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

where all the transition maps are injections.

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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