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9.2.1 Truncated Objects

Let $n$ be an integer. Recall that a Kan complex $X$ is $n$-truncated if, for every integer $m \geq n+2$, every morphism $\operatorname{\partial \Delta }^{m} \rightarrow X$ can be extended to an $m$-simplex of $X$. We now introduce a counterpart of this condition for objects of an arbitrary $\infty $-category.

Definition 9.2.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. We say that an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if, for every object $Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ is an $n$-truncated Kan complex.

Remark 9.2.1.2. In the formulation of Definition 9.2.1.1, we can replace $M = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ by any Kan complex which is homotopy equivalent $M$. For example, we can replace $M$ by the pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(Y,X)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(Y,X)$ (see Proposition 4.6.5.10).

Example 9.2.1.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For $n \leq -2$, an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if and only if it is a final object of $\operatorname{\mathcal{C}}$ (Definition 4.6.7.1). In particular, this condition is independent of $n$, so long as $n \leq -2$. Consequently, in the setting of Definition 9.2.1.1, there is no loss of generality in assuming that $n \geq -2$.

Example 9.2.1.4. Let $X$ be a Kan complex and let $n$ be an integer. The following conditions are equivalent:

$(1)$

The Kan complex $X$ is $n$-truncated, in the sense of Definition 3.5.7.1.

$(2)$

For every Kan complex $Y$, the Kan complex $\operatorname{Fun}(Y,X)$ is $n$-truncated.

$(3)$

For every simplicial set $Y$, the Kan complex $\operatorname{Fun}(Y, X)$ is $n$-truncated.

$(4)$

The Kan complex $X$ is $n$-truncated when regarded as an object of the $\infty $-category $\operatorname{\mathcal{S}}$ (in the sense of Definition 9.2.1.1).

The implications $(3) \Rightarrow (2) \Rightarrow (1)$ are immediate, the implication $(1) \Rightarrow (3)$ follows from Corollary 3.5.9.27, and the equivalence $(2) \Leftrightarrow (4)$ follows from the homotopy equivalence $\operatorname{Fun}(Y,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(Y,X)$ of Remark 5.5.1.5.

Remark 9.2.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$. If $X$ is $n$-truncated and $Y$ is a retract of $X$, then $Y$ is also $n$-truncated. In particular, if $X$ and $Y$ are isomorphic, then $X$ is $n$-truncated if and only if $Y$ is $n$-truncated.

Remark 9.2.1.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $n$ be an integer, and let $X \in \operatorname{\mathcal{C}}$ be an object whose image $F(X)$ is an $n$-truncated object of $\operatorname{\mathcal{D}}$. If the functor $F$ is essentially $(n+1)$-categorical (Definition 4.8.6.1), then $X$ is an $n$-truncated object of $\operatorname{\mathcal{C}}$ (see Proposition 3.5.9.13). In particular, if $F$ is fully faithful, then $X$ is an $n$-truncated object of $\operatorname{\mathcal{C}}$.

Remark 9.2.1.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if and only if the image $Y = F(X)$ is an $n$-truncated object of $\operatorname{\mathcal{D}}$. The “if” direction follows from Remark 9.2.1.6. For the converse, suppose that $X$ is $n$-truncated and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse to $F$. Then $G(Y) \in \operatorname{\mathcal{C}}$ is isomorphic to $X$, and is therefore an $n$-truncated object of $\operatorname{\mathcal{C}}$ (Remark 9.2.1.5). Since $G$ is fully faithful, Remark 9.2.1.6 guarantees that $Y$ is an $n$-truncated object of $\operatorname{\mathcal{D}}$.

Remark 9.2.1.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then $\operatorname{\mathcal{C}}$ is locally $n$-truncated (in the sense of Definition 4.8.2.1) if and only if every object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated (in the sense of Definition 9.2.1.1).

Remark 9.2.1.9 (Monotonicity). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $m \leq n$ be integer. If an object $X \in \operatorname{\mathcal{C}}$ is $m$-truncated, then it is also $n$-truncated (see Remark 3.5.9.6).

