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.

There are also:

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