$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 4.8.5.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is faithful if and only if it satisfies the following pair of conditions:
- $(1)$
The induced functor of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is faithful.
- $(2)$
The diagram of $\infty $-categories
4.82
\begin{equation} \begin{gathered}\label{equation:testing-faithfulness-homotopy} \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}\ar [d]^{F} \ar [r] & \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}) \ar [d]^{ \operatorname{N}_{\bullet }( \mathrm{h} \mathit{F} ) } \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{D}}} ) } \end{gathered} \end{equation}
is a categorical pullback square.
Proof of Proposition 4.8.5.8.
By definition, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is faithful if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ induces a homotopy equivalence from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to a summand of $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) )$. This is equivalent to the following pair of assertions:
- $(1_{X,Y} )$
The map of sets $\pi _0( F_{X,Y} ): \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) )$ is injective.
- $(2_{X,Y})$
The map of Kan complexes
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \times _{ \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) ) } \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
is a homotopy equivalence.
The desired result now follows by allowing the objects $X$ and $Y$ to vary (and applying Remark 4.8.5.9).
$\square$