Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Recall that $F$ is fully faithful if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of morphism spaces
is a homotopy equivalence (Definition 4.6.2.1). It is sometimes convenient to break this into two separate conditions:
Definition 4.8.5.1 (Full Functors). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is full if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map is surjective on connected components.
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
Definition 4.8.5.2 (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 4.8.5.3. 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 4.6.2.1) if and only if it is both full and faithful (in the sense of Definitions 4.8.5.1 and 4.8.5.2).
Example 4.8.5.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Then the functor of $\infty $-categories $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is full (in the sense of Definition 4.8.5.1) if and only if the functor $F$ is full (in the usual category-theoretic sense). Similarly, $\operatorname{N}_{\bullet }(F)$ is faithful (in the sense of Definition 4.8.5.2) if and only if $F$ is faithful. Consequently, we can view Definitions 4.8.5.1 and Definition 4.8.5.2 as generalizations of their classical counterparts.
Remark 4.8.5.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is full (in the sense of Definition 4.8.5.1) if and only if the induced functor of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is full (in the usual category-theoretic sense).
Remark 4.8.5.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. If $G \circ F$ is full and $F$ is essentially surjective, then $G$ is also full.
Exercise 4.8.5.7. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Show that $f$ is full (in the sense of Definition 4.8.5.1) if and only if it satisfies the following pair of conditions:
The map of connected components $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is injective.
For every vertex $x \in X$ having image $y = f(x)$, the map of fundamental groups $\pi _{1}(f): \pi _1(X,x) \rightarrow \pi _{1}(Y,y)$ is surjective.
The counterpart of Remark 4.8.5.5 for faithful functors is slightly more involved.
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:
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.
The diagram of $\infty $-categories
is a categorical pullback square.
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}
Remark 4.8.5.9. In the situation of Proposition 4.8.5.8, the comparison map $\operatorname{\mathcal{D}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{D}}} )$ is automatically an isofibration (Corollary 4.4.1.9). By virtue of Proposition 4.5.2.26, condition $(2)$ is equivalent to the following:
The functor $F$ induces an equivalence of $\infty $-categories $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) \times _{ \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{D}}} )} \operatorname{\mathcal{D}}$.
Note that the functor $F'$ is bijective on objects, and therefore essentially surjective. Using Theorem 4.6.2.21, 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 4.8.5.8. 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:
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 4.8.5.9). $\square$
Note the asymmetry between Remark 4.8.5.5 and Proposition 4.8.5.8: in the higher-categorical setting, fullness is a relatively weak condition which can be tested at the level of homotopy categories, but faithfulness is not. It will therefore be useful to further analyze Definition 4.8.5.2.
Definition 4.8.5.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
We say that $F$ is $0$-full if it is essentially surjective: that is, every object of $\operatorname{\mathcal{D}}$ is isomorphic to $F(X)$, for some object $X \in \operatorname{\mathcal{C}}$ (Definition 4.6.2.12).
We say that $F$ is $1$-full if it is full: that is, for objects $X,Y \in \operatorname{\mathcal{C}}$ having images $\overline{X} = F(X)$ and $\overline{Y} = F(Y)$ in $\operatorname{\mathcal{D}}$, the map
is surjective (Definition 4.8.5.1).
For $n \geq 2$, we say that $F$ is $n$-full if, for every morphism $u: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ having image $\overline{u}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{D}}$, the induced map
is injective for $m = n-2$ and surjective for $m = n-1$.
\[ \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ) ) \]
\[ \pi _{m}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u) \rightarrow \pi _{m}( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y}), \overline{u} ) \]
Remark 4.8.5.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
The functor $F$ is faithful if and only if it is $n$-full for each $n \geq 2$ (see Example 3.5.9.3).
The functor $F$ is fully faithful if and only if it is $n$-full for each $n \geq 1$ (see Remark 4.8.5.3).
The functor $F$ is an equivalence of $\infty $-categories if and only if it is $n$-full for each $n \geq 0$ (see Theorem 4.6.2.21).
