# Kerodon

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### 4.8.4 Higher Homotopy Categories

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. In ยง1.4.5, we constructed the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, and showed that it is characterized (up to isomorphism) by the following universal property: for any category $\operatorname{\mathcal{D}}$, there is a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors of ordinary categories \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}} \} \ar [d]^{\sim } \\ \{ \textnormal{Functors of \infty -categories \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})} \} . }$

This motivates the following:

Definition 4.8.4.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a functor of $\infty$-categories and let $n$ be an integer. We say that $F$ exhibits $\operatorname{\mathcal{C}}'$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:

$(1)$

The $\infty$-category $\operatorname{\mathcal{C}}'$ is an $(n,1)$-category (Definition 4.8.1.8).

$(2)$

For every $(n,1)$-category $\operatorname{\mathcal{D}}$, precomposition with $F$ induces a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors of (n,1)-categories \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}} \} \ar [d]^{\sim } \\ \{ \textnormal{Functors of \infty -categories \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}} \} . }$

Notation 4.8.4.2. Let $n$ be a nonnegative integer. We will see in a moment that for every $\infty$-category $\operatorname{\mathcal{C}}$, there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ which exhibits $\operatorname{\mathcal{C}}'$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$ (Corollary 4.8.4.16). It follows immediately from the definition that the simplicial set $\operatorname{\mathcal{C}}'$ is unique up to (canonical) isomorphism and depends functorially on $\operatorname{\mathcal{C}}$. To emphasize this dependence, we will often denote $\operatorname{\mathcal{C}}'$ by $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ and refer to it as the homotopy $n$-category of $\operatorname{\mathcal{C}}$. For a more explicit description of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ (at least for $n > 0$), see Construction 4.8.4.9 (and Proposition 4.8.4.15).

Example 4.8.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category (Definition 1.4.5.3). Then the comparison map $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ of Construction 1.4.5.6 exhibits $\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ as a homotopy $1$-category of $\operatorname{\mathcal{C}}$, in the sense of Definition 4.8.4.1. This is a reformulation of Proposition 1.4.5.7 (see Example 4.8.1.3). Stated more informally, there is a canonical isomorphism of simplicial sets $\mathrm{h}_{\mathit{\leq 1}}\mathit{(\operatorname{\mathcal{C}})} \simeq \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$. We will sometimes abuse notation by identifying the homotopy $1$-category $\mathrm{h}_{\mathit{\leq 1}}\mathit{(\operatorname{\mathcal{C}})}$ with the ordinary category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Exercise 4.8.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, and let $Q = \pi _0( \operatorname{\mathcal{C}}^{\simeq } )$ denote the collection of isomorphism classes of objects of $\operatorname{\mathcal{C}}$. For each object $X \in \operatorname{\mathcal{C}}$, let $[X] \in Q$ denote its isomorphism class. Show that:

• There is a partial ordering $\leq _{Q}$ on the set $Q$, where $[X] \leq _{Q} [Y]$ if and only if there exists a morphism from $X$ to $Y$ in the $\infty$-category $\operatorname{\mathcal{C}}$.

• There is a unique functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(Q)$ which carries each object $X \in \operatorname{\mathcal{C}}$ to the isomorphism class $[X] \in Q$.

• The functor $F$ exhibits $\operatorname{N}_{\bullet }(Q)$ as a homotopy $0$-category of $\operatorname{\mathcal{C}}$, in the sense of Definition 4.8.4.1.

Example 4.8.4.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then:

• For every integer $n \leq -2$, the unique functor $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ exhibits $\Delta ^0$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$.

• If $\operatorname{\mathcal{C}}$ is nonempty, then $F$ also exhibits $\Delta ^0$ as a homotopy $(-1)$-category of $\operatorname{\mathcal{C}}$.

• If $\operatorname{\mathcal{C}}$ is empty, then the identity map $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \emptyset$ exhibits the empty simplicial set as a homotopy $(-1)$-category of $\operatorname{\mathcal{C}}$.

