Lemma 4.22.9. Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with filtered index category $\mathcal{I}$. The following are equivalent

$M$ is an essentially constant ind-object,

there exists a cocone $(X, \{ M_ i \to X\} _ i)$ such that for any $W$ in $\mathcal{C}$ the map $\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X)$ is bijective,

$X = \mathop{\mathrm{colim}}\nolimits _ i M_ i$ exists and for any $W$ in $\mathcal{C}$ the map $\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X)$ is bijective, and

there exists an $i$ in $\mathcal{I}$ and a morphism $X \to M_ i$ such that for any $W$ in $\mathcal{C}$ the map $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \to \mathop{\mathrm{colim}}\nolimits _{j \geq i} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ j)$ is bijective.

In cases (2), (3), and (4) the value of the essentially constant system is $X$.

**Proof.**
It is clear that (3) implies (2). Assume (2). Then $\text{id}_ X \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, X)$ comes from a morphism $X \to M_ i$ for some $i$, i.e., $X \to M_ i \to X$ is the identity. Then both maps

\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \]

are bijective for all $W$ where the first one is induced by the morphism $X \to M_ i$ we found above, and the composition is the identity. This means that the composition

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \]

is the identity too. Setting $W = M_ j$ and starting with $\text{id}_{M_ j}$ in the colimit, we see that $M_ j \to X \to M_ i \to M_ k$ is equal to $M_ j \to M_ k$ for some $k$ large enough. This proves (1) holds.

Assume (4). Let $k$ be an object of $\mathcal{I}$. Setting $W = M_ k$ we deduce there exists a unique morphism $M_ k \to X$ such that there exists a $j$ and morphisms $k \to j$ and $i \to j$ in $\mathcal{I}$ such that $M_ k \to X \to M_ i \to M_ j$ is equal to $M_ k \to M_ j$. The uniqueness guarantees that we obtain a cocone $(X, \{ M_ k \to X\} )$. In this way we see that (4) implies (2); some details omitted.

We omit the proof that (1) implies the other conditions.
$\square$

## Comments (2)

Comment #8282 by Et on

Comment #8915 by Stacks project on

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