Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Recall that the essential image of $F$ is the full subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ spanned by those objects which are isomorphic to $F(X)$, for some object $X \in \operatorname{\mathcal{C}}$. The functor $F$ then factors as a composition
where the functor on the left is essentially surjective and the functor on the right is fully faithful. It is sometimes useful to consider a different factorization.
Proposition 4.8.7.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Then $F$ factors as a composition where $G$ is faithful and $F'$ is both full and essentially surjective.
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}} \]
Proof. We construct the category $\operatorname{\mathcal{D}}'$ as follows:
The objects of $\operatorname{\mathcal{D}}'$ are the objects of $\operatorname{\mathcal{C}}$. To avoid confusion, for each object $X \in \operatorname{\mathcal{C}}$, we write $\overline{X}$ for the corresponding object of $\operatorname{\mathcal{D}}'$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we take $\operatorname{Hom}_{ \operatorname{\mathcal{D}}' }( \overline{X}, \overline{Y} )$ to be image of the map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$. To avoid confusion, if $u: F(X) \rightarrow F(Y)$ is a morphism of $\operatorname{\mathcal{D}}$ which belongs to the image of $F_{X,Y}$, we write $\overline{u}: \overline{X} \rightarrow \overline{Y}$ for the corresponding morphism of $\operatorname{\mathcal{D}}'$.
For every pair of objects $X,Y, Z \in \operatorname{\mathcal{C}}$, the composition law
is the restriction of the composition law $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(Y), F(Z) ) \times \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Z) )$ for the category $\operatorname{\mathcal{D}}$: that is, it satisfies the formula $\overline{v} \circ \overline{u} = \overline{v \circ u}$.
Let $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$ be the functor which carries each object $X \in \operatorname{\mathcal{C}}$ to the object $\overline{X} \in \operatorname{\mathcal{D}}'$, and each morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ to the morphism $\overline{ F(u) }: \overline{X} \rightarrow \overline{Y}$ of $\operatorname{\mathcal{D}}'$. Let $G: \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}$ be the functor which carries each object $\overline{X} \in \operatorname{\mathcal{D}}'$ to the object $F(X) \in \operatorname{\mathcal{D}}$, and each morphism $\overline{u}: \overline{X} \rightarrow \overline{Y}$ of $\operatorname{\mathcal{D}}'$ to the morphism $u: F(X) \rightarrow F(Y)$ of $\operatorname{\mathcal{D}}$. Then the functor $G$ is faithful, the functor $F'$ is full and essentially surjective, and the composition $G \circ F'$ is equal to $F$. $\square$
Exercise 4.8.7.2 (Uniqueness). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. The proof of Proposition 4.8.7.1 constructs a factorization where $G$ is faithful and $F'$ is both full and bijective on objects. Show that these properties characterize the category $\operatorname{\mathcal{D}}'$ up to (unique) isomorphism.
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}} \]
Our goal in this section is to prove the following $\infty $-categorical generalization of Proposition 4.8.7.1:
Theorem 4.8.7.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Then $F$ admits a factorization $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}}$ with the following properties:
The functor $G$ is $n$-faithful: that is, it is $m$-full for $m > n$.
The functor $F'$ is categorically $n$-connective: that is, it is $m$-full for $m \leq n$.
Example 4.8.7.4. For $n < 0$, Theorem 4.8.7.3 asserts that every functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a factorization $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}}$, where the functor $G$ is an equivalence of $\infty $-categories. This is trivial: we can take $\operatorname{\mathcal{D}}' = \operatorname{\mathcal{D}}$ and $G$ to be the identity functor.
Example 4.8.7.5. When $n = 0$, Theorem 4.8.7.3 asserts that every functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a factorization $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}}$, where the functor $G$ is fully faithful and the functor $F'$ is essentially surjective. For example, we can take $\operatorname{\mathcal{D}}' \subseteq \operatorname{\mathcal{D}}$ to be the essential image of the functor $F$ (Definition 4.8.1.11), and $G: \operatorname{\mathcal{D}}' \hookrightarrow \operatorname{\mathcal{D}}$ to be the inclusion map. See Remark 4.8.1.12.
