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 $\operatorname{\mathcal{D}}' \subseteq \operatorname{\mathcal{D}}$ 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 4.8.8.1. 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 4.8.8.2 (Uniqueness). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. The proof of Proposition 4.8.8.1 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 4.8.8.1:
Theorem 4.8.8.3. 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 essentially $n$-categorical: that is, it is $m$-full for $m \geq n+2$.
The functor $F'$ is categorically $(n+1)$-connective: that is, it is $m$-full for $m \leq n+1$.
Example 4.8.8.4. For $n \leq -2$, Theorem 4.8.8.3 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 4.8.8.5. When $n = -1$, Theorem 4.8.8.3 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 $F$, and $G: \operatorname{\mathcal{D}}' \hookrightarrow \operatorname{\mathcal{D}}$ to be the inclusion map. See Remark 4.6.2.13.
Example 4.8.8.6. When $n = 0$, Theorem 4.8.8.3 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 4.8.8.1. To handle the general case, we can use (the proof of) Proposition 4.8.8.1 to factor the functor $\mathrm{h} \mathit{F}$ as a composition $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \xrightarrow {F'_0} \operatorname{\mathcal{D}}'_0 \xrightarrow {G_0} \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ 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 4.8.8.3, we can take $\operatorname{\mathcal{D}}'$ to be the fiber product $\operatorname{N}_{\bullet }( \operatorname{\mathcal{D}}'_0 ) \times _{ \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{D}}}) } \operatorname{\mathcal{D}}$, 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 4.8.5.8).
Example 4.8.8.7. 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 $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is the homotopy $n$-category constructed in §4.8.4. This factorization satisfies the requirements of Theorem 4.8.8.3: the functor $G$ is essentially $n$-categorical because $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is an $(n,1)$-category (Example 4.8.6.4), and the functor $F'$ is categorically $(n+1)$-connective by Example 4.8.5.12.
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} \xrightarrow {G} \Delta ^0, \]
Remark 4.8.8.8 (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 4.8.8.3 is well-defined up to equivalence. More precisely, if the functor $F$ admits two factorizations where the functors $F'_0$ and $F'_1$ are essentially $n$-categorical, and the functors $G_0$ are $G_1$ are categorically $(n+1)$-connective, then we can find a commutative diagram where the vertical maps are equivalences of $\infty $-categories. To prove this, we can use Corollary 4.5.2.23 to reduce to the case where $F'_0$ is a monomorphism of simplicial sets and $G_1$ is an isofibration. In this case, Corollary 4.8.7.18 (and Remark 4.8.7.19) 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 4.8.7.12 that the functor $H$ is categorically $(n+1)$-connective, and from Remark 4.8.6.7 that $H$ is essentially $n$-categorical. Applying Remark 4.8.5.11, we conclude that $H$ is an equivalence of $\infty $-categories.
\[ \operatorname{\mathcal{C}}\xrightarrow {F'_0} \operatorname{\mathcal{D}}'_0 \xrightarrow {G_0} \operatorname{\mathcal{D}}\quad \quad \operatorname{\mathcal{C}}\xrightarrow {F'_1} \operatorname{\mathcal{D}}'_1 \xrightarrow {G_1} \operatorname{\mathcal{D}} \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F'_0} & \operatorname{\mathcal{D}}'_0 \ar [r]^-{ G_0 } & \operatorname{\mathcal{D}}\\ \operatorname{\mathcal{C}}\ar@ {=}[u] \ar@ {=}[d] \ar [r] & \operatorname{\mathcal{D}}'_{01} \ar [u]_{\sim } \ar [d]^{\sim } \ar [r] & \operatorname{\mathcal{D}}\ar@ {=}[u] \ar@ {=}[d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ F'_1 } & \operatorname{\mathcal{D}}'_{1} \ar [r]^-{ G_1 } & \operatorname{\mathcal{D}}} \]
\[ \operatorname{Fun}( \operatorname{\mathcal{D}}'_0, \operatorname{\mathcal{D}}'_1 ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}'_1 ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) } \operatorname{Fun}( \operatorname{\mathcal{D}}'_0, \operatorname{\mathcal{D}}). \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F'_1} \ar [d]^{F'_0} & \operatorname{\mathcal{D}}'_1 \ar [d]^{G_1} \\ \operatorname{\mathcal{D}}'_0 \ar [r]^-{G_0} \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F'_0} \ar@ {=}[d] & \operatorname{\mathcal{D}}'_0 \ar [r]^-{ G_0 } \ar [d]^{H} & \operatorname{\mathcal{D}}\ar@ {=}[d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ F'_1 } & \operatorname{\mathcal{D}}'_{1} \ar [r]^-{ G_1 } & \operatorname{\mathcal{D}}. } \]
Corollary 4.8.8.9. 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 $n$-truncated and $f'$ is $(n+1)$-connective.
