Kerodon

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Proposition 9.2.3.13. 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:

$(1)$

If $Y$ is an $n$-truncated morphism and $f$ is an $n$-truncated morphism, then $X$ is an $n$-truncated object.

$(2)$

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 9.2.3.13 follows by allowing the object $C$ to vary. $\square$