Example 4.8.5.12. Let $n \geq -2$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{D}}$ (see Definition 4.8.2.9). Then $F$ is $m$-full for $m \leq n+2$. In particular, for any $\infty $-category $\operatorname{\mathcal{C}}$, the canonical maps are $m$-full for $m \leq n+2$.
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}) \quad \quad \operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n+1}^{\circ }(\operatorname{\mathcal{C}}) \quad \quad \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n+1}}\mathit{(\operatorname{\mathcal{C}})} \]
Remark 4.8.5.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq -2$ be an integer. Then $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if the projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is $m$-full for all $m \geq n+3$.
Remark 4.8.5.14 (Symmetry). 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 $n$-full if and only the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is $n$-full.
Remark 4.8.5.15 (Composition). 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 $n \geq 0$ be an integer. If $F$ and $G$ are $n$-full, then the composite functor $G \circ F$ is $n$-full.
Remark 4.8.5.16 (Change of Target). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, where $G$ is fully faithful. For $n \geq 1$, the functor $F$ is $n$-full if and only if the composite functor $G \circ F$ is $n$-full. If $G$ is an equivalence of $\infty $-categories, then this is also true when $n = 0$.
Remark 4.8.5.17 (Isomorphism Invariance). Let $F_0, F_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories which are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). Then $F_0$ is $n$-full if and only if $F_1$ is $n$-full. To see this, let $\operatorname{Isom}(\operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$ spanned by the isomorphisms (Example 4.4.1.14), so that the evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Isom}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$ are equivalences of $\infty $-categories (Corollary 4.4.5.10). The assumption that $F_0$ and $F_1$ are isomorphic guarantees that there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{Isom}(\operatorname{\mathcal{D}})$ satisfying $F_0 = \operatorname{ev}_0 \circ F$ and $F_1 = \operatorname{ev}_1 \circ F$. Using Remark 4.8.5.16, we see that $F_0$ is $n$-full if and only if $F$ is $n$-full. Similarly, $F_1$ is $n$-full if and only if $F$ is $n$-full.
Remark 4.8.5.18 (Change of Source). 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 $n \geq 0$ be an integer. If $F$ is an equivalence of $\infty $-categories, then $G$ is $n$-full if and only if $G \circ F$ is $n$-full. The “only if” direction follows immediately from Remark 4.8.5.15. For the converse, suppose that $G \circ F$ is $n$-full, and let $F^{-1}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse to $F$. Then $F^{-1}$ is an equivalence of $\infty $-categories; in particular, it is $n$-full (Remark 4.8.5.11). Applying Remark 4.8.5.15, we conclude that the composition $G \circ F \circ F^{-1}$ is also $n$-full. This composition is isomorphic to $G$, so $G$ is $n$-full as well (Remark 4.8.5.17).
Remark 4.8.5.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $u: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ having image $\overline{u}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{D}}$. For each integer $n$, the requirement that the map is injective or surjective depends only on the isomorphism class of $u$ (as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$).
\[ \pi _{n}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u) \rightarrow \pi _{n}( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ), \overline{u} ) \]
In the setting of Kan complexes, Definition 4.8.5.10 can be simplified:
Proposition 4.8.5.20. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Then:
The morphism $f$ is $0$-full (in the sense of Definition 4.8.5.10) if and only if the induced map $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is surjective.
For $n \geq 1$, the morphism $f$ is $n$-full if and only if, for every vertex $x \in X$ having image $y = f(x)$, the induced map $\pi _{m}(f): \pi _{m}(X,x) \rightarrow \pi _{m}(Y,y)$ is injective for $m = n-1$ and surjective for $m = n$.
Proof. Assertion $(a)$ is immediate from the definitions. We will prove $(b)$. The case $n=1$ follows from Exercise 4.8.5.7. Let us therefore assume that $n \geq 2$. By definition, $f$ is $n$-full if and only if, for every edge $u: x \rightarrow x'$ of $X$ having image $v: y \rightarrow y'$ in $Y$, the induced map
is injective for $m = n-1$ and surjective for $m = n$. By virtue of Remark 4.8.5.19, it suffices to check this in the special case where $u = \operatorname{id}_{x}$ is a degenerate edge of $X$. Assertion $(b)$ now follows from isomorphisms
of Example 4.6.1.13. $\square$
Remark 4.8.5.21. In the situation of Proposition 4.8.5.20, suppose that $f$ is a Kan fibration. Then assertions $(a)$ and $(b)$ can be reformulated as follows:
The Kan fibration $f$ is $0$-full if and only if, for each vertex $y \in Y$, the fiber $X_{y} = \{ y\} \times _{Y} X$ is nonempty.