Proof. Using Theorem 4.8.8.3, we can factor $f$ as a composition $f'' \circ f'$, where $f'': \operatorname{\mathcal{C}}\rightarrow Z$ is an essentially $n$-categorical functor of $\infty $-categories and $f': X \rightarrow \operatorname{\mathcal{C}}$ is categorically $(n+1)$-connective. If $n \leq -1$, then $f''$ induces an equivalence from $\operatorname{\mathcal{C}}$ to a summand of $Z$, so that $\operatorname{\mathcal{C}}$ is a Kan complex. If $n \geq 0$, Remark 4.8.7.10 guarantees that $\operatorname{\mathcal{C}}$ is a Kan complex. Setting $Y = \operatorname{\mathcal{C}}$, we observe that $f''$ is $n$-truncated (Example 4.8.6.3) and $f'$ is $(n+1)$-connective (Example 4.8.7.3). $\square$
We will prove Theorem 4.8.8.3 in general by reducing to the special case studied in Example 4.8.8.7. For this, we will need a relative version of the construction $\operatorname{\mathcal{C}}\mapsto \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ introduced in §4.8.4.
Construction 4.8.8.10 (Relative Homotopy $n$-Categories). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $n \geq 0$ be an integer. For every $m$-simplex $\sigma $ of $\operatorname{\mathcal{D}}$, let $\operatorname{\mathcal{C}}_{\sigma }$ denote the fiber product $\Delta ^ m \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. We let $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}_{m}$ denote the collection of pairs $(\sigma , \tau )$, where $\sigma $ is an $m$-simplex of $\operatorname{\mathcal{D}}$ and $\tau $ is a section of the projection map If $f: [m'] \rightarrow [m]$ is a nondecreasing function, we let $f^{\ast }: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}_{m} \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}_{m'}$ denote the map given by $f^{\ast }( \sigma , \tau ) = (\sigma ', \tau ' )$, where $\sigma '$ is the composite map $\Delta ^{m'} \xrightarrow {f} \Delta ^{m} \xrightarrow {\sigma } \operatorname{\mathcal{D}}$ and $\tau '$ is given by the composition where the second isomorphism is provided by Proposition 4.8.4.20. By means of this construction, we can view the assignment $[m] \mapsto \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}_{m}$ as a simplicial set, which we will denote by $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$. Note that the construction $(\sigma , \tau ) \mapsto \sigma $ determines a comparison map of simplicial sets $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$. It will be useful to extend this construction to the case where $n < 0$. If $n = -1$, we define $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ to be the full simplicial subset of $\operatorname{\mathcal{D}}$ whose vertices belong to the image of $F$, and we take $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \hookrightarrow \operatorname{\mathcal{D}}$ to be the inclusion map. If $n \leq -2$, we define $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ to be the simplicial set $\operatorname{\mathcal{D}}$, and $G$ to be the identity morphism $\operatorname{id}_{\operatorname{\mathcal{D}}}$.
