Definition 9.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. We say that a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is $n$-truncated if, for every object $C \in \operatorname{\mathcal{C}}$, composition with the homotopy class $[f]$ induces an $n$-truncated morphism of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.3.3 Truncated Morphisms
We now introduce a relative version of Definition 9.3.1.1.
Remark 9.3.3.2. In the situation of Definition 9.3.3.1, the composition map is only well-defined up to homotopy (see Notation 4.6.9.15). However, the condition that $\theta $ is $n$-truncated depends only on its homotopy class (Remark 3.5.9.5).
\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y) \]
Remark 9.3.3.3. Let $f: X \rightarrow Y$ be a morphism in an $\infty $-category $\operatorname{\mathcal{C}}$. The condition that $f$ is $n$-truncated depends only on the homotopy class $[f]$, regarded as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.
Remark 9.3.3.4 (Monotonicity). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $m \leq n$ be integers. If $f: X \rightarrow Y$ is an $m$-truncated morphism of $\operatorname{\mathcal{C}}$, then it is also $n$-truncated.
Example 9.3.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For $n \leq -2$, a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is $n$-truncated if and only if it is an isomorphism. See Example 3.5.9.2.
Example 9.3.3.6. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n$ be an integer. The following conditions are equivalent:
The morphism $f$ is $n$-truncated, in the sense of Definition 3.5.9.1.
For every Kan complex $K$, composition with $f$ induces an $n$-truncated morphism $\operatorname{Fun}(K,X) \rightarrow \operatorname{Fun}(K,Y)$.
For every simplicial set $K$, composition with $f$ induces an $n$-truncated morphism $\operatorname{Fun}(K,X) \rightarrow \operatorname{Fun}(K,Y)$.
The morphism $f$ is $n$-truncated when regarded as a morphism in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces, in the sense of Definition 9.3.3.1.
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 Remark 5.5.1.5.
Proposition 9.3.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be an integer, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then the morphism $f$ is $n$-truncated (in the sense of Definition 9.3.3.1) if and only if it is $n$-truncated when regarded as an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$ (in the sense of Definition 9.3.1.1).
Proof. By definition, $f$ is $n$-truncated as an object of $\operatorname{\mathcal{C}}_{/Y}$ if and only if, for every morphism $g: C \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the morphism space $K = \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{/Y} }( g, f )$ is $n$-truncated. Using Corollary 4.6.9.18, we can identify $K$ with the homotopy fiber of the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$ over the vertex $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$. The desired result now follows from Corollary 3.5.9.12. $\square$
Corollary 9.3.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be an integer, and let $f: X \rightarrow Y$ be an $n$-truncated morphism of $\operatorname{\mathcal{C}}$. Suppose we are given a diagram $F: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{/X}$. If the simplicial set $K$ is $(n+1)$-connective, then $F$ is a limit diagram in $\operatorname{\mathcal{C}}_{/X}$ if and only if the composition is a limit diagram, where $f_{!}$ is the functor given by composition with $f$ (see Example 4.3.6.15). In particular, the functor $f_{!}$ preserves limits indexed by $(n+1)$-connective simplicial sets.
\[ K^{\triangleleft } \rightarrow { F} \operatorname{\mathcal{C}}_{/X} \rightarrow { f_!} \operatorname{\mathcal{C}}_{/Y} \]
Proof. Combine Proposition 9.3.3.7 with Corollary 9.3.1.22. $\square$
Corollary 9.3.3.9 (Homotopy Invariance). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is $n$-truncated if and only if the image $F(f): F(X) \rightarrow F(Y)$ is $n$-truncated.
