Definition 3.4.5.1. Let $f: X \rightarrow Y$ be a morphism of semisimplicial sets. We will say that $f$ is a weak homotopy equivalence if the induced map of simplicial sets $f^{+}: X^{+} \rightarrow Y^{+}$ is a weak homotopy equivalence, in the sense of Definition 3.1.6.12.
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