Kerodon

Remark 7.4.4.5. Let $n \geq -2$ be an integer and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is locally $n$-truncated. Then the limit $\varprojlim (\mathscr {F} )$ is also locally $n$-truncated. This is a special case of Variant 7.1.4.25, since the locally $n$-truncated $\infty $-categories span a reflective subcategory of $\operatorname{\mathcal{QC}}$ (Variant 6.2.2.8). Alternatively, it can be deduced from the description of $\varprojlim (\mathscr {F} )$ supplied by Proposition 7.4.4.1. If $\mathscr {F}$ is the covariant transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, then $U$ is essentially $(n+1)$-categorical (Variant 5.1.5.17). It follows from Corollary 4.8.6.22 that the $\infty $-category of sections $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is locally $n$-truncated, so that the full subcategory $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is also locally $n$-truncated.