Kerodon

Proof. We proceed as in the proof of Proposition 8.7.3.11. Choose a functor $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$. Assume that $\operatorname{\mathcal{C}}$ is locally $n$-truncated; we will show that $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ has the same property (the converse is immediate, since the functor $H$ is fully faithful). Fix objects $\widehat{X},\widehat{Y} \in \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$; we wish to show that the morphism space $\operatorname{Hom}_{ \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) }( \widehat{X}, \widehat{Y})$ is an $n$-truncated Kan complex. Let us first regard the object $\widehat{Y}$ as fixed, and let $\operatorname{\mathcal{D}}$ denote the full subcategory of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ spanned by those objects $\widehat{X}$ for which $\operatorname{Hom}_{ \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})}( \widehat{X}, \widehat{Y})$ is $n$-truncated. Since the collection of $n$-truncated spaces is closed under limits (Remark 7.4.1.5), we see that $\operatorname{\mathcal{D}}$ is closed under colimits in $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ (see Proposition 7.4.1.18). In particular, it is closed under $\lambda $-small $\kappa $-filtered colimits. Consequently, to show that $\operatorname{\mathcal{D}}= \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, it will suffice to show that the mapping space $\operatorname{Hom}_{ \operatorname{Ind}(\operatorname{\mathcal{C}})}( \widehat{X}, \widehat{Y})$ is $n$-truncated in the special case $\widehat{X} = H(X)$ for some object $X \in \operatorname{\mathcal{C}}$.

Let us now regard the object $\widehat{X} = H(X)$ as fixed, and let $\operatorname{\mathcal{D}}' \subseteq \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ denote the full subcategory spanned by those objects $\widehat{Y}$ for which the morphism space $\operatorname{Hom}_{ \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})}( \widehat{X}, \widehat{Y})$ is an $n$-truncated Kan complex. Since $\widehat{X}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, the subcategory $\operatorname{\mathcal{D}}'$ is closed under $\lambda $-small $\kappa $-filtered colimits (Variant 9.1.10.3). Consequently, to prove that $\operatorname{\mathcal{D}}' = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, it will suffice to show that it contains $H(Y)$, for each object $Y \in \operatorname{\mathcal{C}}$. Since $H$ is fully faithful, we are reduced to proving that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $n$-truncated, which follows from our assumption that the $\infty $-category $\operatorname{\mathcal{C}}$ is locally $n$-truncated. $\square$