Theorem 5.1.6.1. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^-{U} & \operatorname{\mathcal{C}}' \ar [d]^-{U'} \\ \operatorname{\mathcal{D}}\ar [r]^-{ \overline{F} } & \operatorname{\mathcal{D}}' } \]
where $U$ is a cartesian fibration of $\infty $-categories, $U'$ is an isofibration of $\infty $-categories, and $\overline{F}$ is an equivalence of $\infty $-categories. Then the functor $F$ is an equivalence of $\infty $-categories if and only if it satisfies the following conditions:
- $(1)$
For every object $D \in \operatorname{\mathcal{D}}$ having image $D' = \overline{F}(D)$ in $\operatorname{\mathcal{D}}'$, the induced functor
\[ F_{D}: \operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \{ D' \} \times _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{C}}' = \operatorname{\mathcal{C}}'_{D'} \]
is an equivalence of $\infty $-categories.
- $(2)$
The functor $F$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to $U'$-cartesian morphisms of $\operatorname{\mathcal{C}}'$.
Moreover, if these conditions are satisfied, then $U'$ is also a cartesian fibration of $\infty $-categories.
Proof of Theorem 5.1.6.1.
Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^-{U} & \operatorname{\mathcal{C}}' \ar [d]^-{U'} \\ \operatorname{\mathcal{D}}\ar [r]^-{ \overline{F} } & \operatorname{\mathcal{D}}' } \]
where $U$ is a cartesian fibration of $\infty $-categories, $U'$ is an isofibration of $\infty $-categories, and $\overline{F}$ is an equivalence of $\infty $-categories. If $F$ satisfies conditions $(1)$ and $(2)$ of Theorem 5.1.6.1, then it is fully faithful (Proposition 5.1.6.7) and essentially surjective (Remark 4.6.2.19), hence an equivalence of $\infty $-categories by virtue of Theorem 4.6.2.20. Conversely, if $F$ is an equivalence of $\infty $-categories, then it satisfies conditions $(1)$ and $(2)$ by virtue of Corollary 4.5.2.32 and Proposition 5.1.6.7, respectively. To complete the proof, we must show that if these conditions are satisfied, then $U'$ is also a cartesian fibration of $\infty $-categories.
Let $Z'$ be an object of $\operatorname{\mathcal{C}}'$ and let $\overline{g}': \overline{Y}' \rightarrow U'(Z')$ be a morphism in $\operatorname{\mathcal{D}}'$; we wish to show that $\overline{g}'$ can be lifted to a $U'$-cartesian morphism $Y' \rightarrow Z'$ in $\operatorname{\mathcal{C}}'$. Since $F$ is essentially surjective, we can choose an object $Z \in \operatorname{\mathcal{C}}$ and an isomorphism $v: F(Z) \rightarrow Z'$ in the $\infty $-category $\operatorname{\mathcal{C}}'$. Since $\overline{F}$ is essentially surjective, we can choose an object $\overline{Y} \in \operatorname{\mathcal{D}}$ and an isomorphism $\overline{u}: \overline{F}(\overline{Y}) \rightarrow \overline{Y}'$ in the $\infty $-category $\operatorname{\mathcal{D}}'$. Since $\overline{F}$ is fully faithful at the level of homotopy categories, we can choose a morphism $\overline{g}: \overline{Y} \rightarrow U(Z)$ in $\operatorname{\mathcal{D}}$ for which the diagram
\[ \xymatrix@R =50pt@C=50pt{ \overline{F}( \overline{Y} ) \ar [r]^-{ \overline{F}( \overline{g} ) } \ar [d]^{ \overline{u} } & \overline{F}( U(Z) ) \ar [d]^-{ U'(v) } \\ \overline{Y}' \ar [r]^-{ \overline{g}' } & \overline{Z}', } \]
commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}'}$, and can therefore be lifted to a commutative diagram $\overline{\sigma }$ in $\infty $-category $\operatorname{\mathcal{D}}'$ (see Exercise 1.5.2.10). Using our assumption that $U$ is a cartesian fibration, we can lift $\overline{g}$ to a $U$-cartesian morphism $g: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}$. Since $U'$ is an isofibration, Corollary 4.4.5.9 guarantees that we can lift $\overline{\sigma }$ to a commutative diagram $\sigma :$
\[ \xymatrix@R =50pt@C=50pt{ F(Y) \ar [r]^-{ F(g) } \ar [d] & F(Z) \ar [d]^-{ v } \\ Y' \ar [r]^-{ g' } & Z' } \]
in the $\infty $-category $\operatorname{\mathcal{C}}'$, where the vertical maps are isomorphisms. To complete the proof, it will suffice to show that the morphism $g'$ is $U'$-cartesian. This follows from Corollary 5.1.2.5, since the morphism $F(g)$ is $U'$-cartesian (Proposition 5.1.6.6).
$\square$