The Stacks project

Lemma 4.22.10. Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with cofiltered index category $\mathcal{I}$. The following are equivalent

  1. $M$ is an essentially constant pro-object, and

  2. $X = \mathop{\mathrm{lim}}\nolimits _ i M_ i$ exists and for any $W$ in $\mathcal{C}$ the map

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

    is bijective.

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

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

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

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

is the identity too. Setting $W = M_ j$ and starting with $\text{id}_{M_ j}$ in the colimit, we see that $M_ k \to M_ i \to X \to M_ j$ is equal to $M_ k \to M_ j$ for some $k$ large enough. This proves (1) holds. The proof of (1) $\Rightarrow $ (2) is omitted. $\square$

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