Definition 4.8.6.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. We say that $F$ is essentially $n$-categorical if it is $m$-full for $m \geq n+2$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
4.8.6 Essentially Categorical Functors
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Combining Corollary 4.8.3.3 with Remark 4.8.5.13, we see that the following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to an $(n,1)$-category.
The projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is $m$-full for $m \geq n+2$.
This motivates the following:
Example 4.8.6.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
The functor $F$ is essentially $0$-categorical if and only if it is faithful.
The functor $F$ is essentially $(-1)$-categorical if and only if it is fully faithful.
The functor $F$ is essentially $(-2)$-categorical if and only if it is an equivalence of $\infty $-categories. In this case, $F$ is also essentially $n$-categorical for any $n \leq -2$.
This is a restatement of Remark 4.8.5.11.
Example 4.8.6.3. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n$ be an integer. Then $f$ is essentially $n$-categorical (in the sense of Definition 4.8.6.1) if and only if it is $n$-truncated (in the sense of Definition 3.5.9.1). See Corollary 4.8.5.24.
Example 4.8.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. The following conditions are equivalent:
The projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is essentially $n$-categorical.
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated. Moreover, if $n \leq -2$, then $\operatorname{\mathcal{C}}$ is nonempty.
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to an $(n,1)$-category.
For $m \geq n+2$, every morphism $\operatorname{\partial \Delta }^{m} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $m$-simplex of $\operatorname{\mathcal{E}}$.
The equivalence $(1) \Leftrightarrow (2)$ follows from Remark 4.8.5.13, the equivalence $(2) \Leftrightarrow (3)$ from Corollary 4.8.3.3, and the equivalence $(2) \Leftrightarrow (4)$ from Corollary 4.8.3.11.
Remark 4.8.6.5 (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 essentially $n$-categorical if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is essentially $n$-categorical. See Remark 4.8.5.14.
Remark 4.8.6.6 (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 essentially $m$-categorical, then it is essentially $n$-categorical.
Remark 4.8.6.7 (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 essentially $n$-categorical, then the composite functor $G \circ F$ is essentially $n$-categorical.
If $G \circ F$ is essentially $n$-categorical and $G$ is essentially $(n+1)$-categorical, then $F$ is essentially $n$-categorical.
If $G \circ F$ is essentially $n$-categorical and $F$ is essentially $(n-1)$-categorical, full, and essentially surjective, then $G$ is essentially $n$-categorical.
Assertion $(a)$ follows from Remark 4.8.5.15, assertion $(b)$ from Proposition 4.8.5.31 (together with Exercise 4.8.5.32 in the case $n \leq -2$), and assertion $(c)$ follows from Proposition 4.8.5.33 (together with Remark 4.5.1.18 in the case $n \leq -2$).
Remark 4.8.6.8. 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 essentially $n$-categorical. Then $F$ is essentially $n$-categorical if and only if $G \circ F$ is essentially $n$-categorical. This follows by combining Remarks 4.8.6.6 and 4.8.6.7.
Example 4.8.6.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq -1$ be an integer. Suppose that $\operatorname{\mathcal{D}}$ is locally $(n-1)$-truncated. Then $F$ is essentially $n$-categorical if and only if $\operatorname{\mathcal{C}}$ is also locally $(n-1)$-truncated. This follows by applying Remark 4.8.6.8 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$ (see Example 4.8.6.4).
Remark 4.8.6.10 (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 essentially $n$-categorical if and only if $G$ is essentially $n$-categorical. If $G$ is an equivalence of $\infty $-categories, then $G \circ F$ is essentially $n$-categorical if and only if $F$ is essentially $n$-categorical. Both assertions are special cases of Remark 4.8.6.7.
Remark 4.8.6.11 (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 essentially $n$-categorical if and only if $F_1$ is essentially $n$-categorical. See Remark 4.8.5.17.
Remark 4.8.6.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq -1$. Then $F$ is essentially $n$-categorical if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of Kan complexes is $(n-1)$-truncated. This follows by combining Example 4.8.6.3 with Corollary 4.8.5.22.
