Definition 4.8.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. We say that $\operatorname{\mathcal{C}}$ is locally $n$-truncated if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $n$-truncated (see Definition 3.5.7.1).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
4.8.2 Locally Truncated $\infty $-Categories
We now formulate a homotopy-invariant counterpart of Definition 4.8.1.8.
Example 4.8.2.2. Let $n$ be an integer. Then every $(n,1)$-category is locally $(n-1)$-truncated (Corollary 4.8.1.20). In particular:
If $Q$ is a partially ordered set, then the nerve $\operatorname{N}_{\bullet }(Q)$ is locally $(-1)$-truncated $\infty $-category (Proposition 4.8.1.15).
If $\operatorname{\mathcal{C}}$ is an ordinary category, then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a locally $0$-truncated $\infty $-category (Example 4.8.1.3).
If $\operatorname{\mathcal{C}}$ is a $2$-category in which every $2$-morphism is an isomorphism, then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is a locally $1$-truncated $\infty $-category (Example 4.8.1.4).
Remark 4.8.2.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories. If $\operatorname{\mathcal{D}}$ is locally $n$-truncated, then $\operatorname{\mathcal{C}}$ is locally $n$-truncated. The converse holds if $F$ is an equivalence of $\infty $-categories. In particular, if $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are equivalent, then $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if $\operatorname{\mathcal{D}}$ is locally $n$-truncated.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Combining Example 4.8.2.2 with Remark 4.8.2.3, we see for $\operatorname{\mathcal{C}}$ to be equivalent to an $(n,1)$-category, it is necessary for $\operatorname{\mathcal{C}}$ to be locally $(n-1)$-truncated. In ยง4.8.3, we will prove that this condition is also sufficient, provided that $n \geq -1$ (Corollary 4.8.3.3).
Example 4.8.2.4. Let $n \geq -1$ be an integer and let $X$ be a Kan complex. Then $X$ is $n$-truncated (in the sense of Definition 3.5.7.1) if and only if it is locally $(n-1)$-truncated when regarded as an $\infty $-category (in the sense of Definition 4.8.2.1). This is reformulation of Example 3.5.9.18. See Corollary 4.8.3.11 for a more general statement.
Remark 4.8.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. For every integer $n$, the following conditions are equivalent:
The morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $n$-truncated.
The left-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ is $n$-truncated.
The right-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ is $n$-truncated.
This follows from Corollary 3.5.7.12, since the pinch inclusion maps
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \hookleftarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \]
are homotopy equivalences (Proposition 4.6.5.10).
Proposition 4.8.2.6. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. For every integer $n$, the following conditions are equivalent:
The homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is locally $n$-truncated.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is $n$-truncated.
Proof. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Theorem 4.6.8.5 supplies a homotopy equivalence from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ to the pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}( X, Y)$. The desired result now follows from Remark 4.8.2.5. $\square$
Variant 4.8.2.7. Let $\operatorname{\mathcal{C}}$ be a differential graded category. For every integer $n \geq -1$, the following conditions are equivalent:
The differential graded nerve $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is locally $n$-truncated.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ is homologically $n$-truncated: that is, the homology groups $\mathrm{H}_{m}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ vanish for $m > n$.
Proof. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Example 4.6.5.15 supplies an isomorphism from of the Eilenberg-MacLane space $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ with the pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{\mathcal{C}})}( X, Y)$. The result now follows by combining Remark 4.8.2.5 with the criterion of Example 3.5.7.10. $\square$
Example 4.8.2.2 admits a slight generalization:
Proposition 4.8.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. If $\operatorname{\mathcal{C}}$ is an $n$-coskeletal simplicial set (Definition 3.5.3.1), then it is locally $(n-2)$-truncated.
Proof. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, our assumption that $\operatorname{\mathcal{C}}$ is $n$-coskeletal guarantees that the pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-coskeletal (Remark 4.6.5.4). In particular, it is $(n-2)$-truncated (Example 3.5.7.2). The desired result now follows from Remark 4.8.2.5. $\square$
Our next goal is to show that every $\infty $-category $\operatorname{\mathcal{C}}$ admits an optimal approximation by a locally $n$-truncated $\infty $-category.
Definition 4.8.2.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. We say that $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
The functor $F$ is essentially surjective (Definition 4.6.2.11).
