The Stacks project

Definition 4.22.1. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram in a category $\mathcal{C}$.

  1. Assume the index category $\mathcal{I}$ is filtered and let $(X, \{ M_ i \to X\} _ i)$ be a cocone for $M$, see Remark 4.14.5. We say $M$ is essentially constant with value $X$ if there exist an $i \in \mathcal{I}$ and a morphism $X \to M_ i$ such that

    1. $X \to M_ i \to X$ is $\text{id}_ X$, and

    2. for all $j$ there exist $k$ and morphisms $i \to k$ and $j \to k$ such that the morphism $M_ j \to M_ k$ equals the composition $M_ j \to X \to M_ i \to M_ k$.

  2. Assume the index category $\mathcal{I}$ is cofiltered and let $(X, \{ X \to M_ i\} _ i)$ be a cone for $M$, see Remark 4.14.5. We say $M$ is essentially constant with value $X$ if there exist an $i \in \mathcal{I}$ and a morphism $M_ i \to X$ such that

    1. $X \to M_ i \to X$ is $\text{id}_ X$, and

    2. for all $j$ there exist $k$ and morphisms $k \to i$ and $k \to j$ such that the morphism $M_ k \to M_ j$ equals the composition $M_ k \to M_ i \to X \to M_ j$.

Please keep in mind Lemma 4.22.3 when using this definition.

Comments (0)

There are also:

  • 10 comment(s) on Section 4.22: Essentially constant systems

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05PU. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 4.22.1 as pdf