Example 4.8.4.6. Let $X$ be a Kan complex, let $n$ be a nonnegative integer, and let $\pi _{\leq n}(X)$ denote the fundamental $n$-groupoid of $X$ (Notation 3.5.6.6). Then the tautological map $u: X \rightarrow \pi _{\leq n}(X)$ exhibits $\pi _{\leq n}(X)$ as a homotopy $n$-category of $X$, in the sense of Definition 4.8.4.1. Since $\pi _{\leq n}(X)$ is an $(n,1)$-category (Example 4.8.1.11), it will suffice that for every $(n,1)$-category $\operatorname{\mathcal{D}}$, precomposition with $u$ induces a bijection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \pi _{\leq n}(X), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{\mathcal{D}})$. By virtue of Proposition 4.4.3.17, we can replace $\operatorname{\mathcal{D}}$ by its core $\operatorname{\mathcal{D}}^{\simeq }$, and thereby reduce to the case where $\operatorname{\mathcal{D}}$ is an $n$-groupoid (Example 4.8.1.17). In this case, the desired result follows from the universal property of Proposition 3.5.6.5.

In the situation of Notation 4.8.4.2, the homotopy $n$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ automatically satisfies a stronger universal property:

Proposition 4.8.4.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a functor of $\infty$-categories and let $n$ be an integer. The following conditions are equivalent:

$(1)$

For every $(n,1)$-category $\operatorname{\mathcal{D}}$, precomposition with $F$ induces a bijection of sets

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}).$
$(2)$

For every $(n,1)$-category $\operatorname{\mathcal{D}}$, precomposition with $F$ induces an isomorphism of $\infty$-categories

$\operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}).$

Proof. Assume that $(1)$ is satisfied; we will prove $(2)$ (the reverse implication follows immediately from the definitions). Let $\operatorname{\mathcal{D}}$ be an $(n,1)$-category; we wish to show that precomposition with $F$ induces an isomorphism of simplicial sets from $\operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}})$ to $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Equivalently, we wish to show that for every simplicial set $K$, the induced map

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, \operatorname{Fun}(\operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ).$

This follows by applying condition $(1)$ to the simplicial set $\operatorname{Fun}(K, \operatorname{\mathcal{D}})$, which is an $(n,1)$-category by virtue of Proposition 4.8.1.22. $\square$

Corollary 4.8.4.8. Let $n \geq -1$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a functor of $\infty$-categories which exhibits $\operatorname{\mathcal{C}}'$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$ (Definition 4.8.4.1). Then $F$ exhibits $\operatorname{\mathcal{C}}'$ as a local $(n-1)$-truncation of $\operatorname{\mathcal{C}}$ (Definition 4.8.2.9).

Proof. Since $\operatorname{\mathcal{C}}'$ is an $(n,1)$-category, it is locally $(n-1)$-truncated (Example 4.8.2.2). It will therefore suffice to show that, for every locally $(n-1)$-truncated $\infty$-category $\operatorname{\mathcal{D}}$, precomposition with $F$ induces an equivalence of $\infty$-categories $\theta : \operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (Proposition 4.8.2.18). By virtue of Corollary 4.8.3.3, we may assume that $\operatorname{\mathcal{D}}$ is an $(n,1)$-category. In this case, Proposition 4.8.4.7 guarantees that $\theta$ is an isomorphism of simplicial sets. $\square$

Our next goal is to show that every $\infty$-category $\operatorname{\mathcal{C}}$ admits a homotopy $n$-category, for every integer $n$. For $n \leq 0$, this follows from Exercise 4.8.4.4 and Example 4.8.4.5. To handle the case $n > 0$, we will use a generalization of Construction 3.5.6.10.

Construction 4.8.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $n$ be a positive integer, and let $\operatorname{cosk}^{\circ }_{n}(\operatorname{\mathcal{C}})$ denote the weak $n$-coskeleton of $\operatorname{\mathcal{C}}$ (Notation 3.5.4.19). For every integer $m \geq 0$, we will identify $m$-simplices of $\operatorname{cosk}^{\circ }_ n(\operatorname{\mathcal{C}})$ with diagrams $\sigma : \operatorname{sk}_{n}( \Delta ^{m} ) \rightarrow \operatorname{\mathcal{C}}$ which can be extended to the $(n+1)$-skeleton of $\Delta ^{m}$ (Remark 3.5.4.21). Given two such morphisms $\sigma , \sigma ': \operatorname{sk}_{n}( \Delta ^ m ) \rightarrow \operatorname{\mathcal{C}}$, we write $\sigma \sim _{m} \sigma '$ if $\sigma$ and $\sigma '$ are isomorphic relative to $\operatorname{sk}_{n-1}( \Delta ^{m} )$ (Definition 4.7.6.1). The construction

$( [m] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, \operatorname{cosk}^{\circ }_{n}(\operatorname{\mathcal{C}}) ) / \sim _{m}$

determines a simplicial set, which we will denote by $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. By construction, it is equipped with an epimorphism of simplicial sets $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) \twoheadrightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$, which determines a comparison map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$.

