Kerodon

Variant 7.4.1.4. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{S}}^{< \kappa }$ denote the $\infty $-category of essentially $\kappa $-small spaces, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a diagram indexed by a simplicial set $\operatorname{\mathcal{C}}$. It follows from the proof of Corollary 7.4.1.2 that $\mathscr {F}$ admits a limit in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$ if and only if the morphism space $M = \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( \underline{\Delta ^0}_{\operatorname{\mathcal{C}}}, \mathscr {F} )$ is essentially $\kappa $-small. Moreover, if this condition is satisfied, then $M$ is a limit of $\mathscr {F}$. In particular, the limit of $\mathscr {F}$ is independent of $\kappa $ (that is, it is preserved by the inclusion functors $\operatorname{\mathcal{S}}^{< \kappa } \hookrightarrow \operatorname{\mathcal{S}}^{< \lambda }$ for $\lambda \geq \kappa $.