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The Stacks project

Lemma 12.30.3. Let $\mathcal{I}$ be a filtered category. Let $\mathcal{A}$ be an additive, Karoubian category. Let $F : \mathcal{I} \to \mathcal{A}$ and $G : \mathcal{I} \to \mathcal{A}$ be functors. The following are equivalent

  1. $F \oplus G : \mathcal{I} \to \mathcal{A}$ is essentially constant, and

  2. $F$ and $G$ are essentially constant.

Proof. Assume (1) holds. In particular $W = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} F \oplus G$ exists and hence by Lemma 12.30.2 we have $W = X \oplus Y$ with $X = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} F$ and $Y = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} G$. A straightforward argument (omitted) using for example the characterization of Categories, Lemma 4.22.9 shows that $F$ is essentially constant with value $X$ and $G$ is essentially constant with value $Y$. Thus (2) holds. The proof that (2) implies (1) is omitted. $\square$

Comments (2)

Comment #542 by Nuno on

I do not see how can we apply Lemma 12.26.1 here, but since we already know that the colimits of and exist, we can use the fact that preserves kernels and apply tag 05PY.

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