Kerodon

Variant 7.4.1.15 (Size Estimates for Limits). Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential cofinality. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, where $\operatorname{\mathcal{C}}$ is a $\kappa $-small simplicial set. Then $\mathscr {F}$ can be identified with the covariant transport representation of a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which is essentially $\lambda $-small. Applying Variant 4.7.9.11, we deduce that the Kan complex $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is also essentially $\lambda $-small, and can therefore be identified with a limit of $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$. In particular, the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ admits $\kappa $-small limits.