Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 3.4.5.5. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. Then $f$ is a weak homotopy equivalence (in the sense of Definition 3.1.6.11) if and only if the underlying morphism of semisimplicial sets is a weak homotopy equivalence (in the sense of Definition 3.4.5.1).

Proof. We have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X^{+} \ar [r]^-{f^{+}} \ar [d]^{ v_{X} } & Y^{+} \ar [d]^{ v_{Y} } \\ X \ar [r]^-{f} & Y, } \]

where the vertical maps are weak homotopy equivalences by virtue of Proposition 3.4.5.4. Invoking Remark 3.1.6.15, we deduce that $f$ is a weak homotopy equivalence if and only if $f^{+}$ is a weak homotopy equivalence. $\square$