Kerodon

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

Corollary 8.1.2.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is fully faithful if and only if the diagram

8.3
\begin{equation} \begin{gathered}\label{equation:twisted-arrow-fully-faithful} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \operatorname{Tw}(F) } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r]^-{ F^{\operatorname{op}} \times F} & \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

is a categorical pullback square.

Proof. Since the vertical maps in the diagram (8.3) are left fibrations (Proposition 8.1.1.11), it is a categorical pullback square if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map

\[ \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \{ F(X) \} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}}) \times _{\operatorname{\mathcal{D}}} \{ F(Y) \} \]

is a homotopy equivalence of Kan complexes (Corollary 5.1.6.15). Using Notation 8.1.2.10, we see that this is equivalent to the requirement that $F$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$. $\square$