Remark 9.2.1.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $n \geq -2$ be an integer. The following conditions are equivalent:

$(1)$

The object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated, in the sense of Definition 9.2.1.1.

$(2)$

The constant map $\operatorname{\partial \Delta }^{n+2} \rightarrow \{ \operatorname{id}_ X \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$ exhibits $X$ as a power of itself by $\operatorname{\partial \Delta }^{n+2}$, in the sense of Definition 7.6.2.1.

$(3)$

The constant map

\[ ( \operatorname{\partial \Delta }^{n+2} )^{\triangleright } \simeq \Lambda ^{n+3}_{n+3} \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}} \]

is a limit diagram in $\operatorname{\mathcal{C}}$, in the sense of Definition 7.1.2.4.

The equivalence $(1) \Leftrightarrow (2)$ follows from Corollary 3.5.9.22, and the equivalence $(2) \Leftrightarrow (3)$ from Remark 7.6.2.6.

Remark 9.2.1.11. In the formulation of Remark 9.2.1.10, we can replace $\operatorname{\partial \Delta }^{n+2}$ by any simplicial set $K$ of the same weak homotopy type (that is, any simplicial set $K$ for which the geometric realization $|K|$ is homotopy equivalent to a sphere of dimension $n+1$). For example, we can take $K$ to be the subdivision $\operatorname{Sd}( \Delta ^{n+1} )$ (see Proposition 3.3.4.8).

Remark 9.2.1.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves finite limits. Then, for every $n$-truncated object $X \in \operatorname{\mathcal{C}}$, the image $F(X)$ is an $n$-truncated object of $\operatorname{\mathcal{D}}$. This follows from the criterion of Remark 9.2.1.10.

Remark 9.2.1.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq -1$ be an integer. Then an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if and only if the right fibration $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ is a locally $(n-1)$-truncated functor. This follows from the criterion of Corollary 5.1.5.18 (together with Remark 9.2.1.2).

Proposition 9.2.1.14 (Limits of Truncated Objects). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then the collection of $n$-truncated objects of $\operatorname{\mathcal{C}}$ is closed under limits. That is, if $F: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$ having the property that $F(a)$ is $n$-truncated for each vertex $a \in A$, then $F$ carries the cone point of $A^{\triangleleft }$ to an $n$-truncated object of $\operatorname{\mathcal{C}}$.

Proof. Fix an object $C \in \operatorname{\mathcal{C}}$ and let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ denote the functor corepresented by $C$. We wish to show that the composition

\[ A^{\triangleleft } \xrightarrow {F} \operatorname{\mathcal{C}}\xrightarrow {h^{C}} \operatorname{\mathcal{S}} \]

carries the cone point of $A^{\triangleleft }$ to an $n$-truncated Kan complex. This follows from Remark 7.4.5.9, since $h^{C} \circ F$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}$ (Corollary 7.4.5.17). $\square$

Remark 9.2.1.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects $Y \in \operatorname{\mathcal{C}}$ for which the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ is $n$-truncated. Since the representable functor

\[ h_{X}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}\quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X) \]

carries colimits in the $\infty $-category $\operatorname{\mathcal{C}}$ to limits in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$, (Corollary 7.4.5.17), Remark 7.4.5.9 guarantees that the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is closed under the formation of colimits. Consequently, if $\operatorname{\mathcal{C}}$ is generated (under the formation of small colimits) by some full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$, then the object $X$ is $n$-truncated if and only if the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ is $n$-truncated for each object $Y \in \operatorname{\mathcal{C}}_0$.

Definition 9.2.1.1 can be reformulated as a filling condition:

Proposition 9.2.1.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $n \geq -2$ be an integer. Then $X$ is $n$-truncated if and only if it satisfies the following condition for each $m \geq n+3$:

$(\ast _ m)$

Every morphism $\sigma : \operatorname{\partial \Delta }^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (m) = X$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.

