Kerodon

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

Proposition 5.3.6.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which has the two-out-of-six property. If $W$ contains every identity morphism of $\operatorname{\mathcal{C}}$, then it contains every isomorphism of $\operatorname{\mathcal{C}}$.

Proof of Proposition 5.3.6.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be an isomorphism in $\operatorname{\mathcal{C}}$. Then $f$ admits a homotopy inverse $g: Y \rightarrow X$. Let $\sigma $ be a $2$-simplex of $\operatorname{\mathcal{C}}$ which witnesses $\operatorname{id}_{X}$ as a composition $f$ and $g$, and let $\sigma '$ be a $2$-simplex of $\operatorname{\mathcal{C}}$ which witnesses $\operatorname{id}_{Y}$ as a composition of $g$ and $f$. Then the triple $(\sigma ', s_0(f), \bullet , \sigma )$ can be regarded as a morphism of simplicial sets $\tau _0: \Lambda ^{3}_{2} \rightarrow \operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is an $\infty $-category, we can extend $\tau _0$ to a $3$-simplex $\tau : \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$, whose restriction to the $1$-skeleton of $\Delta ^3$ is indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [r]^-{g} \ar [drr]_{\operatorname{id}_ Y} & X \ar [dr]^{ f} & \\ X \ar [ur]^{f} \ar [urr]_{ \operatorname{id}_{X} } \ar [rrr]^{f} & & & Y. } \]

It follows that if $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $\operatorname{id}_{X}$, $\operatorname{id}_{Y}$, and has the two-out-of-six property, then $W$ also contains the isomorphism $f$. $\square$