free cocompletion of C). The category of ind-objects of C is written ind-C or Ind(C).
Here, “ind” is short for “inductive system”, as in the inductive systems used to define directed colimits, and as contrasted with “pro” in the dual notion of pro-object corresponding to “projective system”.
Their ind-categories contain then also the infinite versions of these objects as limits of sequences of inclusions of finite objects of ever increasing size.
Moreover, ind-categories allow one to handle “big things in terms of small things” also in another important sense: many large categories are actually (equivalent to) ind-categories of small categories.
This means that, while large, they are for all practical purposes controlled by a small category (see the description of the hom-set of Ind(C) in terms of that of C below). Such large categories equivalent to ind-categories are therefore called accessible categories.
8. References
Ind-categories were introduced in
http://sage.math.washington.edu/home/wstein/www/home/craigcitro/sga4/Grothendieck/SGA4/sga41.pdf
Alexander Grothendieck, Jean-Louis Verdier in SGA4 Exp. 1 pdf file
つづく
192(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/04/05(金) 13:59:00.32 ID:VayTWyHw(10/15) AAS
>>191
つづき
and the dual notion of pro-object in
http://archive.numdam.org/ARCHIVE/SB/SB_1958-1960__5_/SB_1958-1960__5__369_0/SB_1958-1960__5__369_0.pdf
A. Grothendieck, Techniques de descente et theoremes d’existence en geometrie algebrique, II: le theoreme d’existence en theorie formelle des modules, Seminaire Bourbaki 195, 1960, (pdf).
https://ncatlab.org/nlab/show/Categories+and+Sheaves
Masaki Kashiwara, Pierre Schapira, section 6 of Categories and Sheaves , Grundlehren der mathematischen Wissenschaften 332 (2006).
(引用終わり)
以上
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