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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

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