Remark 4.8.4.10. In the situation of Construction 4.8.4.9, the relation $\sigma \sim _{m} \sigma '$ implies that $\sigma = \sigma '$ whenever $m < n$. It follows that the tautological map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is bijective on simplices of dimension $< n$, and surjective on simplices of dimension $n$.

Proposition 4.8.4.11. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $n$ be a positive integer, and let $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ be the simplicial set of Construction 4.8.4.9. Then, for every simplicial set $A$, the comparison map

$\theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( A, \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} )$

is surjective. Moreover, if $f_0, f_1: A \rightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ are morphisms of simplicial sets which correspond to diagrams $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}$, then $\theta (f_0) = \theta (f_1)$ if and only if $u_0$ and $u_1$ are isomorphic relative to $\operatorname{sk}_{n-1}(A)$.

Proof. We proceed as in the proof of Proposition 3.5.6.12. Fix a morphism of simplicial sets $g: A \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. Using Remark 4.8.4.10 (and Proposition 1.1.4.12), we see that $g|_{ \operatorname{sk}_{n}(A) }$ can be lifted to a morphism of simplicial sets $u: \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}$. We will show that $u$ can be extended to the $(n+1)$-skeleton of $A$ (and is therefore classified by a morphism $f: A \rightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ satisfying $\theta (f) = g$; see Remark 3.5.4.21). By virtue of Proposition 1.1.4.12, this is equivalent to the assertion that for every $(n+1)$-simplex $\sigma$ of $A$ having restriction $\sigma _0 = \sigma |_{ \operatorname{\partial \Delta }^{n+1} }$, the composition $(g \circ \sigma _0): \operatorname{\partial \Delta }^{n+1} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $(n+1)$-simplex of $\operatorname{\mathcal{C}}$. Choose a lift of $g(\sigma )$ to an $(n+1)$-simplex of $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$, which we can identify with a diagram $\tau _0: \operatorname{\partial \Delta }^{n+1} \rightarrow \operatorname{\mathcal{C}}$ which admits an extension $\tau : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$. By construction, $g \circ \sigma _0$ and $\tau _0$ coincide after composing with the comparison map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. Using Proposition 1.1.4.12 again, we see that $g \circ \sigma _0$ and $\tau _0$ are isomorphic relative to $\operatorname{sk}_{n-1}( \Delta ^{n+1} )$. The desired result now follows from Corollary 4.4.5.3. This completes the proof that $\theta$ is surjective.

Now suppose that we are given a pair of morphisms $f_0, f_1: A \rightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ satisfying $\theta (f_0) = \theta (f_1)$. We wish to show that the associated maps $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}$ are isomorphic relative to $\operatorname{sk}_{n-1}(A)$ (the converse is immediate from the definitions). Using Remark 4.8.4.10, we deduce that $u_0$ and $u_1$ coincide on $\operatorname{sk}_{n-1}(A)$. By virtue of Proposition 1.1.4.12, we are reduced to showing that for every nondegenerate $n$-simplex $\sigma$ of $A$, the compositions $u_0 \circ \sigma$ and $u_1 \circ \sigma$ are isomorphic relative to $\operatorname{\partial \Delta }^{n}$. This follows from our assumption that the maps $\theta (f_0), \theta (f_1): A \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ coincide on the simplex $\sigma$. $\square$

Remark 4.8.4.12. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $n$ be a positive integer, and let $A$ be a simplicial set. Stated more informally, Proposition 4.8.4.11 asserts that $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} )$ can be viewed as a subquotient of the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(A), \operatorname{\mathcal{C}})$:

• A diagram $u: \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}$ determines a morphism from $A$ to $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ if and only if $u$ can be extended to the $(n+1)$-skeleton of $A$.

• Two diagrams $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}$ determine the same morphism $A$ to $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ if and only if they are isomorphic relative to the $(n-1)$-skeleton of $A$.

Compare with Remark 3.5.6.13.

Corollary 4.8.4.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n$ be a positive integer. Then the comparison map $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) \twoheadrightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ of Construction 4.8.4.9 is a trivial Kan fibration.

