Lemma 14.18.3. Let $f : U \to V$ be a morphism of simplicial sets. Suppose that (a) the image of every nondegenerate simplex of $U$ is a nondegenerate simplex of $V$ and (b) the restriction of $f$ to a map from the set of nondegenerate simplices of $U$ to the set of nondegenerate simplices of $V$ is injective. Then $f_ n$ is injective for all $n$. Same holds with “injective” replaced by “surjective” or “bijective”.
Proof. Under hypothesis (a) we see that the map $f$ preserves the disjoint union decompositions of the splitting of Lemma 14.18.2, in other words that we get commutative diagrams
And then (b) clearly shows that the left vertical arrow is injective (resp. surjective, resp. bijective). $\square$
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