Definition 4.8.3.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is fully faithful if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is a homotopy equivalence of Kan complexes.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
4.8.3 Faithful Functors
Recall that a functor of categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is fully faithful if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of sets
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
is a bijection. This condition has a counterpart in the $\infty $-categorical setting:
Example 4.8.3.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory (Definition 4.1.2.15). Then the inclusion map $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is fully faithful. In fact, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}'$, the inclusion $\iota $ induces an isomorphism of simplicial sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.
Example 4.8.3.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $F$ is fully faithful if and only if the induced map $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is fully faithful (in the sense of Definition 4.8.3.1). Consequently, we can regard Definition 4.8.3.1 as a generalization of the classical notion of fully faithful functor.
Remark 4.8.3.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories, so that $F$ induces a functor of homotopy categories $\operatorname {h}\! \mathit{F}: \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$. If $F$ is fully faithful, then $\operatorname {h}\! \mathit{F}$ is also fully faithful (see Remark 4.6.1.12). Beware that the converse is false in general.
Proposition 4.8.3.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories. Then $F$ is conservative (Definition 4.4.2.7). That is, if $u: X \rightarrow Y$ is a morphism in $\operatorname{\mathcal{C}}$ for which $F(u)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$, then $u$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.
Proof. Let $\overline{v}: F(Y) \rightarrow F(X)$ be a homotopy inverse to $F(u)$. Since $F$ is fully faithful, the natural map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(Y), F(X))$ is a homotopy equivalence. We may therefore assume without loss of generality that $\overline{v} = F(v)$, for some morphism $v: Y \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Let $v \circ u$ be a composition of $u$ and $v$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $F(u)$ is homotopy inverse to $F(v)$, the morphism $F( v \circ u)$ is homotopic to $\operatorname{id}_{ F(X) } = F( \operatorname{id}_{X} )$. Since the map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(X))$ is a homotopy equivalence, it follows that $v \circ u$ is homotopic to $\operatorname{id}_{X}$: that is, $v$ is a left homotopy inverse to $u$. A similar argument (with the roles of $u$ and $v$ reversed) shows that $v$ is also a right homotopy inverse to $u$, so that $u$ is an isomorphism. $\square$
Remark 4.8.3.6. Suppose we are given a commutative diagram of $\infty $-categories Combining Remark 4.6.1.15 with Corollary 3.4.1.6, we see that the following conditions are equivalent:
The diagram (4.81) induces a fully faithful functor from $\operatorname{\mathcal{C}}_{01}$ to the homotopy fiber product $\operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$.
For every object $X_{01} \in \operatorname{\mathcal{C}}_{01}$ having images $X_0 \in \operatorname{\mathcal{C}}_0$, $X_1 \in \operatorname{\mathcal{C}}_1$, $X \in \operatorname{\mathcal{C}}$ and every object $Y_{01} \in \operatorname{\mathcal{C}}_{01}$ having images $Y_0 \in \operatorname{\mathcal{C}}_0$, $Y_1 \in \operatorname{\mathcal{C}}_1$, $Y \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes
is a homotopy pullback square.
4.81
\begin{equation} \begin{gathered}\label{equation:morphism-space-categorical-pullback-square} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{01}}( X_{01}, Y_{01} ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}_0}( X_0, Y_0 ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{\mathcal{C}}_1}( X_1, Y_1 ) \ar [r] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X, Y) } \]
Proposition 4.8.3.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. Then, for every functor of $\infty $-categories $F: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, the inclusion map is fully faithful.
\[ \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{E}} \]
Proof. For every pair of objects $X,Y \in \operatorname{\mathcal{E}}$, the functor $U$ induces a Kan fibration of morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Y) )$ (Proposition 4.6.1.22). The desired result now follows from the criterion of Remark 4.8.3.6 (together with Example 3.4.1.3). $\square$
As in classical category theory, it will be useful to break Definition 4.8.3.1 into two separate conditions.
Definition 4.8.3.8 (Faithful Functors). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is faithful if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map is a homotopy equivalence from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to a summand of $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$.
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
Remark 4.8.3.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is fully faithful (in the sense of Definition 4.8.3.1) if and only if it is both full (in the sense of Definitions 4.8.2.1) and faithful (in the sense of Definition 4.8.3.8).
Example 4.8.3.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $F$ is faithful (in the usual sense) if and only if $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a faithful functor of $\infty $-categories.