\[ \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_{\sigma })} \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\Delta ^ m)} \simeq \Delta ^{m}. \]
\begin{eqnarray*} \Delta ^{m'} & \simeq & \Delta ^{m'} \times _{ \Delta ^{m} } \Delta ^{m} \\ & \xrightarrow {(\operatorname{id}, \tau )} & \Delta ^{m'} \times _{ \Delta ^ m } \mathrm{h}_{\mathit{\leq n}}\mathit{( \operatorname{\mathcal{C}}_{\sigma } )} \\ & \simeq & \mathrm{h}_{\mathit{\leq n}}\mathit{( \Delta ^{m'} \times _{\Delta ^{m}} \operatorname{\mathcal{C}}_{\sigma } )} \\ & \simeq & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_{\sigma '} )}, \end{eqnarray*}
Example 4.8.8.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n$ be an integer. Then there is a comparison map from the simplicial set $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ to the homotopy $n$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. For $n \geq 0$, this map carries an $m$-simplex $(\sigma ,\tau )$ of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ to the $m$-simplex of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ given by the composite map If $\operatorname{\mathcal{D}}$ is an $(n,1)$-category, then this comparison map is an isomorphism (Proposition 4.8.4.20).
\[ \Delta ^ m \xrightarrow {\tau } \mathrm{h}_{\mathit{\leq n}}\mathit{( \operatorname{\mathcal{C}}_{\sigma } )} \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}. \]
Example 4.8.8.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, so that the projection map $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is an inner fibration. Since $\Delta ^0$ is an $(n,1)$-category, Example 4.8.8.11 supplies an isomorphism of simplicial sets $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\Delta ^0)} \simeq \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$.
Remark 4.8.8.13 (Base Change). Suppose we are given a pullback diagram of simplicial sets where the vertical maps are inner fibrations. Then, for every integer $n$, the simplicial set $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}'/\operatorname{\mathcal{D}}')}$ can be identified with the fiber product $\operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} {\, }\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$. In particular, for every vertex $D \in \operatorname{\mathcal{D}}$, we have a canonical isomorphism
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}' \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d] \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]
\[ \{ D\} \times _{\operatorname{\mathcal{D}}} \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \simeq \mathrm{h}_{\mathit{\leq n}}\mathit{( \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}})}. \]
Proposition 4.8.8.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $n$ be an integer. Then the comparison map $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ of Construction 4.8.8.10 is an $n$-categorical inner fibration (see Definition 4.8.6.24).
Proof. For $n < 0$, this is immediate from the construction. We may therefore assume without loss of generality that $n \geq 0$. Using Remarks 4.8.6.33 and 4.8.8.13, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In particular, $\operatorname{\mathcal{D}}$ is an $(n,1)$-category. In this case, Example 4.8.8.11 guarantees that the simplicial set $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \simeq \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{D}})}$ is an $(n,1)$-category. The desired result now follows from Proposition 4.8.6.32. $\square$
Remark 4.8.8.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $n$ be an integer. Then the comparison map $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ of Construction 4.8.8.10 fits into a commutative diagram For $n \geq 0$, the morphism $F'$ carries each $m$-simplex of $\operatorname{\mathcal{C}}$ to the $m$-simplex $( F(\sigma ), \tau )$ of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$, where $\tau $ is the composite map
\[ \xymatrix@C =50pt@R=50pt{ & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \ar [dr]^{G} & \\ \operatorname{\mathcal{C}}\ar [ur]^{F'} \ar [rr]^{F} & & \operatorname{\mathcal{D}}. } \]
\[ \Delta ^{m} \xrightarrow {(\operatorname{id},\sigma )} \Delta ^{m} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_{\sigma } \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{( \operatorname{\mathcal{C}}_{\sigma } )}. \]
