Definition 4.22.2 (05PV)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.22: Essentially constant systems
Definition 4.22.2 (cite)

Definition 4.22.2. Let $\mathcal{C}$ be a category. A directed system $(M_ i, f_{ii'})$ is an essentially constant system if $M$ viewed as a functor $I \to \mathcal{C}$ defines an essentially constant diagram. A directed inverse system $(M_ i, f_{ii'})$ is an essentially constant inverse system if $M$ viewed as a functor $I^{opp} \to \mathcal{C}$ defines an essentially constant inverse diagram.

Comments (2)

Comment #6722 by Alejandro González Nevado on November 14, 2021 at 01:22

Shouldn't both appearences of $I$ be substitute by $\mathcal{I}$ as we see functors going between categories and $\mathcal{I}$ is the category associated to the set $I$ , which are not exactly the same thing? If this is not the case I apologize but then I did not see before in the text the definition of $I^\mbox{opp}$ for a preordered set $I$ (although it is obviously a clear thing).

Comment #6917 by Johan on January 19, 2022 at 11:47

The relationship between systems over $I$ and diagrams whose index category is the category associated to $I$ is discussed in Section 4.21. I think the definition as stated here is sufficiently clear, although of course a machine wouldn't be able to read it. One way to change the definition would be to remove the equation $I \to \mathcal{C}$ and just say " $M$ viewed as a functor". I don't think that would be so helpful.

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