Proof. Using Corollary 4.6.4.19, we see that $F$ induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{D}}_{ / F(Y) }$. The desired result now follows by combining Proposition 9.3.3.7 with Remark 9.3.1.7. $\square$
Corollary 9.3.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n \geq 0$ be an integer, let $\operatorname{\raise {0.1ex}{\square }}^{n+1}$ denote the simplicial cube of dimension $n+1$ (Notation 2.4.5.2), and let $y \in \operatorname{\raise {0.1ex}{\square }}^{n+1}$ be the final vertex. Let $Q: \operatorname{\raise {0.1ex}{\square }}^{n+1} \rightarrow \Delta ^1$ be the morphism given on vertices by Then a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is $(n-2)$-truncated if and only if the composite map is a limit diagram in $\operatorname{\mathcal{C}}$.
\[ Q(v) = \begin{cases} 1 & \text{ if $v = y$ } \\ 0 & \text{ otherwise. } \end{cases} \]
\[ \operatorname{\raise {0.1ex}{\square }}^{n+1} \xrightarrow {Q} \Delta ^1 \xrightarrow {f} \operatorname{\mathcal{C}} \]
Proof. Let us identify $\operatorname{\raise {0.1ex}{\square }}^{n+1}$ with the iterated join $\{ x \} \star \operatorname{Sd}( \operatorname{\partial \Delta }^ n ) \star \{ y\} $, where $\operatorname{Sd}(\operatorname{\partial \Delta }^ n)$ denotes the subdivision of $\operatorname{\partial \Delta }^ n$ (see Proposition 3.3.3.17). Using Remark 7.1.3.11, we see that $f \circ Q$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if the constant map
\[ \{ x\} \star \operatorname{Sd}(\operatorname{\partial \Delta }^ n) \rightarrow \{ f \} \hookrightarrow \operatorname{\mathcal{C}}_{/Y} \]
is a limit diagram in the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. The desired result now follows by combining Proposition 9.3.3.7 with Remark 9.3.1.11. $\square$
Corollary 9.3.3.11. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right fibration of $\infty $-categories, let $n$ be an integer, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is $n$-truncated if and only if $U(f)$ is an $n$-truncated morphism of $\operatorname{\mathcal{D}}$.
Corollary 9.3.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then the collection of $n$-truncated morphisms of $\operatorname{\mathcal{C}}$ is closed under retracts (in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$).
Corollary 9.3.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $n \geq -2$ be an integer. Then $f$ is $n$-truncated if and only if it satisfies the following condition for every positive integer $m \geq n+4$:
If $\sigma : \Lambda ^{m}_{m} \rightarrow \operatorname{\mathcal{C}}$ is a diagram having the property that the composite map
is equal to $f$, then $\sigma $ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ m-1 < m \} ) \hookrightarrow \Lambda ^{m}_{m} \xrightarrow {\sigma } \operatorname{\mathcal{C}} \]
Proposition 9.3.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be an integer, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then:
If $Y$ is an $n$-truncated morphism and $f$ is an $n$-truncated morphism, then $X$ is an $n$-truncated object.
If $X$ is an $n$-truncated object and $Y$ is an $(n+1)$-truncated object, then $f$ is an $n$-truncated morphism.
Proof. Let $C \in \operatorname{\mathcal{C}}$ be an object and let $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$ be given by composition with the homotopy class $[f]$. Invoking Proposition 3.5.9.13, we obtain:
- $(1_ C)$
If the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$ is $n$-truncated and $\theta $ is $n$-truncated, then the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is $n$-truncated.
- $(2_ C)$
If the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is $n$-truncated and the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$ is $(n+1)$-truncated, then $\theta $ is $n$-truncated.
Proposition 9.3.3.14 follows by allowing the object $C$ to vary. $\square$
Corollary 9.3.3.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be an integer, and let $Y$ be an $n$-truncated object of $\operatorname{\mathcal{C}}$. Then a morphism $f: X \rightarrow Y$ is $n$-truncated if and only if the object $X$ is $n$-truncated.
Proof. Combine Proposition 9.3.3.14 with Remark 9.3.3.4. $\square$
Example 9.3.3.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which contains a final object $Y$. Then every object $X \in \operatorname{\mathcal{C}}$ admits a morphism $f: X \rightarrow Y$ which is uniquely determined up to homotopy. In this case, the object $X$ is $n$-truncated (in the sense of Definition 9.3.1.1) if and only if the morphism $f$ is $n$-truncated (in the sense of Definition 9.3.3.1).