Example 4.8.7.6. When $n = 1$, Theorem 4.8.7.3 asserts that every functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a factorization $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}}$, where the functor $G$ is faithful and the functor $F'$ is both full and essentially surjective. When $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are (nerves of) ordinary categories, this follows from Proposition 4.8.7.1. To handle the general case, we can use (the proof of) Proposition 4.8.7.1 to factor the functor $\operatorname {h}\! \mathit{F}$ as a composition $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} \xrightarrow {F'_0} \operatorname{\mathcal{D}}'_0 \xrightarrow {G_0} \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$ where $G_0$ is a faithful functor and $F'_0$ is a full functor which is essentially surjective (or even bijective on objects). To prove Theorem 4.8.7.3, we can take $\operatorname{\mathcal{D}}'$ to be the fiber product $\operatorname{N}_{\bullet }( \operatorname{\mathcal{D}}'_0 ) \times _{ \operatorname{N}_{\bullet }(\operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}) } \operatorname{\mathcal{D}}$, and $G: \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}$ to be the functor given by projection onto the second factor (which is faithful by virtue of Proposition 4.8.4.11).
Example 4.8.7.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then the projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ factors as a composition where $\operatorname {h}_{\mathit{\leq {}n-1}}{\mathit{(\operatorname{\mathcal{C}})}}$ is the homotopy $(n-1)$-category constructed in ยง4.7.9. This factorization satisfies the requirements of Theorem 4.8.7.3. Assume for simplicity that $n \geq 0$ (for $n < 0$ the result follows from Example 4.8.7.4). Then the functor $G$ is $n$-faithful because $\operatorname {h}_{\mathit{\leq {}n-1}}{\mathit{(\operatorname{\mathcal{C}})}}$ is locally $(n-2)$-truncated (Example 4.8.3.14), and the functor $F'$ is categorically $n$-connective by Example 4.8.5.19.
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname {h}_{\mathit{\leq {}n-1}}{\mathit{(\operatorname{\mathcal{C}})}} \xrightarrow {G} \Delta ^0, \]
Corollary 4.8.7.8. Let $f: X \rightarrow Z$ be a morphism of Kan complexes and let $n$ be an integer. Then $f$ factors as a composition $X \xrightarrow {f'} Y \xrightarrow {f''} Z$, where $f''$ is $(n-1)$-truncated and $f'$ is $n$-connective.
Proof. Using Theorem 4.8.7.3, we can factor $f$ as a composition $f'' \circ f'$, where $f'': \operatorname{\mathcal{C}}\rightarrow Z$ is an $n$-faithful functor of $\infty $-categories and $f': X \rightarrow \operatorname{\mathcal{C}}$ is categorically $n$-connective. If $n \leq 0$, then $f''$ induces an equivalence from $\operatorname{\mathcal{C}}$ to a summand of $Z$, so that $\operatorname{\mathcal{C}}$ is a Kan complex. If $n > 0$, Remark 4.8.5.13 guarantees that $\operatorname{\mathcal{C}}$ is a Kan complex. Setting $Y = \operatorname{\mathcal{C}}$, we observe that $f''$ is $(n-1)$-truncated (Example 4.8.3.14) and $f'$ is $n$-connective (Example 4.8.5.3). $\square$
Proof of Theorem 4.8.7.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. We wish to show that $F$ factors as a composition $G \circ F'$, where $G$ is $n$-faithful and $F'$ is categorically $n$-connective. Using Proposition 4.1.3.2, we can reduce to the case where $F$ is an inner fibration. In this case, the factorization $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname {h}_{\mathit{\leq {}n-1}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \xrightarrow {G} \operatorname{\mathcal{D}}$ of Remark 4.8.6.18 has the desired properties: Proposition 4.8.6.17 guarantees that $G$ is an $(n-1)$-categorical inner fibration (and therefore $n$-faithful by virtue of Proposition 4.8.6.12), and Corollary 4.8.6.19 guarantees that $F'$ is categorically $n$-connective. $\square$
Remark 4.8.7.9 (Uniqueness). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then, for every integer $n$, the factorization of Theorem 4.8.7.3 is well-defined up to equivalence. More precisely, if the functor $F$ admits two factorizations where the functors $F'_0$ and $F'_1$ are categorically $n$-connective and the functors $G_0$ and $G_1$ are $n$-faithful, then we can find a commutative diagram where the vertical maps are equivalences of $\infty $-categories. To prove this, we can use Corollary 4.5.3.24 to reduce to the case where $F'_0$ is a monomorphism of simplicial sets and $G_1$ is an isofibration. In this case, Corollary 4.8.5.23 and Remark 4.8.5.24 guarantee that the functors $F'_0$ and $G_1$ induce a trivial Kan fibration In particular, this map is surjective on vertices, so the lifting problem has a solution. A choice of solution determines a commutative diagram It follows from Proposition 4.8.5.15 that the functor $H$ is categorically $n$-connective, and from Remark 4.8.3.26 that $H$ is $n$-faithful. Applying Theorem 4.8.4.1, we conclude that $H$ is an equivalence of $\infty $-categories.
