Kerodon

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

Proposition 4.6.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. Then the projection map $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration and the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.

Proof. We will prove the second assertion; the first follows by a similar argument. Let $A \hookrightarrow B$ be a right anodyne morphism of simplicial sets; we wish to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}} \]

admits a solution. Unwinding the definitions, we are reduced to showing that a map of simplicial sets

\[ \sigma _0: B \coprod _{A} (A \diamond \{ X\} ) \rightarrow \operatorname{\mathcal{C}} \]

can be extended to a map $\sigma : B \diamond \{ X\} \rightarrow \operatorname{\mathcal{C}}$ (see Notation 4.5.8.3). By virtue of Lemma 4.5.5.2, it will suffice to show that the inclusion map

\[ \iota : B \coprod _{A} ( A \diamond \{ X\} ) \hookrightarrow B \diamond \{ X\} \]

is a categorical equivalence of simplicial sets, which follows from Corollary 4.5.8.14. $\square$