Our first goal in this section is to prove the following result:
Theorem 4.8.4.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
The functor $F$ is an equivalence of $\infty $-categories.
The functor $F$ is essentially surjective and fully faithful.
The functor $F$ is essentially surjective, full, and faithful.
The functor $F$ is $n$-full for every integer $n \geq 0$.
Example 4.8.4.2. Suppose that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are the nerves of categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{D}}_0$, so that $F$ is obtained from a functor $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$. In this case, Theorem 4.8.4.1 reduces to the assertion that $F_0$ is an equivalence of categories if and only if it is essentially surjective, full, and faithful. See Example 4.8.2.8.
Example 4.8.4.3. Suppose that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are Kan complexes. In this case, the equivalence $(1) \Leftrightarrow (4)$ of Theorem 4.8.4.1 reduces to the combinatorial version of Whitehead's theorem: a morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a homotopy equivalence if and only if it is bijective on connected components and induces an isomorphism of homotopy groups (Theorem 3.2.7.1). See Proposition 4.8.2.20.
Corollary 4.8.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{D}}' \subseteq \operatorname{\mathcal{D}}$ be the essential image of $F$ (Definition 4.8.1.11). Then $F$ is fully faithful if and only if it induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$.
We will deduce Theorem 4.8.4.1 from the following more general result, which asserts the compatibility between Definitions 4.8.0.4 and 4.8.2.7:
Proposition 4.8.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq 1$ be an integer. Then $F$ is $n$-full if and only if every lifting problem admits a solution. If $F$ is an isofibration, then this is also true in the case $n = 0$.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]
Proof. The case $n=0$ follows from Remark 4.8.1.6 and the case $n=1$ from Remark 4.8.2.4. We may therefore assume without loss of generality that $n \geq 2$. Using Proposition 4.8.2.25, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In this case, the functor $F$ is $n$-full if and only if, for every morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the set $\pi _{n-2}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u )$ consists of a single element (Corollary 4.8.2.24). The desired result now follows from Corollary 4.7.6.25. $\square$
Proof of Theorem 4.8.4.1. The equivalences $(2) \Leftrightarrow (3) \Leftrightarrow (4)$ are immediate from the definitions (see Remarks 4.8.3.9 and 4.8.3.16) and the implication $(1) \Rightarrow (4)$ follows from Example 4.8.2.16. We will complete the proof by showing that $(4)$ implies $(1)$. Assume that $F$ is $n$-full for every integer $n \geq 0$; we wish to show that $F$ is an equivalence of $\infty $-categories. Using Corollary 4.5.3.24, we can factor $F$ as a composition $\operatorname{\mathcal{C}}\xrightarrow {T} \overline{\operatorname{\mathcal{C}}} \xrightarrow {\overline{F}} \operatorname{\mathcal{D}}$, where $T$ is an equivalence of $\infty $-categories and $\overline{F}$ is an isofibration. Using Remark 4.8.2.18, we deduce that $\overline{F}$ is $n$-full for every integer $n$. It follows from Proposition 4.8.4.5 that $\overline{F}$ is a trivial Kan fibration. In particular, $\overline{F}$ is an equivalence of $\infty $-categories (Proposition 4.5.4.11), so that $F = \overline{F} \circ T$ is also an equivalence of $\infty $-categories. $\square$
Let us record some applications of Theorem 4.8.4.1. Recall that, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories, then the induced map of cores $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a homotopy equivalence of Kan complexes (Remark 4.5.1.19). The converse assertion is not true in general. For example, the inclusion map $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ induces an isomorphism on cores, but is never an equivalence of $\infty $-categories unless $\operatorname{\mathcal{C}}$ is a Kan complex. However, we have the following slightly weaker result:
Theorem 4.8.4.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is an equivalence of $\infty $-categories if and only if the induced map of Kan complexes $\theta : \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})^{\simeq }$ is a homotopy equivalence.
Proof. Assume that $\theta $ is a homotopy equivalence; we will show that $F$ is an equivalence of $\infty $-categories (the reverse implication is a special case of Proposition 4.5.1.22). Note that the map of cores $\operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a retract of $\theta $, and is therefore also a homotopy equivalence (Proposition 3.2.8.3). In particular, it is surjective on connected components, so that $F$ is essentially surjective. By virtue of Theorem 4.8.4.1, it will suffice to show that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is a homotopy equivalence. This follows by applying Proposition 3.3.7.1 to the commutative diagram of Kan complexes
where the horizontal maps are homotopy equivalences (by assumption) and the vertical maps are Kan fibrations (Corollary 4.4.5.4). $\square$
Corollary 4.8.4.7. A commutative diagram of $\infty $-categories is a categorical pullback square if and only if the induced diagram of Kan complexes is a homotopy pullback square.
