Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proof. If the vertex $C$ is $F$-initial, then it is an initial object of $\operatorname{\mathcal{C}}_{D}$ by virtue of Remark 7.1.5.8 and Example 7.1.5.4 (note that this implication does not require the assumption that $F$ is a cartesian fibration). Conversely, suppose that $C$ is an initial object of $\operatorname{\mathcal{C}}_ D$. To show that $C$ is $F$-initial, we must show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n} \ar [r]^-{\overline{\sigma } } \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]

admits a solution, provided that $n > 0$ and $\sigma _0(0) = C$. Replacing $F$ by the projection map $\Delta ^ n \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$, we may assume without loss of generality that $\operatorname{\mathcal{D}}= \Delta ^ n$ is a standard simplex and that $\overline{\sigma }$ is the identity map (so that $D = 0$ is the initial vertex of $\operatorname{\mathcal{D}}= \Delta ^ n$). In this case, the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category. We will complete the proof by showing that $C$ is an initial object of $\operatorname{\mathcal{C}}$, so that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ (note that the equality $F(\sigma ) = \overline{\sigma }$ is then automatically satisfied). Fix an object $Y \in \operatorname{\mathcal{C}}$; we wish to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$ is contractible. Since $F$ is a cartesian fibration, we can choose an $F$-cartesian morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ satisfying $F(X) = D$. It follows that composition with the homotopy class $[u]$ induces an isomorphism

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{D}}(C,X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,Y) \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (Corollary 5.1.2.3). Our assumption that $C$ is an initial object of $\operatorname{\mathcal{C}}_{D}$ guarantees that $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{D}}(C,X)$ is a contractible Kan complex, so that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,Y)$ is also contractible. $\square$