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Proposition 5.4.5.14. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then:
- $(1)$
The projection map $\pi : \operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ is a cartesian fibration of $\infty $-categories. Moreover, a morphism $u$ of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is $\pi $-cartesian if and only if, for every vertex $z \in K$, the composite map
\[ \Delta ^2 \simeq \Delta ^1 \star \{ z\} \hookrightarrow \Delta ^1 \star K \xrightarrow {u} \operatorname{\mathcal{C}} \]
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
- $(2)$
The projection map $\pi ': \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ is a cocartesian fibration of $\infty $-categories. Moreover, a morphism $v$ of $\operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is $\pi '$-cocartesian if and only if, for every vertex $x \in K$, the composite map
\[ \Delta ^2 \simeq \{ x\} \star \Delta ^1 \hookrightarrow K \star \Delta ^1 \xrightarrow {v} \operatorname{\mathcal{C}} \]
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
Proof.
We will prove $(1)$; the proof of $(2)$ is similar. It follows from Remark 5.4.2.4 that $\pi $ is an interior fibration. Since $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Proposition 5.4.5.6), it is an inner fibration of $\infty $-categories (Example 5.4.2.2). Let us say that a morphism $u$ of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is special if, for every vertex $z \in K$, the composite map
\[ \Delta ^2 \simeq \Delta ^1 \star \{ z\} \hookrightarrow \Delta ^1 \star K \xrightarrow {u} \operatorname{\mathcal{C}} \]
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$. Let $\overline{\pi }: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ be the projection map. It follows from Corollary 5.4.4.2 that every special morphism of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is $\overline{\pi }$-cartesian when viewed as a morphism of $\operatorname{\mathcal{C}}_{/f}$, and therefore also $\pi $-cartesian (Remark 5.1.1.12). Conversely, any $\pi $-cartesian morphism of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is locally $\overline{\pi }$-cartesian when viewed as a morphism of $\operatorname{\mathcal{C}}_{/f}$, and therefore special (again by Corollary 5.4.4.2). To complete the proof, it will suffice to show that if $Y$ is an object of $\operatorname{\mathcal{C}}_{/f}$, then any morphism $\overline{u}: \overline{X} \rightarrow q( \overline{Y} )$ in $\operatorname{Pith}( \operatorname{\mathcal{C}})$ can be lifted to a special morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$, which follows from Proposition 5.4.3.9.
$\square$