The simplicial set $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ of Construction 4.8.8.10 can be characterized by a universal mapping property:
Proposition 4.8.8.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $n$ be an integer. Then, for every $n$-categorical inner fibration $\operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}$, the comparison map of Remark 4.8.8.15 induces an isomorphism of simplicial sets
\[ \theta : \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}, \operatorname{\mathcal{D}}' ) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}' ). \]
Proof. We may assume without loss of generality that $n \geq 0$ (otherwise, the result follows immediately from the construction). For every morphism of simplicial sets $K \rightarrow \operatorname{\mathcal{D}}$, Remark 4.8.8.15 determines a comparison map
We will prove that each $\theta _{K}$ is an isomorphism of simplicial sets; Proposition 4.8.8.16 then follows by taking $K = \operatorname{\mathcal{D}}$. Note that the construction $K \mapsto \theta _{K}$ carries colimits (in the category of simplicial sets with a morphism to $\operatorname{\mathcal{D}}$) to limits (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$). By virtue of Remark 1.1.3.13, we can assume without loss of generality that $K$ is a standard simplex. Replacing $F$ by the projection map $K \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow K$ and $\operatorname{\mathcal{D}}'$ by the fiber product $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}'$, we are reduced to proving Proposition 4.8.8.16 in the special case where $\operatorname{\mathcal{D}}$ is a standard simplex: in particular, it is an $(n,1)$-category. In this case, $\operatorname{\mathcal{D}}'$ is also an $(n,1)$-category (Proposition 4.8.6.32), and we can identify $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ with the homotopy $n$-category of $\operatorname{\mathcal{C}}$ (Example 4.8.8.11). Applying Proposition 4.8.4.7, we see that the horizontal maps in the commutative diagram
are isomorphisms. The desired result now follows by passing to fibers of the vertical maps. $\square$
Remark 4.8.8.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets, let $n$ be an integer, and let $A \subseteq B$ be simplicial sets. If $B$ has dimension $\leq n+1$, then every lifting problem has a solution. Moreover, if $B$ has dimension $\leq n-1$, then the solution is unique. To prove this, we can assume without loss of generality that $B = \Delta ^ m$ is a standard simplex for some $m \leq n+1$, and that $A = \operatorname{\partial \Delta }^ m$ is its boundary (see Proposition 1.1.4.12). The case $n \leq -2$ is vacuous, and the case $n = -1$ is immediate from the definition. We may therefore assume that $n \geq 0$. Replacing $F$ by the projection map $\Delta ^ m \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^ m$, we can reduce to the case where $\operatorname{\mathcal{D}}$ is a standard simplex, so that $U'$ identifies $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ with the homotopy $n$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ (Example 4.8.8.11). In this case, the desired result follows from Corollary 4.8.4.17.
\[ \xymatrix@C =50pt@R=50pt{ A \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F'} \\ B \ar [r] \ar@ {-->}[ur] & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} } \]
Proposition 4.8.8.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $n$ be an integer. Then the comparison map $F': \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ of Remark 4.8.8.15 is an inner fibration.
Proof. Without loss of generality, we may assume that $n \geq 0$. Using Remarks 4.1.1.13 and 4.8.8.13, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In this case, $F'$ identifies with the tautological map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ (Example 4.8.8.11), so the desired result follows from Corollary 4.8.4.16. $\square$
Corollary 4.8.8.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n$ be an integer. Then the simplicial set $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ is an $\infty $-category. Moreover, the functor $F': \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ of Remark 4.8.8.15 is categorically $(n+1)$-connective.
