Kerodon

Remark 7.4.1.5 (Limits of Truncated Spaces). Let $n$ be an integer and let $\operatorname{\mathcal{S}}_{\leq n}$ denote the full subcategory of $\operatorname{\mathcal{S}}$ spanned by the $n$-truncated spaces (Definition 3.5.9.1). Then $\operatorname{\mathcal{S}}_{\leq n}$ is a reflective subcategory of $\operatorname{\mathcal{S}}$ (Example 6.2.2.7). Combining Corollary 7.4.1.3 with Variant 7.1.4.25, we conclude that the $\infty $-category $\operatorname{\mathcal{S}}_{\leq n}$ admits small limits, which are preserved by the inclusion functor $\operatorname{\mathcal{S}}_{\leq n} \hookrightarrow \operatorname{\mathcal{S}}$. In other words, if $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a diagram having the property that each of the Kan complexes $\mathscr {F}(C)$ is $n$-truncated, then the limit $\varprojlim (\mathscr {F}) \simeq \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \underline{ \Delta ^0 }_{\operatorname{\mathcal{C}}}, \mathscr {F} )$ is also $n$-truncated.