Proof. This is a special case of Proposition 4.8.6.19, since the object $X$ is $n$-truncated if and only if the right fibration $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ is essentially $(n+1)$-categorical (Remark 9.2.1.13). $\square$

We close this section by classifying the truncated objects of the $\infty $-category $\operatorname{\mathcal{QC}}$ of (small) $\infty $-categories (Construction 5.5.4.1).

Proposition 9.2.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is $n$-truncated when viewed as an object of $\operatorname{\mathcal{QC}}$, in the sense of Definition 9.2.1.1.

$(2)$

The Kan complex $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$ is $n$-truncated.

$(3)$

The core $\operatorname{\mathcal{C}}^{\simeq }$ is an $n$-truncated Kan complex. Moreover, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is also an $n$-truncated Kan complex.

Proof. By virtue of Proposition , the $\infty $-category $\operatorname{\mathcal{QC}}$ is generated under colimits by the object $\Delta ^1 \in \operatorname{\mathcal{QC}}$. The equivalence $(1) \Leftrightarrow (2)$ now follows by combining Remarks 9.2.1.15 and 5.5.4.5. We next show that, if condition $(2)$ is satisfied, then the core $\operatorname{\mathcal{C}}^{\simeq }$ is an $n$-truncated Kan complex. Let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms of $\operatorname{\mathcal{C}}$. Then the diagonal map

\[ \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Isom}(\operatorname{\mathcal{C}}) \quad \quad X \mapsto \operatorname{id}_{X} \]

is an equivalence of $\infty $-categories (Corollary 4.5.3.13), and therefore restricts to a homotopy equivalence of Kan complexes $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq }$. We are therefore reduced to showing that the Kan complex $\operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq }$ is $n$-truncated. Since $\operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq }$ is a summand of the Kan complex $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$, this follows immediately from assumption $(2)$ if $n \geq -1$. The case $n \leq -2$ then follows from the additional observation that if $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$ is nonempty, then the $\infty $-category $\operatorname{\mathcal{C}}$ is nonempty, so $\operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq }$ is nonempty.

We now complete the proof by showing that $(2)$ and $(3)$ are equivalent. By virtue of the preceding argument, we may assume that the core $\operatorname{\mathcal{C}}^{\simeq }$ is $n$-truncated, so the product $\operatorname{\mathcal{C}}^{\simeq } \times \operatorname{\mathcal{C}}^{\simeq }$ is $n$-truncated (Remark 3.5.7.6). Using Proposition 3.5.9.13, we see that condition $(2)$ is satisfied if and only if the map of Kan complexes

\[ U: \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{\mathcal{C}}^{\simeq } \times \operatorname{\mathcal{C}}^{\simeq } \quad \quad (f: X \rightarrow Y) \mapsto (X,Y) \]

is $n$-truncated. Since $U$ is a Kan fibration (Corollary 4.4.5.4), this is equivalent to the requirement that each fiber of $U$ is an $n$-truncated Kan complex (Proposition 3.5.9.8), which is a restatement of $(3)$. $\square$

Remark 9.2.1.18. If $n \geq -1$, we can reformulate condition $(3)$ of Proposition 9.2.1.17 as follows:

$(3')$

For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $n$-truncated. Moreover, the summand $\operatorname{Isom}_{\operatorname{\mathcal{C}}}(X,Y) \subseteq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ spanned by the isomorphisms from $X$ to $Y$ is $(n-1)$-truncated.

See Example 3.5.9.18.

Corollary 9.2.1.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then:

  • If $\operatorname{\mathcal{C}}$ is an $n$-truncated object of $\operatorname{\mathcal{QC}}$ (in the sense of Definition 9.2.1.1), then it is locally $n$-truncated (in the sense of Definition 4.8.2.1).

  • If $n \geq -1$ and $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated, then it is an $n$-truncated object of $\operatorname{\mathcal{QC}}$.