Lemma 14.21.5. Let U be a simplicial set. Let n \geq 0 be an integer. The morphism i_{n!} \text{sk}_ n U \to U identifies i_{n!} \text{sk}_ n U with the simplicial set U' \subset U defined in Lemma 14.18.4.
Proof. By Lemma 14.21.4 the only nondegenerate simplices of i_{n!} \text{sk}_ n U are in degrees \leq n. The map i_{n!} \text{sk}_ n U \to U is an isomorphism in degrees \leq n. Combined we conclude that the map i_{n!} \text{sk}_ n U \to U maps nondegenerate simplices to nondegenerate simplices and no two nondegenerate simplices have the same image. Hence Lemma 14.18.3 applies. Thus i_{n!} \text{sk}_ n U \to U is injective. The result follows easily from this. \square
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