Proof. Since $\operatorname{\mathcal{D}}$ is an $\infty $-category and the comparison map $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ is an inner fibration (Proposition 4.8.8.14), the simplicial set $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ is also an $\infty $-category (Remark 4.1.1.9). Fix an integer $m \leq n+1$; we wish to show that the functor $F'$ is $m$-full. For $n = -2$, there is nothing to prove. If $n = -1$, then $U'$ is surjective on objects (by construction) and therefore essentially surjective. We may therefore assume without loss of generality that $n \geq 0$. Since $U'$ is an inner fibration (Proposition 4.8.8.18), it will suffice to show that for every morphism $\Delta ^1 \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})}$, the projection map $\Delta ^{1} \times _{ \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})} } \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is $m$-full (Proposition 4.8.5.27. Using Remark 4.8.8.13, we can replace $F$ by the projection map $\Delta ^1 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1$, and thereby reduce to the situation where $\operatorname{\mathcal{D}}= \Delta ^1$ is an $(n,1)$-category. In this case, the functor $F'$ exhibits $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$ (Example 4.8.8.11), so the desired result follows from Example 4.8.5.12. $\square$
Proof of Theorem 4.8.8.3. 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 essentially $n$-categorical and $F'$ is categorically $(n+1)$-connective. Using Proposition 4.1.3.2, we can reduce to the case where $F$ is an inner fibration. In this case, the factorization
of Remark 4.8.8.15 has the desired properties: Proposition 4.8.8.14 guarantees that $G$ is an $n$-categorical inner fibration (and is therefore essentially $n$-categorical, by virtue of Proposition 4.8.6.35), and Corollary 4.8.8.19 guarantees that $F'$ is categorically $(n+1)$-connective. $\square$
Warning 4.8.8.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. In the case $n = 0$, our proof of Theorem 4.8.8.3 shows that $F$ factors as a composition where $F'$ is fully faithful and essentially surjective, and $G$ is a $0$-categorical inner fibration (in particular, $G$ is faithful). Beware that generally does not coincide with the factorization constructed in Example 4.8.8.6. 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 $\mathrm{h}_{\mathit{\leq 0}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$. Consequently, the functor $F'$ is generally not bijective on objects. A related phenomenon occurs in the case $n = -1$. By construction, $\mathrm{h}_{\mathit{\leq -1}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ 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 $U$ is not an isofibration, this subcategory might be smaller than the essential image of $F$.
\[ \operatorname{\mathcal{E}}\xrightarrow {F'} \mathrm{h}_{\mathit{\leq 0}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \xrightarrow {G} \operatorname{\mathcal{D}}, \]
We close this section with a few additional observations about Construction 4.8.8.10.
Proposition 4.8.8.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets, let $n$ be an integer, and let $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ be the comparison map of Construction 4.8.8.10. Then:
If $F$ is a left fibration, then $G$ is a left fibration.
If $F$ is a right fibration, then $G$ is a right fibration.
If $F$ is a Kan fibration, then $G$ is a Kan fibration.
If $F$ is an isofibration of $\infty $-categories, then $G$ is an isofibration of $\infty $-categories.
Proof. We first prove $(1)$. Assume that $F$ is a left fibration, and suppose we are given integers $0 \leq i < n$; we wish to show that every lifting problem
admits a solution. If $m \leq n+2$, then $\sigma _0$ can be lifted to a morphism $\Lambda ^{m}_{i} \rightarrow \operatorname{\mathcal{C}}$ (Remark 4.8.8.17), so the desired result follows from our assumption that $F$ is a left fibration. We may therefore assume that $m \geq n+3$. If $n = -2$, then $G$ is an isomorphism and there is nothing to prove. If $n = -1$, then $G$ identifies $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ with a full simplicial subset of $\operatorname{\mathcal{D}}$, and the desired result follows from the observation that $\Lambda ^{m}_{i}$ contains every vertex of $\Delta ^ m$. We may therefore assume that $n \geq 0$. Replacing $F$ by the projection map $\Delta ^ m \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^ m$, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In this case, $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ is an $(n,1)$-category (Example 4.8.8.11). In particular, it is an $(n+1)$-coskeletal simplicial set, so the lifting problem (4.88) has a unique solution (since $\Lambda ^{m}_{i}$ contains the $(n+1)$-skeleton of $\Delta ^ m$).
