Lemma 4.22.9 (05PY)—The Stacks project

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

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

is bijective.

Proof. Assume (2) holds. 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. The proof of (1) $\Rightarrow $ (2) is omitted. $\square$

Comments (2)

Comment #8282 by Et on March 26, 2023 at 10:39

It seems aa though in (2) => (1) you don't actually use the colimit aasumption, right? That is to say it is enough to assume we have a cocone X for which the map in (2) is bijective.

The reason I'm asking is that in lemma 13.14.6, you spend the last part of the proof proving (in a purely formal manner) the colimit assumption on the object denoted C in that lemma, and it seems as though that may be unecessary.

Comment #8915 by Stacks project on April 10, 2024 at 12:48

OK, I upgraded this lemma to deal with the use case you mentioned and I fixed the proof in derived categories as you suggested. It still isn't optimal perhaps because it could be formulated on the level of functors, but we can add that if we ever need it. Thanks! Changes are here.

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

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