4.82
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square29} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d]^{U} \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
4.83
\begin{equation} \begin{gathered}\label{equation:bellie} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_{01} )^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_{0} )^{\simeq } \ar [d] \\ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_{1} )^{\simeq } \ar [r] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } } \end{gathered} \end{equation}
Proof. We proceed as in the proof of Proposition 4.5.3.15. By definition, the diagram (4.82) is a categorical pullback square if and only if the induced map $\theta : \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is an equivalence of $\infty $-categories. Using the criterion of Theorem 4.8.4.6, we see that this is equivalent to the requirement that $\theta $ induces a homotopy equivalence of Kan complexes $\rho : \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{01} )^{\simeq } \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1)^{\simeq }$. Using Remarks 4.5.3.6 and 4.5.3.7, we can identify $\rho $ with the map
determined by the commutative diagram (4.83). The desired result now follows from the criterion of Corollary 3.4.1.6. $\square$
Definition 4.8.4.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a simplicial functor.
We say that $F$ is weakly fully faithful if, for every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )_{\bullet }$ is a weak homotopy equivalence of simplicial sets.
We say that $F$ is weakly essentially surjective if the induced functor of homotopy categories $\operatorname {h}\! \mathit{F}: \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$ is essentially surjective (that is, every object of $\operatorname{\mathcal{D}}$ is homotopy equivalent to an object of the form $F(X)$, for some $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$).
We say that $F$ is a weak equivalence of simplicial categories if it is weakly fully faithful and weakly essentially surjective.
Corollary 4.8.4.9. 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:
The functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is fully faithful (in the sense of Definition 4.8.3.1) if and only if the simplicial functor $F$ is weakly fully faithful (in the sense of Definition 4.8.4.8).
The functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is essentially surjective (in the sense of Definition 4.8.1.1) if and only if the simplicial functor $F$ is weakly essentially surjective (in the sense of Definition 4.8.4.8).
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 4.8.4.8).
Proof. For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, we have a commutative diagram of simplicial sets
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 2.4.6.9. Assertion $(3)$ follows by combining $(1)$ and $(2)$ with the criterion of Theorem 4.8.4.1. $\square$
Proposition 4.8.4.10. Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
The functor $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$ (Definition 4.7.6.13).
The functor $F$ is essentially surjective and, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ exhibits the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ as an $n$-truncation of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.
Proof. We may assume without loss of generality that $n \geq -2$ and that the $\infty $-category $\operatorname{\mathcal{D}}$ is locally $n$-truncated (since this follows from both $(1)$ and $(2)$). It follows from Proposition 4.7.6.19 that there exists a functor $\overline{F}: \operatorname{\mathcal{C}}\rightarrow \overline{\operatorname{\mathcal{D}}}$ which is bijective on objects and exhibits $\overline{\operatorname{\mathcal{D}}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$ (for example, we can take $\overline{\operatorname{\mathcal{D}}}$ to be the coskeleton $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$). Replacing $F$ by an isomorphic functor if necessary, we may assume that it factors as a composition $\operatorname{\mathcal{C}}\xrightarrow {\overline{F}} \overline{\operatorname{\mathcal{D}}} \xrightarrow {G} \operatorname{\mathcal{D}}$. In this case, condition $(1)$ is satisfied if and only if $G$ is an equivalence of $\infty $-categories. By virtue of Theorem 4.8.4.1, we can rewrite this condition as follows:
The functor $G: \overline{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ is fully faithful and essentially surjective.
Since $\overline{F}$ is bijective on objects, the functor $G$ is essentially surjective if and only if $F$ is essentially surjective. It will therefore suffice to show that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the following conditions are equivalent:
The functor $G$ induces a homotopy equivalence $G_{X,Y}: \operatorname{Hom}_{ \overline{\operatorname{\mathcal{D}}} }( \overline{F}(X), \overline{F}(Y) ) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}(F(X), F(Y) )$.
The morphism $F_{X,Y}$ exhibits $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ as an $n$-truncation of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.
The equivalence of these conditions follows from the observation that the map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{ \overline{\operatorname{\mathcal{D}}} }( \overline{F}(X), \overline{F}(Y) )$ exhibits $\operatorname{Hom}_{ \overline{\operatorname{\mathcal{D}}} }( \overline{F}(X), \overline{F}(Y) )$ as an $n$-truncation of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (Corollary 4.7.6.24). $\square$
Recall that the condition of faithfulness cannot be tested at the level of homotopy categories. Instead, we have the following:
Proposition 4.8.4.11. 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:
The induced functor of homotopy categories $\operatorname {h}\! \mathit{F}: \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$ is faithful.
The diagram of $\infty $-categories
is a categorical pullback square.
4.84
\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 }(\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}) \ar [d]^{ \operatorname{N}_{\bullet }( \operatorname {h}\! \mathit{F} ) } \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{N}_{\bullet }( \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}} ) } \end{gathered} \end{equation}
Remark 4.8.4.12. In the situation of Proposition 4.8.4.11, the comparison map $\operatorname{\mathcal{D}}\rightarrow \operatorname{N}_{\bullet }( \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}} )$ is automatically an isofibration (Corollary 4.4.1.9). By virtue of Proposition 4.5.3.27, condition $(2)$ is equivalent to the following:
The functor $F$ induces an equivalence of $\infty $-categories $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} ) \times _{ \operatorname{N}_{\bullet }( \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}} )} \operatorname{\mathcal{D}}$.
