Kerodon

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

Corollary 4.6.7.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be locally Kan simplicial categories, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a simplicial functor, and let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F): \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$ be the induced functor of $\infty $-categories. Then:

$(1)$

The functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is fully faithful (in the sense of Definition 4.6.2.1) if and only if the simplicial functor $F$ is weakly fully faithful (in the sense of Definition 4.6.7.7).

$(2)$

The functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is essentially surjective (in the sense of Definition 4.6.2.9) if and only if the simplicial functor $F$ is weakly essentially surjective (in the sense of Definition 4.6.7.7).

$(3)$

The functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is an equivalence of $\infty $-categories (in the sense of Definition 4.5.1.10) if and only if $F$ is a weak equivalence of simplicial categories (in the sense of Definition 4.6.7.7).

Proof. For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, we have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [r]^-{F} \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) )_{\bullet } \ar [d] \\ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X, Y) \ar [r]^-{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(F) } & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})}( F(X), F(Y) ), } \]

where the vertical maps are the homotopy equivalences supplied by Remark 4.6.7.6. It follows that the upper horizontal map is a homotopy equivalence if and only if the lower horizontal map is a homotopy equivalence. This proves $(1)$. Assertion $(2)$ follows from Proposition 2.4.6.8. Assertion $(3)$ follows by combining $(1)$ and $(2)$ with the criterion of Theorem 4.6.2.17. $\square$