Warning 4.8.3.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $F$ is faithful (in the sense of Definition 4.8.3.8), then the induced functor of ordinary categories $\operatorname {h}\! \mathit{F}: \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$ is faithful (in the usual category-theoretic sense). Beware that the converse is usually false: see Proposition 4.8.4.11.
It will be useful to consider a common generalization of Definitions 4.8.3.1 and 4.8.3.8.
Definition 4.8.3.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 0$ be an integer. We will say that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $n$-faithful if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of Kan complexes is $(n-2)$-truncated (Definition 3.5.9.1).
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
Example 4.8.3.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is faithful (in the sense of Definition 4.8.3.8) if and only if it is $1$-faithful (in the sense of Definition 4.8.3.12). Similarly, $F$ is fully faithful (in the sense of Definition 4.8.3.1) if and only if it is $0$-faithful. See Examples 3.5.9.3 and 3.5.9.2.
Example 4.8.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$. Then the projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is $n$-faithful (in the sense of Definition 4.8.3.12) if and only if $\operatorname{\mathcal{C}}$ is locally $(n-2)$-truncated (in the sense of Definition 3.5.9.1).
Variant 4.8.3.15. It will be sometimes be useful to extend Definition 4.8.3.12 to the case where $n < 0$. In this case, we say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $n$-faithful if it is both $0$-faithful (that is, fully faithful) and essentially surjective. In ยง4.8.4, we will see that this condition is satisfied if and only if $F$ is an equivalence of $\infty $-categories (Theorem 4.8.4.1).
Remark 4.8.3.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Then $F$ is $n$-faithful if and only if it is $m$-full for every nonnegative integer $m > n$.
Example 4.8.3.17. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n$ be an integer. Then $f$ is $n$-faithful (when regarded as a functor of $\infty $-categories) if and only if it is $(n-1)$-truncated (in the sense of Definition 3.5.9.1). This follows by combining Remark 4.8.3.16 with Proposition 4.8.2.20. In particular, $f$ is fully faithful if and only if it induces a homotopy equivalence from $X$ to a summand of $Y$.
Remark 4.8.3.18 (Symmetry). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Then $F$ is $n$-faithful if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is $n$-faithful. See Remark 4.8.2.9.
Remark 4.8.3.19 (Monotonicity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $m \leq n$ be integers. If $F$ is $m$-faithful, then it is $n$-faithful.
Remark 4.8.3.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 0$ be an integer. Then $F$ is $(n-1)$-faithful if and only if it is both $n$-faithful and $n$-full (see Remark 4.8.3.16).
Remark 4.8.3.21 (Homotopy Invariance). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $n$ be an integer. If $F$ is an equivalence of $\infty $-categories, then $G \circ F$ is $n$-faithful if and only if $G$ is $n$-faithful. If $G$ is an equivalence of $\infty $-categories, then $G \circ F$ is $n$-faithful if and only if $F$ is $n$-faithful. See Remark 4.8.3.26.
Remark 4.8.3.22 (Isomorphism Invariance). Let $F_0, F_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories which are isomorphic (when regarded as objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})$). Then $F_0$ is $n$-faithful if and only if $F_1$ is $n$-faithful. See Remark 4.8.2.17.
Remark 4.8.3.23 (Products). Let $n$ be an integer, let $\{ F_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}_ i \} _{i \in I}$ be a collection of functors of $\infty $-categories, and let $F: \prod _{i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \prod _{i \in I} \operatorname{\mathcal{D}}_ i$ be their product. If each of the functors $F_ i$ is $n$-faithful, then $F$ is $n$-faithful. The converse holds if each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ is nonempty. See Remark 4.8.2.10.
Remark 4.8.3.24 (Coproducts). Let $n$ be an integer, let $\{ F_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}_ i \} _{i \in I}$ be a collection of functors of $\infty $-categories, and let $F: \coprod _{i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \coprod _{i \in I} \operatorname{\mathcal{D}}_ i$ be their coproduct. Then $F$ is $n$-faithful if and only if each of the functors $F_{i}$ is $n$-faithful. See Remark 4.8.2.11.
Remark 4.8.3.25. Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which can be realized as the colimit of a filtered diagram $\{ F_{\alpha }: \operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{D}}_{\alpha } \} $ in the category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$. If each $F_{\alpha }$ is an $n$-faithful functor of $\infty $-categories, then $F$ is an $n$-faithful functor of $\infty $-categories. See Remark 4.8.2.12.