For $n \geq 1$, the Kan fibration $f$ is $n$-full if and only if, for every vertex $x \in X$ having image $y = f(x)$, the set $\pi _{n-1}( X_{y}, x)$ consists of a single element.
See Corollary 3.2.6.8 (and Variant 3.2.6.9 for the case $n = 1$).
Corollary 4.8.5.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 1$. Then $F$ is $n$-full if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of morphism spaces is $(n-1)$-full.
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
Remark 4.8.5.23. Stated more informally, Corollary 4.8.5.22 asserts that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $n$-full if it is surjective up to homotopy on $n$-morphisms (having fixed source and target). For an alternative formulation of this heuristic, see Proposition 4.8.5.30 below.
Corollary 4.8.5.24. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n$ be an integer. Then:
Corollary 4.8.5.25. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq 2$ be an integer. Then $F$ is $n$-full if and only if, for every morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ having image $\overline{u}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{D}}$, the set $\pi _{n-2}(\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\overline{u}}, u )$ consists of a single element. Here $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\overline{u}}$ denotes the fiber $\{ \overline{u} \} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} )} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.
Proof. By virtue of Corollary 4.8.5.22, the functor $F$ is $n$-full if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of Kan complexes
is $(n-1)$-full. Since $F$ is an inner fibration, $F_{X,Y}$ is a Kan fibration (Proposition 4.6.1.21). The desired result now follows from Remark 4.8.5.21. $\square$
Variant 4.8.5.26. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. Then $F$ is full if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map is surjective on vertices.
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
Proposition 4.8.5.27. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq 1$ be an integer. The following conditions are equivalent:
The functor $F$ is $n$-full.
For every pullback diagram of $\infty $-categories
the functor $F'$ is $n$-full.
For every pullback diagram of $\infty $-categories
the functor $F'$ is $n$-full.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^1 \ar [r] & \operatorname{\mathcal{D}}, } \]
Proof. For $n \geq 2$, this follows from the criterion of Corollary 4.8.5.25. For $n = 1$, it follows from the criterion of Variant 4.8.5.26. $\square$
Corollary 4.8.5.28. Let $n > 0$ be an integer and suppose we are given a commutative diagram of $\infty $-categories where the vertical maps are inner fibrations. Then $F$ is $n$-full if and only if, for every morphism $u$ of $\operatorname{\mathcal{E}}$, the induced map is $n$-full.
\[ \xymatrix { \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr] & & \operatorname{\mathcal{D}}\ar [dl] \\ & \operatorname{\mathcal{E}}, & } \]
\[ F_{u}: \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}} \]
Proof. Using Proposition 4.1.3.2, we can reduce to the case where $F$ is an inner fibration, in which case it follows from the criterion of Proposition 4.8.5.27. $\square$
Corollary 4.8.5.29. Let $n \geq 0$ be an integer, and suppose we are given a categorical pullback diagram of $\infty $-categories If $F$ is $n$-full, then $F'$ is $n$-full. The converse holds if $G$ is full and essentially surjective.
