Kerodon

Variant 7.4.4.14. Let $\lambda $ be an uncountable cardinal of exponential cofinality $\kappa = \mathrm{ecf}(\lambda )$ and suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{< \lambda }$, where $\operatorname{\mathcal{C}}$ is a $\kappa $-small simplicial set. It follows from the proof of Proposition 7.4.4.1 (together with Remark 7.4.4.13) that the diagram $\mathscr {F}$ admits a limit $\varprojlim (\mathscr {F} )$, which can be identified with $\operatorname{Fun}^{\operatorname{CCart}}_{ /\operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ for any cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ with covariant transport representation $\mathscr {F}$ (for example. we can take $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$). In particular, the $\infty $-category $\operatorname{\mathcal{QC}}^{< \lambda }$ admits $\kappa $-small limits, which are preserved by the inclusion functors $\operatorname{\mathcal{QC}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{QC}}^{< \mu }$ for every $\mu \geq \lambda $.