Remark 4.8.3.26 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $n$ be an integer. Then:
If $F$ and $G$ are $n$-faithful, then the composite functor $G \circ F$ is $n$-faithful.
If $G \circ F$ is $n$-faithful and $G$ is $(n+1)$-faithful, then $F$ is $n$-faithful.
If $G \circ F$ is $n$-faithful and $F$ is $(n-1)$-faithful, full, and essentially surjective, then $G$ is $n$-faithful.
Assertion $(a)$ follows from Remark 4.8.2.14, assertion $(b)$ from Proposition 4.8.2.28 (together with Exercise 4.8.2.29 and Proposition 4.8.3.5 for the case $n < 0$), and assertion $(c)$ follows from Proposition 4.8.2.30 and Remark 4.8.2.31.
Example 4.8.3.27. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Then:
If $F$ and $G$ are fully faithful, then $G \circ F$ is fully faithful.
If $G \circ F$ is fully faithful and $G$ is faithful, then $F$ is fully faithful.
If $G \circ F$ is fully faithful and $F$ is an equivalence of $\infty $-categories, then $G$ is fully faithful.
Example 4.8.3.28. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 0$ be an integer. Then:
If $F$ is $n$-faithful and $\operatorname{\mathcal{D}}$ is locally $(n-2)$-truncated, then $\operatorname{\mathcal{C}}$ is locally $(n-2)$-truncated.
If $\operatorname{\mathcal{C}}$ is locally $(n-2)$-truncated and $\operatorname{\mathcal{D}}$ is locally $(n-1)$-truncated, then $F$ is $n$-faithful.
If $\operatorname{\mathcal{C}}$ is locally $(n-2)$-truncated and $F$ is $(n-1)$-faithful, full, and essentially surjective, then $\operatorname{\mathcal{D}}$ is locally $(n-2)$-truncated.
This follows by combining Remark 4.8.3.26 (in the special case $\operatorname{\mathcal{E}}= \Delta ^0$) with Example 4.8.3.13.
Remark 4.8.3.29. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $n$ be an integer. Suppose that $G$ is $n$-faithful. Then $F$ is $n$-faithful if and only if $(G \circ F)$ is $n$-faithful. This follows by combining Remarks 4.8.3.19 and 4.8.3.26.
Example 4.8.3.30. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 0$ be an integer. Suppose that $\operatorname{\mathcal{D}}$ is locally $(n-2)$-truncated. Then $F$ is $n$-faithful if and only if $\operatorname{\mathcal{C}}$ is locally $(n-2)$-truncated. This follows by Remark 4.8.3.29 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$ (see Example 4.8.3.14).
Remark 4.8.3.31. Let $n$ be an integer and suppose we are given a categorical pullback diagram of $\infty $-categories If $F$ is $n$-faithful, then $F'$ is $n$-faithful. In particular, if $F$ is faithful or fully faithful, then $F'$ has the same property. See Corollary 4.8.2.27.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d]^{F'} \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r]^-{G} & \operatorname{\mathcal{D}}. } \]
Variant 4.8.3.32. Suppose we are given a pullback diagram of $\infty $-categories where either $F$ or $G$ is an inner fibration. Then, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}'$, the diagram of Kan complexes is a pullback square where either the horizontal or vertical maps are Kan fibrations (Proposition 4.6.1.22), and therefore a homotopy pullback square (Example 3.4.1.3). Applying Corollary 3.5.9.11, we see that if $F$ is $n$-faithful for some integer $n \geq 0$, then $F'$ is also $n$-faithful. Beware that this conclusion does not hold for $n < 0$.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d]^{F'} \ar [r]^-{G'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r]^-{G} & \operatorname{\mathcal{D}}, } \]
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G'(X), G'(Y) ) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}'}( F'(X), F'(Y) ) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( FG'(X), FG'(Y) ) } \]
Remark 4.8.3.33. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. For $n > 0$, the functor $F$ is $n$-faithful if and only if the diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is $(n-1)$-faithful. This follows from Corollary 3.5.9.18, since $F$ induces a Kan fibration $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ (Proposition 4.6.1.22).