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
Remark 4.8.6.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. For $n \geq 0$, $F$ is essentially $n$-categorical if and only if the diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is essentially $(n-1)$-categorical. This follows by combining Remark 4.8.6.12 with Corollary 3.5.9.17, 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 4.6.1.21).
Warning 4.8.6.14. Remark 4.8.6.13 is generally false in the case $n = -1$, 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 $\operatorname{\mathcal{D}}$). The diagonal $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is an isomorphism of simplicial sets, and therefore an equivalence of $\infty $-categories. However, $F$ need not be fully faithful.
Variant 4.8.6.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. For $n \geq 0$, the functor $F$ is essentially $n$-categorical if and only if the composite map is essentially $(n-1)$-categorical. To prove this, we can use Corollaries 4.5.2.23 and 4.5.2.20 to reduce to the situation where $F$ is an isofibration. In this case, the desired result is a reformulation of Remark 4.8.6.13 (see Corollary 4.5.2.28).
\[ \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}^{\mathrm{h}} \operatorname{\mathcal{C}} \]
Remark 4.8.6.16. Let $n$ be an integer and suppose we are given a categorical pullback diagram of $\infty $-categories If $F$ is essentially $n$-categorical, then $F'$ is essentially $n$-categorical. The converse holds if $G$ is full and essentially surjective. See Corollary 4.8.5.29.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d]^{F'} \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r]^-{G} & \operatorname{\mathcal{D}}. } \]
Proposition 4.8.6.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq 0$ be an integer. The following conditions are equivalent:
The functor $F$ is essentially $n$-categorical.
For every pullback diagram of $\infty $-categories
the functor $F'$ is essentially $n$-categorical.
For every pullback diagram of $\infty $-categories
where $\operatorname{\mathcal{D}}'$ is locally $(n-1)$-truncated, the $\infty $-category $\operatorname{\mathcal{C}}'$ is also locally $(n-1)$-truncated.
For every pullback diagram of $\infty $-categories
the $\infty $-category $\operatorname{\mathcal{C}}'$ is locally $(n-1)$-truncated.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^1 \ar [r] & \operatorname{\mathcal{D}}, } \]
Proof. Combine Proposition 4.8.5.27 with the criterion of Example 4.8.6.9. $\square$
Warning 4.8.6.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq -1$ be an integer. If $F$ is essentially $n$-categorical, then each fiber $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ of $F$ is a locally $(n-1)$-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 4.8.5.28, we immediately obtain the following:
Variant 4.8.6.19. Let $n \geq -1$ be an integer and suppose we are given a commutative diagram of $\infty $-categories where the vertical maps are inner fibrations. Then $F$ is essentially $n$-categorical if and only if, for every morphism $u$ of $\operatorname{\mathcal{E}}$, the induced functor is essentially $n$-categorical. In particular, $F$ is fully faithful if and only if each $F_ u$ is fully faithful.
\[ \xymatrix { \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr] & & \operatorname{\mathcal{D}}\ar [dl] \\ & \operatorname{\mathcal{E}}, & } \]
\[ F_{u}: \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}} \]
Proposition 4.8.6.20. 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 essentially $n$-categorical.
For every integer $m \geq n+2$, every lifting problem
admits a solution.
For every simplicial set $B$ and every simplicial subset $A \subseteq B$ which contains the $(n+1)$-skeleton of $B$, every lifting problem
admits a solution.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{m} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]
\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]
Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 4.8.5.30. The implication $(3) \Rightarrow (2)$ is immediate, and the reverse implication follows from Proposition 1.1.4.12. $\square$
Corollary 4.8.6.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset. If $F$ is essentially $n$-categorical, then the induced functor $F': \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ is also essentially $n$-categorical.
