Kerodon

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

Theorem 4.6.2.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functor of $\infty $-categories. Then $F$ is an equivalence of $\infty $-categories if and only if it is fully faithful and essentially surjective.

Proof of Theorem 4.6.2.17. Assume first that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories. Then $F$ induces a homotopy equivalence of Kan complexes $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ (Remark 4.5.1.19). Passing to connected components, we conclude that the induced map $\pi _0( \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \pi _0( \operatorname{\mathcal{D}}^{\simeq } )$ is bijective. In particular, $F$ is essentially surjective. We have a commutative diagram of Kan complexes

4.30
\begin{equation} \begin{gathered}\label{equation:restriction-to-endpoint-diagram} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \ar [r]^-{\theta } \ar [d] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})^{\simeq } \ar [d] \\ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})^{\simeq } \ar [r]^-{\theta _0} & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{D}})^{\simeq },} \end{gathered} \end{equation}

where the horizontal maps are homotopy equivalences (Theorem 4.5.4.1) and the vertical maps are Kan fibrations (Corollary 4.4.5.4). Applying Proposition 3.2.7.1, we conclude that for every vertex $(X,Y) \in \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})^{\simeq }$, the induced map of fibers

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & = & \{ (X,Y) \} \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})^{\simeq } } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \\ & \rightarrow & \{ (X,Y) \} \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{D}})^{\simeq } } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})^{\simeq } \\ & = & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \end{eqnarray*}

is a homotopy equivalence. It follows that $F$ is fully faithful.

Now suppose that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories which is fully faithful and essentially surjective. Using Corollary 4.6.2.8 and Remark 4.6.2.12, we see that the induced map $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is also fully faithful and essentially surjective, and is therefore a homotopy equivalence of Kan complexes (Lemma 4.6.2.18). It follows that the morphism $\theta _0$ in (4.30) is a homotopy equivalence of Kan complexes. Combining our assumption that $F$ is fully faithful with Proposition 3.2.7.1, we conclude that $\theta $ is also a homotopy equivalence. Applying Theorem 4.5.4.1, we conclude that $F$ is an equivalence of $\infty $-categories. $\square$