Proof. Fix an integer $m \geq 0$; we wish to show that every lifting problem

4.81
$$\begin{gathered}\label{equation:coskeleton-vs-homotopy} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ m \ar [d] \ar [r]^-{\sigma _0} & \operatorname{cosk}^{\circ }_{n}(\operatorname{\mathcal{C}}) \ar [d]^{q} \\ \Delta ^ m \ar@ {-->}[ur] \ar [r]^-{\overline{\sigma }} & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} } \end{gathered}$$

Let $\sigma$ be any $m$-simplex of $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ satisfying $q( \sigma ) = \overline{\sigma }$. By virtue of Remark 4.8.4.10, the commutativity of the diagram (4.81) guarantees that $\sigma _0$ and $\sigma$ coincide on the $(n-1)$-skeleton of $\operatorname{\partial \Delta }^{m}$. Consequently, if $m \leq n$, then $\sigma$ is a solution to the the lifting problem (4.81). We will therefore assume that $m > n$. In this case, the boundary $\operatorname{\partial \Delta }^{m}$ contains the $n$-skeleton of $\Delta ^{m}$. It will therefore suffice to show that $\sigma _0$ can be extended to an $m$-simplex $\sigma '$ of $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$: the commutativity of the diagram (4.81) guarantees that any such extension satisfies the identity $q( \sigma ' ) = \overline{\sigma }$ (Proposition 4.8.4.11). If $m \geq n+2$, then the existence of $\sigma '$ is automatic (since $\operatorname{cosk}^{\circ }_{n}(\operatorname{\mathcal{C}})$ is $(n+1)$-coskeletal). It will therefore suffice to treat the case $m = n+1$. In this case, we can identify $\sigma _0$ with a diagram $\tau _0: \operatorname{\partial \Delta }^{n+1} \rightarrow \operatorname{\mathcal{C}}$, and we wish to show that $\tau _0$ can be extended to an $(n+1)$-simplex of $\operatorname{\mathcal{C}}$. Note that $\sigma |_{ \operatorname{\partial \Delta }^{n+1} }$ determines another diagram $\tau _1: \operatorname{\partial \Delta }^{n+1} \rightarrow \operatorname{\mathcal{C}}$. Moreover, the commutativity of the diagram (4.81) guarantees that $\tau _0$ and $\tau _1$ are isomorphic relative to $\operatorname{sk}_{n-1}( \Delta ^{n+1} )$ (Proposition 4.8.4.11). Using Corollary 4.4.5.3, we are reduced to showing that $\tau _1$ can be extended to an $(n+1)$-simplex of $\operatorname{\mathcal{C}}$, which follows from the existence of $\sigma$. $\square$

Corollary 4.8.4.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n$ be a positive integer. Then the simplicial set $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ of Construction 4.8.4.9 is an $(n,1)$-category.

Proof. By virtue of Proposition 4.8.3.6, the weak $n$-coskeleton $\operatorname{cosk}^{\circ }_{n}(\operatorname{\mathcal{C}})$ is an $\infty$-category. Combining Corollary 4.8.4.13 with Proposition 1.5.5.11, we conclude that $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is also an $\infty$-category. To complete the proof, it will suffice to show that if $\sigma$ and $\tau$ are $m$-simplices of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ for some $m > n$ which satisfy $\sigma |_{ \Lambda ^{m}_{i} } = \tau |_{ \Lambda ^{m}_{i} }$ for some $0 < i < m$, then $\sigma = \tau$. Choose maps $\widetilde{\sigma }, \widetilde{\tau }: \operatorname{sk}_{n}( \Delta ^{m} ) \rightarrow \operatorname{\mathcal{C}}$ representing $\sigma$ and $\tau$. Using Proposition 4.8.4.11, we can choose an isomorphism $\alpha$ from $\widetilde{\sigma }|_{ \operatorname{sk}_{n}( \Lambda ^{m}_{i}) }$ to $\widetilde{\tau }|_{ \operatorname{sk}_{n}(\Lambda ^{m}_{i} ) }$ whose image in $\operatorname{Fun}( \operatorname{sk}_{n-1}( \Lambda ^{m}_{i} ), \operatorname{\mathcal{C}})$ is an identity morphism. If $m \geq n+2$, then $\alpha$ is also an isomorphism from $\widetilde{\sigma }|_{ \operatorname{sk}_{n}( \Delta ^ m) }$ to $\widetilde{\tau }|_{ \operatorname{sk}_{n}( \Delta ^ m ) }$, so that $\sigma = \tau$ as desired. In the case $m = n+1$, the morphisms $\widetilde{\sigma }$ and $\widetilde{\tau }$ can be extended to diagrams $\overline{\sigma }, \overline{\tau }: \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$. Using Proposition 1.5.7.6, we can extend $\alpha$ to an isomorphism of $\overline{\sigma }$ with $\overline{\tau }$. Restricting to the $n$-skeleton of $\Delta ^{m}$, we again conclude that $\sigma = \tau$. $\square$

Proposition 4.8.4.15. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n$ be a positive integer. Then the comparison map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ of Construction 4.8.4.9 exhibits $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$.

