$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.4.5.11. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{F} \ar [d]^{U} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}' } \]
with the following properties:
- $(1)$
The morphisms $U$ and $U'$ are cartesian fibrations.
- $(2)$
The morphism $F$ carries $U$-cartesian edges of $\operatorname{\mathcal{E}}$ to $U'$-cartesian edges of $\operatorname{\mathcal{E}}'$.
- $(3)$
The morphism $\overline{F}$ is a Morita equivalence of simplicial sets
- $(4)$
For each vertex $C \in \operatorname{\mathcal{C}}$ having image $C' = \overline{F}(C)$, the functor
\[ F_ C: \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \{ C' \} \times _{ \operatorname{\mathcal{C}}' } \operatorname{\mathcal{E}}' = \operatorname{\mathcal{E}}'_{ C' } \]
is a Morita equivalence of $\infty $-categories.
Then $F$ is a Morita equivalence of simplicial sets.
Proof.
Using Corollary 5.6.7.3, we can choose a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r]^-{G} \ar [d]^{U'} & \operatorname{\mathcal{E}}'' \ar [d]^{ U'' } \\ \operatorname{\mathcal{C}}' \ar [r]^-{\overline{G}} & \operatorname{\mathcal{C}}'', } \]
where $U''$ is a cartesian fibration, $\overline{G}$ is inner anodyne, and $\operatorname{\mathcal{C}}''$ is an $\infty $-category. It follows from Corollary 5.6.7.6 that $G$ is a categorical equivalence of simplicial sets; in particular, it is a Morita equivalence (Example 9.4.5.2). Consequently, to show that $F$ is a Morita equivalence, it will suffice to show that the composite map $G \circ F$ is a Morita equivalence (Remark 9.4.5.5). We may therefore replace $U'$ by $U''$, and thereby reduce to proving Proposition 9.4.5.11 in the special case where $\operatorname{\mathcal{C}}'$ is an $\infty $-category. Similarly, we may assume that $\operatorname{\mathcal{C}}$ is an $\infty $-category. To complete the proof, it will suffice to show that the functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ satisfies the conditions of Proposition 9.4.5.8:
- $(a)$
Since $\overline{F}$ is a Morita equivalence of $\infty $-categories, it is fully faithful. Similarly, for each object $C \in \operatorname{\mathcal{C}}$ having image $C' = \overline{F}(C)$, condition $(4)$ guarantees that the functor $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C'}$ is fully faithful. Using condition $(2)$ and Proposition 5.1.6.7, we conclude that $F$ is fully faithful.
- $(b)$
Let $Y$ be an object of $\operatorname{\mathcal{E}}'$; we wish to show that $Y$ is a retract of $F(X)$, for some object $X \in \operatorname{\mathcal{E}}$. Set $\overline{Y} = U'(Y)$. Since $\overline{F}$ is a Morita equivalence, $\overline{Y}$ is a retract of $C' = \overline{F}( C )$, for some object $C \in \operatorname{\mathcal{C}}$. Choose a retraction diagram $\overline{\sigma }:$
\[ \xymatrix@R =50pt@C=50pt{ & C' \ar [dr]^{ \overline{r} } & \\ \overline{Y} \ar [ur]^{ \overline{i} } \ar [rr]^-{ \operatorname{id}} & & \overline{Y} } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$. Our assumption that $U'$ is a cartesian fibration guarantees that we can lift $\overline{\sigma }$ to a retraction diagram
\[ \xymatrix@R =50pt@C=50pt{ & X' \ar [dr]^{r} & \\ Y \ar [rr]^-{\operatorname{id}} \ar [ur]^{i} & & Y } \]
in the $\infty $-category $\operatorname{\mathcal{E}}'$. Since the functor $F_{ C}$ is a Morita equivalence, there exists an object $X \in \operatorname{\mathcal{E}}_{C}$ such that $X'$ is a retract of $F(X)$ in the $\infty $-category $\operatorname{\mathcal{E}}'_{C'}$, and therefore also in the $\infty $-category $\operatorname{\mathcal{E}}'$. Applying Remark 8.5.1.6, we conclude that $Y$ is a retract of $F(X)$.
$\square$