Proof. If $n \leq -2$, then $F$ is an equivalence of $\infty $-categories; it then follows from Corollary 4.5.2.30 that $F'$ is also an equivalence of $\infty $-categories. We may therefore assume that $n \geq -1$. Using Proposition 3.1.7.1, we can reduce to the case where $F$ is an inner fibration, so that $F'$ is also an inner fibration (Proposition 4.1.4.1). By virtue of Proposition 4.8.6.20, it will suffice to show that for every simplicial set $B'$ and every simplicial subset $A' \subseteq B'$ which contains the $(n+1)$-skeleton of $B$, every lifting problem
4.84
\begin{equation} \begin{gathered}\label{equation:locally-truncated-exponentiation} \xymatrix@C =50pt@R=50pt{ A' \ar [r] \ar [d] & \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \ar [d]^{F'} \\ B' \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}}) } \end{gathered} \end{equation}
Unwinding the definitions, we can rewrite (4.84) as a lifting problem
\[ \xymatrix@C =50pt@R=50pt{ (A \times B') \coprod _{ (A \times A') } (B \times A') \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \times B' \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}. } \]
The existence of a solution now follows from Proposition 4.8.6.20, since $F$ is essentially $n$-categorical and $(A \times B') \coprod _{ (A \times A') } (B \times A')$ contains the $(n+1)$-skeleton of $B \times B'$. $\square$
Corollary 4.8.6.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $B$ be a simplicial set, and let $n$ be an integer. If $F$ is essentially $n$-categorical, then the induced functor $\operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{D}})$ is also essentially $n$-categorical.
Proof. Apply Corollary 4.8.6.21 in the special case $A = \emptyset $. $\square$
Corollary 4.8.6.23. Let $n \geq -1$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an essentially $n$-categorical inner fibration of $\infty $-categories. Then, for every diagram $B \rightarrow \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{D}}}(B, \operatorname{\mathcal{C}})$ is locally $(n-1)$-truncated.
Proof. It follows from Corollary 4.1.4.2 that $F$ induces an inner fibration $F': \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})$, and from Corollary 4.8.6.22 that $F'$ is essentially $n$-categorical. In particular, every fiber of $F$ is locally $(n-1)$-truncated. $\square$
We now study a special class of essentially $n$-categorical functors.
Definition 4.8.6.24. Let $n$ be a positive integer. We say that a morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an $n$-categorical inner fibration if it satisfies the following condition:
For every pair of integers $0 < i < m$, every lifting problem
admits a solution. Moreover, if $m > n$, then the solution is unique.
\[ \xymatrix@C =50pt@R=50pt{ \Lambda ^{m}_{i} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]
It will sometimes be useful to extend Definition 4.8.6.24 to allow $n$ to be an arbitrary integer.
Variant 4.8.6.25. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets.
We say that $U$ is a $0$-categorical inner fibration if, for every morphism $\Delta ^{m} \rightarrow \operatorname{\mathcal{D}}$, the fiber product $\Delta ^ m \times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is isomorphic to the nerve of a partially ordered set.
We say that $F$ is a $(-1)$-categorical inner fibration if it induces an isomorphism from $\operatorname{\mathcal{C}}$ to a full simplicial subset of $\operatorname{\mathcal{D}}$ (Definition 4.1.2.15).
For $n \leq -2$, we say that $F$ is an $n$-categorical inner fibration if it is an isomorphsm of simplicial sets.
Example 4.8.6.26. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ be the projection map. Then $F$ is an $n$-categorical inner fibration if and only if $\operatorname{\mathcal{C}}$ is an $(n,1)$-category.
Example 4.8.6.27. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $F$ is a $1$-categorical inner fibration (Definition 4.8.6.24) if and only if it is an inner covering map (Definition 4.1.5.1).
Remark 4.8.6.28. Let $m \leq n$ be integers. If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an $m$-categorical inner fibration, then it is also an $n$-categorical inner fibration (see Remark 4.8.1.12). In particular, $F$ is an inner fibration.
Remark 4.8.6.29. Suppose we are given a pullback diagram of simplicial sets If $F$ is an $n$-categorical inner fibration, then $F'$ is also an $n$-categorical inner fibration.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{ F } \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}. } \]
Remark 4.8.6.30. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. It follows from Example 4.8.6.26 and Remark 4.8.6.29 that if $F$ is an $n$-categorical inner fibration, then the fiber $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is an $(n,1)$-category for each vertex $D \in \operatorname{\mathcal{D}}$. Beware that the converse is generally false.
Remark 4.8.6.31 (Symmetry). Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an $n$-categorical inner fibration of simplicial sets. Then the opposite map $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is also an $n$-categorical inner fibration.