Proof. By virtue of Corollary 4.8.4.14, it will suffice to show that for every $(n,1)$-category $\operatorname{\mathcal{D}}$, the composite map

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}, \operatorname{\mathcal{D}}) \xrightarrow {\theta } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \xrightarrow {\theta '} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$

is a bijection. Since $\operatorname{\mathcal{D}}$ is weakly $n$-coskeletal, the map $\theta '$ is a bijection. By construction, $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is a quotient of the weak coskeleton $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$, so $\theta$ is an injection. We will complete the proof by showing that $\theta$ is also a surjection: that is, every diagram $F: \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ factors through $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. Let $\sigma$ and $\sigma '$ be $m$-simplices of $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ satisfying $\sigma \sim _{m} \sigma '$ (see Construction 4.8.4.9); we wish to show that $F(\sigma ) = F(\sigma ')$. This follows from the observation that $\operatorname{\mathcal{D}}$ is minimal in dimension $n$ (Proposition 4.8.1.7). $\square$

Corollary 4.8.4.16. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. For every integer $n$, there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ which exhibits $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$. Moreover:

$(1)$

The functor $F$ is bijective on $m$-simplices for $m < n$.

$(2)$

The functor $F$ factors (uniquely) as a composition $\operatorname{\mathcal{C}}\xrightarrow { F' } \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) \xrightarrow {F''} \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$, where $F'$ is the inner fibration of Proposition 4.8.3.6.

$(3)$

The functor $F''$ is a trivial Kan fibration.

$(4)$

If $n \geq -1$, the functor $F$ exhibits $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ as a local $(n-1)$-truncation of $\operatorname{\mathcal{C}}$. In particular, $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated if and only if $F$ is an equivalence of $\infty$-categories.

$(5)$

The functor $F$ is an isofibration.

Proof. The existence of $F$ follows from Example 4.8.4.5 (in the case $n < 0$), Exercise 4.8.4.4 (in the case $n = 0$), and Proposition 4.8.4.15 (in the case $n > 0$). Assertion $(1)$ is vacuous for $n \leq 0$, and follows from Construction 4.8.4.9 for $n > 0$. Since $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is an $(n,1)$-category, it is weakly $n$-coskeletal, so that assertion $(2)$ follows from Proposition 3.5.4.18.

We next prove $(3)$. For $n < 0$, the morphism $F''$ is an isomorphism (see Example 4.8.4.5) and there is nothing to prove. For $n > 0$, the desired result follows from Corollary 4.8.4.13. We may therefore assume that $n = 0$. We wish to show that every lifting problem

$\xymatrix@C =50pt@R=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) \ar [d]^{F''} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}' }$

admits a solution. For $m \geq 2$, this is automatic (since $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ and $\operatorname{\mathcal{C}}'$ are both $1$-coskeletal). The cases $m = 0$ and $m=1$ follow immediately from the construction of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ given in Exercise 4.8.4.4.

Assertion $(4)$ follows by combining $(3)$ with Proposition 4.8.3.6. We now prove $(5)$. For $n \neq 0$, the morphism $F'$ is an isofibration (Proposition 4.8.3.6), so the desired result follows from $(3)$. In the case $n = 0$, $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is isomorphic to the nerve of a partially ordered set, so the result is automatic (Example 4.4.1.6). $\square$

Corollary 4.8.4.17. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $n$ be an integer, and let $A \subseteq B$ be simplicial sets. If $B$ has dimension $\leq n+1$, then every lifting problem

$\xymatrix@C =50pt@R=50pt{ A \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d] \\ B \ar [r] \ar@ {-->}[ur] & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} }$

has a solution. If $B$ has dimension $\leq n-1$, then the solution is unique.

