Lemma 14.3.6. Let $\mathcal{C}$ be a category. Let $U$ be a simplicial object of $\mathcal{C}$. Each of the morphisms $s^ n_ i : U_ n \to U_{n + 1}$ has a left inverse. In particular $s^ n_ i$ is a monomorphism.

**Proof.**
This is true because $d_ i^{n + 1} \circ s^ n_ i = \text{id}_{U_ n}$.
$\square$

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