Corollary 9.3.3.17 (Composition). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a $2$-simplex and let $n$ be an integer. Then:
If the morphisms $f$ and $g$ are $n$-truncated, then the morphism $h$ is $n$-truncated.
If the morphism $h$ is $n$-truncated and the morphism $g$ is $(n+1)$-truncated, then the morphism $f$ is $n$-truncated.
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{ g } & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z, } \]
Proof. Apply Proposition 9.3.3.14 to the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Z}$ (see Proposition 9.3.3.7). $\square$
Proposition 9.3.3.18 (Pullbacks of Truncated Morphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pullback diagram and let $n$ be an integer. If $f$ is $n$-truncated, then $f'$ is also $n$-truncated.
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r] & Y } \]
Proof. Let $C \in \operatorname{\mathcal{C}}$ be an object. Applying Proposition 7.4.1.18, we obtain a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X') \ar [r] \ar [d]^{ \theta '} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \ar [d]^{ \theta } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y') \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y) } \]
in the $\infty $-category of spaces. Corollary 7.6.3.11 guarantees that if $\theta $ is $n$-truncated, then $\theta '$ is also $n$-truncated. Proposition 9.3.3.18 now follows by allowing the object $C$ to vary. $\square$
Proposition 9.3.3.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n \geq -1$ be an integer, and let $X$ be an object of $\operatorname{\mathcal{C}}$ for which there exists a product $X \times X$. Then $X$ is $n$-truncated if and only if the diagonal map $\delta _{X}: X \rightarrow X \times X$ is $(n-1)$-truncated.
Proof. For each object $C \in \operatorname{\mathcal{C}}$, Example 3.5.9.18 shows that the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is $n$-truncated if and only if the diagonal map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \]
is $(n-1)$-truncated. The desired result now follows by allowing the object $C$ to vary. $\square$
Corollary 9.3.3.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n \geq -1$ be an integer, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ for which there exists a fiber product $X \times _{Y} X$. Then $f$ is $n$-truncated if if and only if the relative diagonal $\delta _{X/Y}: X \rightarrow X \times _{Y} X$ is $(n-1)$-truncated (see Notation 7.6.2.15).
Proof. Let us identify the morphism $f$ with an object $\overline{X}$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. By virtue of Proposition 7.6.2.14, there exists a product $\overline{X} \times \overline{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$, whose image in $\operatorname{\mathcal{C}}$ is the fiber product $X \times _{Y} X$. Moreover, the relative diagonal $\delta _{X/Y}$ can be identified with the image of the diagonal map $\delta _{ \overline{X} }: \overline{X} \rightarrow \overline{X} \times \overline{X}$ under the forgetful functor $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$. Applying Corollary 9.3.3.11, we see that $\delta _{X/Y}$ is an $(n-1)$-truncated morphism of $\operatorname{\mathcal{C}}$ if and only if $\delta _{ \overline{X}}$ is an $(n-1)$-truncated morphism of $\operatorname{\mathcal{C}}_{/Y}$. By virtue of Proposition 9.3.3.19, this is equivalent to the requirement that $\overline{X}$ is $n$-truncated as an object of $\operatorname{\mathcal{C}}_{/Y}$. The desired result now follows from the criterion of Proposition 9.3.3.7. $\square$
Corollary 9.3.3.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose that $\operatorname{\mathcal{C}}$ admits pullbacks and that the functor $F$ preserves pullbacks. Then, for every integer $n$, the functor $F$ carries $n$-truncated morphisms of $\operatorname{\mathcal{C}}$ to $n$-truncated morphisms of $\operatorname{\mathcal{D}}$.
Proof. For $n \leq -2$, a morphism is $n$-truncated if and only if it is an isomorphism (Example 9.3.3.5), so the desired result follows from Remark 1.5.1.6. The general case follows by induction on $n$, using Corollary 9.3.3.20. $\square$