Remark 4.8.4.18. Let $n$ be an integer. Then, for every collection of $\infty$-categories $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$, the canonical map

$\mathrm{h}_{\mathit{\leq n}}\mathit{( \prod _{i \in I} \operatorname{\mathcal{C}}_ i )} \rightarrow \prod _{i \in I } \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_ i)}$

is an isomorphism. This follows by inspecting the explicit descriptions supplied by Construction 4.8.4.9 (for the case $n > 0$), Exercise 4.8.4.4 (for the case $n=0$) and Example 4.8.4.5 (for the case $n < 0$).

Remark 4.8.4.19. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then, for every integer $n \geq -1$, the homotopy category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_0)}$ can be identified with the full subcategory of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ spanned by the images of objects which belong $\operatorname{\mathcal{C}}_0$.

Proposition 4.8.4.20. Let $n$ be an integer and suppose we are given a pullback diagram of $\infty$-categories

$\xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r]^-{G_0} \ar [d]^{G_1} & \operatorname{\mathcal{C}}_0 \ar [d]^{F_0} \\ \operatorname{\mathcal{C}}_1 \ar [r]^-{F_1} & \operatorname{\mathcal{C}}. }$

If $\operatorname{\mathcal{C}}$ is an $(n,1)$-category, then the diagram

$\xymatrix@C =50pt@R=50pt{ \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_{01})} \ar [r] \ar [d] & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_0)} \ar [d] \\ \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_1)} \ar [r] & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} }$

is also a pullback square.

Proof. If $n \leq 0$, then we can identify $\operatorname{\mathcal{C}}_{01}$ with the full subcategory of $\operatorname{\mathcal{C}}_0 \times \operatorname{\mathcal{C}}_1$ spanned by those objects $(C_0, C_1)$ satisfying $F_0(C_0) = F_1(C_1)$. In this case, the desired result follows by combining Remarks 4.8.4.18 and 4.8.4.19. We may therefore assume without loss of generality that $n > 0$. Fix a simplicial set $A$; we wish to show that the tautological map

$\theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_{01})} ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_0)} ) \times _{ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} )} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_1)} ).$

is a monomorphism. We first show that $\theta$ is injective. Suppose we are given a pair of maps $u,u': A \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_{01})}$ satisfying $\theta (u) = \theta (u')$; we wish to show that $u = u'$. Using Remark 4.8.4.12, we can choose representatives of $u$ and $u'$ by morphisms $\widetilde{u}, \widetilde{u}': \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}_{01}$. Our assumption that $\theta (u) = \theta (u')$ guarantees that there are natural isomorphisms

$\alpha _0: G_0 \circ \widetilde{u} \rightarrow G_0 \circ \widetilde{u}' \quad \quad \alpha _1: G_1 \circ \widetilde{u} \rightarrow G_1 \circ \widetilde{u}'$

which are the identity when restricted to $\operatorname{sk}_{n-1}(A)$. It follows from the proof of Proposition 4.8.1.7 shows that $\alpha _0$ and $\alpha _1$ have the same image in $\operatorname{Fun}( \operatorname{sk}_{n}(A), \operatorname{\mathcal{C}})$. We can therefore identify the pair $(\alpha _0, \alpha _1)$ with an isomorphism from $\widetilde{u}$ to $\widetilde{u}'$ (which is the identity when restricted to $\operatorname{sk}_{n-1}(A)$), which proves that $u = u'$.

We now prove that $\theta$ is surjective. Choose an element $(u_0, u_1)$ of the fiber product $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_0)} ) \times _{\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} )} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_1)} )$. Using Remark 4.8.4.12, we can choose representatives of $u_0$ and $u_1$ by morphisms $\widetilde{u}_0: \operatorname{sk}_{n+1}(A) \rightarrow \operatorname{\mathcal{C}}_0$ and $\widetilde{u}_1: \operatorname{sk}_{n+1}(A) \rightarrow \operatorname{\mathcal{C}}_1$. Since $\operatorname{\mathcal{C}}$ is an $(n,1)$-category, the tautological map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is an isomorphism. It follows that $F_0 \circ \widetilde{u}_0$ coincides with $F_1 \circ \widetilde{u}_1$, so that the pair $( \widetilde{u}_0, \widetilde{u}_1 )$ determines a morphism $\widetilde{u}: \operatorname{sk}_{n+1}(A) \rightarrow \operatorname{\mathcal{C}}$. This represents a morphism $u: A \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ satisfying $\theta (u) = (u_0, u_1)$. $\square$