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of Kan complexes
exhibits $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ as an $n$-truncation of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, in the sense of Definition 3.5.7.19.
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
Remark 4.8.2.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Suppose that $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{D}}$ is locally $n$-truncated: that is, for every pair of objects $\overline{X}, \overline{Y} \in \operatorname{\mathcal{D}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(\overline{X}, \overline{Y})$ is $n$-truncated. To prove this, we can use the essential surjectivity of $F$ to reduce to the case where $\overline{X} =F(X)$ and $\overline{Y} = F(Y)$ for some objects $X,Y \in \operatorname{\mathcal{C}}$. In this case, the assertion follows from the observation that $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(\overline{X}, \overline{Y})$ is an $n$-truncation of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. Conversely, if $\operatorname{\mathcal{D}}$ is locally $n$-truncated, then $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$ if and only if it is essentially surjective and satisfies the following weaker version of condition $(2)$:
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of Kan complexes
is $(n+1)$-connective.
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
Example 4.8.2.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its homotopy category. Then the tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ exhibits $\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ as a local $0$-truncation of $\operatorname{\mathcal{C}}$.
Remark 4.8.2.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, where $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$. Then $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ exhibits $\operatorname{\mathcal{E}}$ as an $n$-truncation of $\operatorname{\mathcal{C}}$ if and only if $G$ is an equivalence of $\infty $-categories.
Proposition 4.8.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n \geq 0$ be an integer, and let $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ denote the $n$-coskeleton of $\operatorname{\mathcal{C}}$ (Notation 3.5.3.18). Then:
The simplicial set $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ is an $\infty $-category.
The tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ exhibits $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ as a local $(n-2)$-truncation of $\operatorname{\mathcal{C}}$.
Proof. We first prove $(1)$. We proceed as in the proof of Proposition 3.5.3.23. Fix integers $0 < i < m$ and a morphism of simplicial sets $\sigma _0: \Lambda ^{m}_{i} \rightarrow \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$; we wish to show that $\sigma _0$ can be extended to an $m$-simplex of $\operatorname{cosk}_ n(\operatorname{\mathcal{C}})$. Using Remark 3.5.3.21, we can identify $\sigma _0$ with a morphism of simplicial sets $f_0: \operatorname{sk}_{n}( \Lambda ^{m}_{i} ) \rightarrow \operatorname{\mathcal{C}}$; we wish to show that $f_0$ can be extended to the $n$-skeleton of $\Delta ^{m}$. If $n < m-1$, then $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \operatorname{sk}_{n}( \Delta ^{m} )$ and there is nothing to prove. We may therefore assume that $n \geq m-1$, so that $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \Lambda ^{m}_{i}$. In this case, our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category guarantees that $f_0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$.
We now prove $(2)$. By construction, the tautological map $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ is bijective on objects, and therefore essentially surjective. By virtue of Proposition 4.6.5.10, it will suffice to show that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of pinched morphism spaces
\[ \theta : \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( X, Y ) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{cosk}_{n}(\operatorname{\mathcal{C}}) }(X,Y) \]
exhibits $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{cosk}_{n}(\operatorname{\mathcal{C}}) }(X,Y)$ as an $(n-2)$-truncation of $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( X, Y )$. This is a special case of Example 3.5.7.23, since $\theta $ exhibits $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{cosk}_{n}(\operatorname{\mathcal{C}}) }(X,Y)$ as an $(n-1)$-coskeleton of $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( X, Y )$ (Remark 4.6.5.4). $\square$
Proposition 4.8.2.14. Let $n$ 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 $\operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if $F$ is an equivalence of $\infty $-categories.
Proof. By assumption, $F$ is essentially surjective. It follows from Theorem 4.6.2.20 that $F$ is an equivalence of $\infty $-categories if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of Kan complexes
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
is a homotopy equivalence. Since $F_{X,Y}$ exhibits $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ as an $n$-truncation of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, this is equivalent to the requirement that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $n$-truncated. $\square$
Corollary 4.8.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $0$-truncated.
The comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}})$ is an equivalence of $\infty $-categories.
The comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is a trivial Kan fibration.
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to (the nerve of) an ordinary category.
