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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