Kerodon

Definition 9.2.3.1. Let $\kappa $ be a regular cardinal. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-sequentially cocomplete if it admits $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } )$-indexed colimits. Here $\mathrm{Ord}_{< \kappa }$ denotes the collection of ordinals which are strictly smaller than $\kappa $ (so that $\mathrm{Ord}_{< \kappa }$ is a linearly ordered set of order type $\kappa $: see Proposition 4.7.1.22). If this condition is satisfied, then we say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\kappa $-sequential colimits if it preserves $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } )$-indexed colimits.