\[ \operatorname{\mathcal{C}}\xrightarrow {F'_0} \operatorname{\mathcal{D}}'_0 \xrightarrow {G_0} \operatorname{\mathcal{D}}\quad \quad \operatorname{\mathcal{C}}\xrightarrow {F'_1} \operatorname{\mathcal{D}}'_1 \xrightarrow {G_1} \operatorname{\mathcal{D}} \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F'_0} & \operatorname{\mathcal{D}}'_0 \ar [r]^-{ G_0 } & \operatorname{\mathcal{D}}\\ \operatorname{\mathcal{C}}\ar@ {=}[u] \ar@ {=}[d] \ar [r] & \operatorname{\mathcal{D}}'_{01} \ar [u]_{\sim } \ar [d]^{\sim } \ar [r] & \operatorname{\mathcal{D}}\ar@ {=}[u] \ar@ {=}[d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ F'_1 } & \operatorname{\mathcal{D}}'_{1} \ar [r]^-{ G_1 } & \operatorname{\mathcal{D}}} \]
\[ \operatorname{Fun}( \operatorname{\mathcal{D}}'_0, \operatorname{\mathcal{D}}'_1 ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}'_1 ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) } \operatorname{Fun}( \operatorname{\mathcal{D}}'_0, \operatorname{\mathcal{D}}). \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F'_1} \ar [d]^{F'_0} & \operatorname{\mathcal{D}}'_1 \ar [d]^{G_1} \\ \operatorname{\mathcal{D}}'_0 \ar [r]^-{G_0} \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F'_0} \ar@ {=}[d] & \operatorname{\mathcal{D}}'_0 \ar [r]^-{ G_0 } \ar [d]^{H} & \operatorname{\mathcal{D}}\ar@ {=}[d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ F'_1 } & \operatorname{\mathcal{D}}'_{1} \ar [r]^-{ G_1 } & \operatorname{\mathcal{D}}. } \]
Warning 4.8.7.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. In the case $n = 1$, our proof of Theorem 4.8.7.3 shows that $F$ factors as a composition where $F'$ is full and essentially surjective, and $G$ is a $0$-categorical inner fibration (in particular, $G$ is faithful). Beware that this generally does not coincide with the factorization constructed in Example 4.8.7.6. If $u: X \rightarrow Y$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$ having the property that $F(u)$ is an identity morphism in $\operatorname{\mathcal{D}}$, then the functor $F'$ carries $X$ and $Y$ to the same object of $\operatorname {h}_{\mathit{\leq {}0}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$. Consequently, the functor $F'$ is generally not bijective on objects. A related phenomenon occurs in the case $n = 0$. By construction, $\operatorname {h}_{\mathit{\leq {}-1}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ is the full subcategory of $\operatorname{\mathcal{D}}$ spanned by objects of the form $F(X)$, where $X$ is an object of $\operatorname{\mathcal{C}}$. If the inner fibration $F$ is not an isofibration, this subcategory might be smaller than the essential image of $F$.
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname {h}_{\mathit{\leq {}0}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \xrightarrow {G} \operatorname{\mathcal{D}}, \]