Warning 4.8.3.34. Remark 4.8.3.33 is generally false in the case $n = 0$, even if we assume that $F$ is an isofibration. For example, let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ be a subcategory. Then the inclusion map $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an inner fibration (which is even an isofibration, if $\operatorname{\mathcal{C}}$ is a replete subcategory of $\operatorname{\mathcal{D}}$). In this case, diagonal $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is an isomorphism of simplicial sets. However, $F$ is fully faithful only if $\operatorname{\mathcal{C}}$ is a full subcategory of $\operatorname{\mathcal{D}}$.
Variant 4.8.3.35. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. For $n > 0$, the functor $F$ is $n$-faithful if and only if the composite map is $(n-1)$-faithful. To prove this, we can use Corollaries 4.5.3.24 and 4.5.3.21 to reduce to the situation where $F$ is an isofibration. In this case, the desired result is a reformulation of Remark 4.8.3.33 (see Corollary 4.5.3.29).
\[ \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}^{\mathrm{h}} \operatorname{\mathcal{C}} \]
Proposition 4.8.3.36. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n > 0$ be an integer. The following conditions are equivalent:
The functor $F$ is $n$-faithful.
For every pullback diagram of $\infty $-categories
the functor $F'$ is $n$-faithful.
For every pullback diagram of $\infty $-categories
where $\operatorname{\mathcal{D}}'$ is locally $(n-2)$-truncated, the $\infty $-category $\operatorname{\mathcal{C}}'$ is also locally $(n-2)$-truncated.
For every pullback diagram of $\infty $-categories
the $\infty $-category $\operatorname{\mathcal{C}}'$ is locally $(n-2)$-truncated.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^1 \ar [r] & \operatorname{\mathcal{D}}, } \]
Moreover, the equivalence $(1) \Leftrightarrow (2)$ also holds for $n = 0$.
Proof. Combine Proposition 4.8.2.25 with the criterion of Example 4.8.3.30. $\square$
Warning 4.8.3.37. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq 0$ be an integer. If $F$ is $n$-faithful, then each fiber $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ of $F$ is a locally $(n-2)$-truncated $\infty $-category. Beware that the converse is false in general, even if $F$ is an isofibration. However, it holds under additional assumptions: see Variant 5.1.5.17.
Using Corollary 4.8.2.26, we immediately obtain the following:
Variant 4.8.3.38. Let $n \geq 0$ be an integer and suppose we are given a commutative diagram of $\infty $-categories where the vertical maps are inner fibrations. Then $F$ is $n$-faithful if and only if, for every morphism $u$ of $\operatorname{\mathcal{E}}$, the induced functor is $n$-faithful. In particular, $F$ is fully faithful if and only if each $F_ u$ is fully faithful.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr] & & \operatorname{\mathcal{D}}\ar [dl] \\ & \operatorname{\mathcal{E}}, & } \]
\[ F_{u}: \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}} \]
Corollary 4.8.3.39. Suppose we are given a commutative diagram of $\infty $-categories where $U$ and $V$ are inner fibrations. Assume that the functors $F$ and $\overline{F}$ are $n$-faithful for some integer $n \geq 0$. Then, for every object $\overline{X} \in \overline{\operatorname{\mathcal{C}}}$, the map of fibers $F_{\overline{X}}: \operatorname{\mathcal{C}}_{\overline{X}} \rightarrow \operatorname{\mathcal{D}}_{ \overline{F}(\overline{X}) }$ is also $n$-faithful. In particular, if $F$ and $\overline{F}$ are fully faithful, then $F_{\overline{X}}$ is fully faithful.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{U} & \operatorname{\mathcal{D}}\ar [d]^{V} \\ \overline{\operatorname{\mathcal{C}}} \ar [r]^-{\overline{F}} & \overline{\operatorname{\mathcal{D}}} } \]
Proof. Note that the functor $F$ factors as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \overline{\operatorname{\mathcal{C}}} \times _{ \overline{\operatorname{\mathcal{D}}} } \operatorname{\mathcal{D}}\xrightarrow {F''} \operatorname{\mathcal{D}}, \]
where $F''$ is $n$-faithful by virtue of Variant 4.8.3.32. Applying Remark 4.8.3.29, we conclude that $F'$ is $n$-faithful. The desired result now follows by applying Variant 4.8.3.38 to the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^{F'} \ar [dr] & & \overline{\operatorname{\mathcal{C}}} \times _{ \overline{\operatorname{\mathcal{D}}} } \operatorname{\mathcal{D}}\ar [dl] \\ & \overline{\operatorname{\mathcal{C}}}. & } \]
$\square$