# Kerodon

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Proposition 10.2.2.11 (Homotopy Invariance). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty$-categories and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. Then $f$ is a quotient morphism if and only if $F(f)$ is a quotient morphism in the $\infty$-category $\operatorname{\mathcal{D}}$.

Proof. Let $\operatorname{\mathcal{C}}^{0}_{ / Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be the sieve generated by $f$, and let $\operatorname{\mathcal{D}}^{0}_{ / F(Y) } \subseteq \operatorname{\mathcal{D}}^{0}_{ / F(Y) }$ be the sieve generated by $F(f)$. Since $F$ is fully faithful, $\operatorname{\mathcal{C}}^{0}_{/Y}$ is the inverse image of $\operatorname{\mathcal{D}}^{0}_{ / F(Y) }$ under the functor $F_{/Y}: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{D}}_{ / F(Y) }$ induced by $F$. Corollary 4.6.4.19 guarantees that $F_{/Y}$ is an equivalence of $\infty$-categories, and therefore induces an equivalence $F_{/Y}^{0}: \operatorname{\mathcal{C}}_{/Y}^{0} \rightarrow \operatorname{\mathcal{D}}^{0}_{/F(Y)}$ (Corollary 4.5.2.29). In particular, $F_{/Y}^{0}$ is right cofinal (Corollary 7.2.1.13). Applying Corollary 7.2.2.3, we deduce that $F(f)$ is a quotient morphism if and only if the composite functor

$( \operatorname{\mathcal{C}}_{ / Y}^{0} )^{\triangleright } \hookrightarrow ( \operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$

is a colimit diagram in $\operatorname{\mathcal{D}}$. By virtue of Proposition 7.1.3.9, this is equivalent to the requirement that $f$ is a quotient morphism in $\operatorname{\mathcal{C}}$. $\square$