Proof. The implication $(1) \Rightarrow (2)$ follows from Example 4.8.2.11 and Proposition 4.8.2.14. Since the comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an isofibration (Corollary 4.4.1.9), the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 4.5.5.20. The implication $(2) \Rightarrow (4)$ is clear, and the implication $(4) \Rightarrow (1)$ follows from Example 4.8.2.2. $\square$
Exercise 4.8.2.16. Show that an $\infty $-category $\operatorname{\mathcal{C}}$ is locally $(-1)$-truncated if and only if there is an equivalence of $\infty $-categories $u: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(Q)$, for some partially ordered set $Q$. In this case, the morphism $u$ is automatically a trivial Kan fibration (see Example 4.4.1.6 and Proposition 4.5.5.20).
Corollary 4.8.2.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$ be an integer. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $(n-2)$-truncated.
The tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories.
There exists an $n$-coskeletal $\infty $-category $\operatorname{\mathcal{D}}$ which is equivalent to $\operatorname{\mathcal{C}}$.
Proof. The implication $(1) \Rightarrow (2)$ follows from Propositions 4.8.2.13 and 4.8.2.14. The implication $(2) \Rightarrow (3)$ is clear (since $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ is an $\infty $-category), and the implication $(3) \Rightarrow (1)$ follows from Proposition 4.8.2.8 (together with Remark 4.8.2.3). $\square$
Local $n$-truncations can be characterized by a universal mapping property:
Proposition 4.8.2.18. Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{D}}$ is locally $n$-truncated. The following conditions are equivalent:
The functor $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$, in the sense of Definition 4.8.2.9.
For every locally $n$-truncated $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
For every locally $n$-truncated $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection
\[ \pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ). \]
Proof. Without loss of generality we may assume that $n \geq -2$. We first show that $(1)$ implies $(2)$. By virtue of Corollary 4.8.2.17, we can assume without loss of generality that $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are $(n+2)$-coskeletal. In this case, $F$ factors (uniquely) as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}) \xrightarrow {F''} \operatorname{\mathcal{D}}$, where $F'$ exhibits $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$ (Proposition 4.8.2.13). Applying Remark 4.8.2.12, we see that $F''$ is an equivalence of $\infty $-categories. We may therefore replace $\operatorname{\mathcal{D}}$ by $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$ and thereby reduce to the case where $F$ exhibits $\operatorname{\mathcal{D}}$ as an $(n+2)$-coskeleton of $\operatorname{\mathcal{C}}$. In this case, Proposition 3.5.3.17 guarantees that the precomposition functor
\[ \operatorname{Fun}( \operatorname{\mathcal{D}},\operatorname{\mathcal{E}}) \xrightarrow {\circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]
is an isomorphism of simplicial sets (and therefore an equivalence of $\infty $-categories).
The implication $(2) \Rightarrow (3)$ follows immediately from the definitions. We will complete the proof by showing that $(3)$ implies $(1)$. As before, we may assume that $\operatorname{\mathcal{D}}= \operatorname{cosk}_{n+2}(\operatorname{\mathcal{D}})$ is $(n+2)$-coskeletal from Proposition 4.5.1.22, so that $F$ factors (uniquely) as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}) \xrightarrow {F''} \operatorname{\mathcal{D}}$; we wish to show that the homotopy class $[F'']$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$. Since $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$ and $\operatorname{\mathcal{D}}$ are locally $n$-truncated, it will suffice to show that for every locally $n$-truncated $\infty $-category $\operatorname{\mathcal{E}}$, the horizontal map in the diagram
\[ \xymatrix@R =50pt@C=50pt{ \pi _0(\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \ar [rr] \ar [dr] & & \pi _0(\operatorname{Fun}( \operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})^{\simeq }) \ar [dl] \\ & \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ) & } \]
is a bijection. This is clear, since the vertical maps are bijections. $\square$
Let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote the homotopy category of $\infty $-categories (Construction 4.5.1.1). For every integer $n \geq -1$, we let $\mathrm{h} \mathit{\operatorname{QCat}}^{\leq n}$ denote the full subcategory of $\mathrm{h} \mathit{\operatorname{QCat}}$ spanned by the those $\infty $-categories $\operatorname{\mathcal{C}}$ which are locally $(n-1)$-truncated. We then have the following:
Corollary 4.8.2.19. Let $n \geq -1$ be an integer. Then the inclusion functor $\mathrm{h} \mathit{\operatorname{QCat}}^{\leq n} \hookrightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ admits a left adjoint, given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{cosk}_{n+1}(\operatorname{\mathcal{C}})$.