Lemma 12.30.3 (0A2G)—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

$F \oplus G : \mathcal{I} \to \mathcal{A}$ is essentially constant, and
$F$ 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 April 08, 2014 at 03:00

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

Comment #552 by Johan on April 09, 2014 at 01:00

Many thanks! Fixed here.

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