Assertion $(2)$ follows by applying $(1)$ to the opposite inner fibration $U^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$. Assertion $(3)$ follows by combining $(1)$ and $(2)$ with Example 4.2.1.5. It remains to prove $(4)$. Fix an object $Y \in \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ and an isomorphism $\overline{e}: \overline{X} \rightarrow V(Y)$ in the $\infty $-category $\operatorname{\mathcal{D}}$; we wish to show that $\overline{e}$ can be lifted to an isomorphism $e: X \rightarrow Y$ of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$. If $n \leq -2$, then $G$ is an isomorphism and the result is obvious. Otherwise, the comparison map $F': \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ is surjective on vertices, so we can choose an object $\widetilde{Y} \in \operatorname{\mathcal{C}}$ satisfying $F'( \widetilde{Y} ) = Y$. If $F$ is an isofibration, then there exists an isomorphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ of $\operatorname{\mathcal{C}}$ satisfying $F( \widetilde{e} ) = \overline{e}$. It follows that $e = F'( \widetilde{e} )$ is an isomorphism in $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ satisfying $G(e) = \overline{e}$. $\square$
Proposition 4.8.8.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:
The functor $F$ is essentially $n$-categorical.
The comparison map $F': \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ of Remark 4.8.8.15 is an equivalence of $\infty $-categories.
Proof. It follows from Proposition 4.8.8.14 (and Proposition 4.8.6.35) that the comparison map $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ is essentially $n$-categorical. By virtue of Remark 4.8.6.8, we can replace $(1)$ by the following condition:
The functor $F'$ is essentially $n$-categorical: that is, it is $m$-full for $m \geq n+2$.
Since $F'$ is also $m$-full for $m \leq n+1$ (Corollary 4.8.8.19), the equivalence $(1') \Leftrightarrow (2)$ follows from Remark 4.8.5.11. $\square$
Corollary 4.8.8.23. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. For every integer $n$, the following conditions are equivalent:
The functor $F$ is essentially $n$-categorical.
The functor $F$ factors as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}}$, where $F'$ is an equivalence of $\infty $-categories and $G$ is an $n$-categorical isofibration.
Proof. The implication $(2) \Rightarrow (1)$ follows from Proposition 4.8.6.35 (together with Remark 4.8.5.18). To prove the converse, we may assume without loss of generality that $F$ is an isofibration (Corollary 4.5.2.23). In this case, the factorization $\operatorname{\mathcal{C}}\xrightarrow {F'} \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \xrightarrow {G} \operatorname{\mathcal{D}}$ of Remark 4.8.8.15 has the desired properties: Proposition 4.8.8.22 guarantees that $F'$ is an equivalence of $\infty $-categories, Proposition 4.8.8.21 guarantees that $G$ is an isofibration, and Proposition 4.8.8.14 guarantees that $G$ is $n$-categorical. $\square$
Proposition 4.8.8.24. Let $n$ be an integer, and suppose we are given a commutative diagram of $\infty $-categories where the vertical maps are inner fibrations. If $F$ is categorically $(n+1)$-connective, then it induces an equivalence of $\infty $-categories $F': \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{E}})} \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{D}}/\operatorname{\mathcal{E}})}$.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr] & & \operatorname{\mathcal{D}}\ar [dl] \\ & \operatorname{\mathcal{E}}& } \]
Proof. We have a commutative diagram of $\infty $-categories
Here $F$ is categorically $(n+1)$-connective by assumption, and the vertical maps are categorically $(n+1)$-connective by virtue of Corollary 4.8.8.19. Applying Proposition 4.8.7.12, we see that the functor $F'$ is also categorically $(n+1)$-connective. We also have a commutative diagram
where the vertical maps are essentially $n$-categorical (Proposition 4.8.8.14). Using Remark 4.8.6.8, we see that $F'$ is also essentially $n$-categorical. Using Remark 4.8.5.11, we see that $F'$ is an equivalence of $\infty $-categories. $\square$
Corollary 4.8.8.25. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:
The comparison map $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories.
The functor $F$ is categorically $(n+1)$-connective.
Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 4.8.8.14 and Remark 4.8.5.16. The reverse implication follows by applying Proposition 4.8.8.24 in the special case $\operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}$. $\square$