4.83
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-locally-truncated} \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}}. } \end{gathered} \end{equation}
Proof. The case $n = 0$ follows from Remark 4.6.2.19. We will therefore assume that $n \geq 1$. Using Corollary 4.5.2.23, we can reduce to the case where $F$ and $G$ are isofibrations. In this case, our assumption that (4.83) is a categorical pullback square guarantees that the induced map $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is an equivalence of $\infty $-categories (Proposition 4.5.2.26). Using Remark 4.8.5.18, $\operatorname{\mathcal{C}}'$ by the fiber product $\operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ and thereby reduce to the case where the diagram (4.83) is a pullback square. Note that, if the functor $G$ is full and essentially surjective, then every morphism of $\operatorname{\mathcal{D}}$ can be lifted to a morphism of $\operatorname{\mathcal{D}}'$. The desired result now follows from the criterion of Proposition 4.8.5.27. $\square$
Proposition 4.8.5.30. 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 $n=0$ reduces to the observation that an isofibration is essentially surjective if and only if it is surjective on objects. The case $n = 1$ is a reformulation of Variant 4.8.5.26. We may therefore assume without loss of generality that $n \geq 2$. Using Proposition 4.8.5.27, 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 4.8.5.25). The desired result now follows from Corollary 4.8.3.10. $\square$
For later use, we record a few variants of Remark 4.8.5.15.
Proposition 4.8.5.31. 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 $n \geq 1$. If $G \circ F$ is $n$-full and $G$ is $(n+1)$-full, then $F$ is $n$-full.
Proof. We first treat the case $n = 1$. Fix a pair of objects $X,Y \in \operatorname{\mathcal{C}}$ having images $\overline{X}$ and $\overline{Y}$ in $\operatorname{\mathcal{D}}$. We wish to show that every morphism $\overline{u}: \overline{X} \rightarrow \overline{Y}$ is homotopic to $F(v)$, for some morphism $v: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$. Our assumption that $(G \circ F)$ is $1$-full guarantees that we can choose $v$ such that $(G \circ F)(v)$ is homotopic to $G( \overline{u} )$. Since $G$ is $2$-full, the map $\pi _0( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ) ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{E}}}( G( \overline{X}), G( \overline{Y} ) )$ is injective, so that $F(v)$ is homotopic to $\overline{u}$ as desired.
We now treat the case $n \geq 2$. Without loss of generality, we may assume that $F$ and $G$ are inner fibrations. Using Proposition 4.8.5.27, we can further reduce to the case where $\operatorname{\mathcal{E}}$ is a standard simplex. Fix a morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ having image $\overline{u}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{D}}$. We wish to show that the map
is injective for $m = n-2$ and surjective for $m = n-1$. This is clear: our assumption that $G \circ F$ is $n$-full guarantees that the set $\pi _{n-2}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u )$ consists of a single element (so $\theta _{n-2}$ is automatically injective), and our assumption that $G$ is $(n+1)$-full guarantees that the set $\pi _{n-1}( \operatorname{Hom}_{\operatorname{\mathcal{D}}}(\overline{X}, \overline{Y}), \overline{u} )$ consists of a single element (so that $\theta _{n-1}$ is automatically surjective). $\square$
Exercise 4.8.5.32. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories where $G$ is full and conservative. Show that if $G \circ F$ is essentially surjective, then $F$ is also essentially surjective. Beware that the hypothesis that $G$ is conservative cannot be omitted.
Proposition 4.8.5.33. 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 $n \geq 2$. Assume that $G \circ F$ is $n$-full and that $F$ is essentially surjective, full, and $(n-1)$-full. Then $G$ is $n$-full.
Remark 4.8.5.34. For $n = 0$ and $n=1$, we have stronger versions of Proposition 4.8.5.33:
Proof of Proposition 4.8.5.33. Without loss of generality, we may assume that $F$ and $G$ are inner fibrations. Using Proposition 4.8.5.27, we can reduce to the case where $\operatorname{\mathcal{E}}$ is a standard simplex. In this case, we must show that for every morphism $\overline{u}: \overline{X} \rightarrow \overline{Y}$ of $\operatorname{\mathcal{D}}$, the set $\pi _{n-2}( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ), \overline{u} )$ consists of a single element. Since $F$ is full and essentially surjective, we can assume without loss of generality that $\overline{u} = F(u)$ for some morphism $u: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$. In this case, our assumption that $F$ is $(n-1)$-full guarantees that the map
is surjective. It will therefore suffice to show that the set $\pi _{n-2}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u)$ consists of a single element, which follows from our assumption that $G \circ F$ is $n$-full. $\square$