Proposition 4.8.6.32. Let $n$ be an integer, let $\operatorname{\mathcal{D}}$ be an $(n,1)$-category, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. Then $F$ is $n$-categorical if and only if $\operatorname{\mathcal{C}}$ is an $(n,1)$-category.
Proof. For $n \neq 0$, the desired result follows from immediately from the definitions. Let us therefore assume that $n = 0$, so that $\operatorname{\mathcal{D}}$ is isomorphic to the nerve of a partially ordered set. If $\operatorname{\mathcal{C}}$ is also isomorphic to the nerve of a partially ordered set, then any fiber product $\Delta ^ m \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ has the same property (since the formation of nerves commutes with fiber products). Conversely, suppose that $F$ is a $0$-categorical inner fibration. In this case, we claim that $\operatorname{\mathcal{C}}$ satisfies the criteria of *** snip ***
- $(a)$
The simplicial set $\operatorname{\mathcal{C}}$ is a $(1,1)$-category: this follows by applying Proposition 4.8.6.32 in the case $n = 1$.
- $(b)$
Let $u,u': X \rightarrow Y$ be morphisms in $\operatorname{\mathcal{C}}$ having the same source and target; we wish to show that $f = f'$. Our assumption that $\operatorname{\mathcal{D}}$ is a $(0,1)$-category guarantees that $F(u) = F(u')$. The desired result now follows from the observation that the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is a $(0,1)$-category.
- $(c)$
Let $X$ and $Y$ be isomorphic objects of $\operatorname{\mathcal{C}}$; we wish to show that $X = Y$. Fix morphisms $u: X \rightarrow Y$ and $v: Y \rightarrow X$. Since $\operatorname{\mathcal{D}}$ is a $(1,0)$-category, we have $F(u) = \operatorname{id}_{D} = F(v)$ for some object $D \in \operatorname{\mathcal{D}}$. In this case, we can regard $u$ and $v$ as morphisms of the $\infty $-category $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. Our assumption that $F$ is a $0$-categorical inner fibration guarantees that $\operatorname{\mathcal{C}}_{D}$ is a $(0,1)$-category, so that $X = Y$.
Remark 4.8.6.33. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $n$ be an integer. Then $F$ is an $n$-categorical inner fibration if and only if, for every pullback diagram of simplicial sets the projection map $F'$ is an $n$-categorical inner fibration. For $n \geq 0$, this is equivalent to the requirement that $\operatorname{\mathcal{C}}'$ is an $(n,1)$-category (Proposition 4.8.6.32).
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{ F } \\ \Delta ^{m} \ar [r] & \operatorname{\mathcal{D}}, } \]
Corollary 4.8.6.34 (Transitivity). Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be inner fibrations of simplicial sets, where $G$ is $n$-categorical. Then $F$ is $n$-categorical if and only if $G \circ F$ is $n$-categorical.
Proof. For $n < 0$, this follows immediately from the definitions. We may therefore assume that $n \geq 0$. Using Remark 4.8.6.33, we can reduce to the case where $\operatorname{\mathcal{E}}= \Delta ^ m$ is a standard simplex. In this case, our assumption on $G$ guarantees that $\operatorname{\mathcal{D}}$ is an $(n,1)$-category. We wish to show that $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if and only if the inner fibration $F$ is $n$-categorical, which follows from Proposition 4.8.6.32. $\square$
Proposition 4.8.6.35. Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an $n$-categorical inner fibration of $\infty $-categories. Then $F$ is essentially $n$-categorical.
For a partial converse, see Corollary 4.8.8.23.
Proof of Proposition 4.8.6.35. If $n = -2$, then $F$ is an isomorphism of simplicial sets and therefore an equivalence of $\infty $-categories (Example 4.5.1.11). If $n = -1$, then $F$ is an isomorphism from $\operatorname{\mathcal{C}}$ onto a full subcategory of $\operatorname{\mathcal{D}}$, and therefore fully faithful (Example 4.6.2.2). We may therefore assume without loss of generality that $n \geq 0$. By virtue of Proposition 4.8.6.17, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is a standard simplex; in this case, we wish to show that $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated. This follows from Example 4.8.2.2, since $\operatorname{\mathcal{C}}$ is an $(n,1)$-category (Proposition 4.8.6.32). $\square$