Kerodon

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

Corollary 5.3.7.5. Let $f: K \rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. Then $f$ factors as a composition $K \xrightarrow {j} \operatorname{\mathcal{C}}\xrightarrow {U} \operatorname{\mathcal{D}}$, where $U$ is an isofibration of $\infty $-categories and $j$ is both a monomorphism and a categorical equivalence.

Proof. Using Proposition 4.1.3.2, we can factor $f$ as a composition $K \xrightarrow {i} \operatorname{\mathcal{K}}\xrightarrow {F} \operatorname{\mathcal{D}}$, where $i$ is inner anodyne and $F$ is an inner fibration. Note that the simplicial set $\operatorname{\mathcal{K}}$ is an $\infty $-category (Remark 4.1.1.9), and that $i$ is a categorical equivalence of simplicial sets (Corollary 4.5.3.14). We may therefore replace $f$ by $F$, and thereby reduce to the special case where $K = \operatorname{\mathcal{K}}$ is an $\infty $-category. Let $\operatorname{\mathcal{C}}$ denote the homotopy fiber product $\operatorname{\mathcal{K}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. Then $F$ factors as a composition

\[ \operatorname{\mathcal{K}}\xrightarrow { \delta } \operatorname{\mathcal{K}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\xrightarrow {U} \operatorname{\mathcal{D}}, \]

where the diagonal embedding $\delta $ is an equivalence of $\infty $-categories (Proposition 5.3.7.4) and $U$ is an isofibration (see Remark 4.5.2.2). $\square$