Note that the functor $F'$ is bijective on objects, and therefore essentially surjective. Using Theorem 4.8.4.1, we can reformulate $(2')$ as follows:
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functor $F$ induces a homotopy equivalence
Proof of Proposition 4.8.4.11. 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:
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.
The map of Kan complexes
is a homotopy equivalence.
The desired result now follows by allowing the objects $X$ and $Y$ to vary (and applying Remark 4.8.4.12). $\square$
We now record another consequence of Proposition 4.8.4.5:
Proposition 4.8.4.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq 0$ be an integer. The following conditions are equivalent:
The functor $F$ is $n$-faithful.
For every integer $m > n$, every lifting problem
admits a solution.
For every simplicial set $B$ and every simplicial subset $A \subseteq B$ which contains the $n$-skeleton of $B$, every lifting problem
admits a solution.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{m} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]
\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]
Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 4.8.4.5. The implication $(3) \Rightarrow (2)$ is immediate, and the reverse implication follows from Proposition 1.1.4.12. $\square$
Corollary 4.8.4.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset. If $F$ is $n$-faithful for some integer $n$, then the induced map $F': \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ is also $n$-faithful. Moreover, if $A$ contains the $n$-skeleton of $B$, then $F'$ is an equivalence of $\infty $-categories.
Proof. If $n < 0$, then $F$ is an equivalence of $\infty $-categories (Theorem 4.8.4.1); it then follows from Corollary 4.5.3.34 that $F'$ is also an equivalence of $\infty $-categories. We may therefore assume that $n \geq 0$. Using Proposition 3.1.8.1, we can reduce to the case where $F$ is an inner fibration, so that $F'$ is also an inner fibration (Proposition 4.1.4.1). By virtue of Proposition 4.8.4.13, it will suffice to show that for every simplicial set $B'$ and every simplicial subset $A' \subseteq B'$ every lifting problem
admits a solution provided either that $A$ contains the $n$-skeleton of $B$ or $A'$ contains the $n$-skeleton of $B'$. Unwinding the definitions, we can rewrite (4.85) as a lifting problem
The existence of a solution now follows from Proposition 4.8.4.13, since $F$ is $n$-faithful and $(A \times B') \coprod _{ (A \times A') } (B \times A')$ contains the $n$-skeleton of $B \times B'$. $\square$
Corollary 4.8.4.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $B$ be a simplicial set, and let $n$ be an integer. If $F$ is $n$-faithful, then the induced functor $F': \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{D}})$ is also $n$-faithful. In particular, if $F$ is fully faithful, then $F'$ is fully faithful .
Proof. Apply Corollary 4.8.4.14 in the special case $A = \emptyset $. $\square$
Corollary 4.8.4.16. Let $n \geq 0$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an $n$-faithful functor of $\infty $-categories. Then, for every diagram $B \rightarrow \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{D}}}(B, \operatorname{\mathcal{C}})$ is locally $(n-2)$-truncated.
Proof. It follows from Corollary 4.1.4.2 that $F$ induces an inner fibration $F': \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ and from Corollary 4.8.4.15 that $F'$ is $n$-faithful. Applying Proposition 4.8.3.36, we deduce that every fiber of $F'$ is locally $(n-2)$-truncated. $\square$
Corollary 4.8.4.17. Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. If $G$ is $n$-faithful for some integer $n$, then the induced maps are also $n$-faithful. In particular, if $G$ is fully faithful, then $G'$ and $G''$ are also fully faithful.
\[ G': \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{D}}_{/(G \circ F)} \quad \quad G'': \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{D}}_{(G \circ F)/} \]
Proof. If $n < 0$, then $G$ is an equivalence of $\infty $-categories (Theorem 4.8.4.1) and the result follows from Corollary 4.6.4.21. We may therefore assume without loss of generality that $n \geq 0$. We will show that $G'$ is $n$-faithful; the proof for $G''$ is similar. Using Proposition 3.1.8.1, we can reduce to the case where $G$ is an inner fibration, so that $G'$ is also an inner fibration (Corollary 4.3.6.10). By virtue of Proposition 4.8.4.13, it will suffice to show that for every simplicial set $B$ and every simplicial subset $A \subseteq B$ which contains the $n$-skeleton of $B$, every lifting problem
admits a solution. Let us identify the upper horizontal map with a diagram $\overline{F}: A \star K \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{F}|_{K} = F$. In this case, we can rewrite (4.86) as a lifting problem
Since $G$ is an $n$-faithful inner fibration and $A \star K$ contains the $n$-skeleton of $B \star K$, this lifting problem admits a solution (Proposition 4.8.4.13). $\square$