Lemma 14.22.5. Let $\mathcal{A}$ be an abelian category. For any simplicial object $V$ of $\mathcal{A}$ we have

$V = \mathop{\mathrm{colim}}\nolimits _ n i_{n!}\text{sk}_ n V$

where all the transition maps are injections.

Proof. This is true simply because each $V_ m$ is equal to $(i_{n!}\text{sk}_ n V)_ m$ as soon as $n \geq m$. See also Lemma 14.21.10 for the transition maps. $\square$

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