Kerodon

Variant 5.5.6.18. Let $\kappa $ be an uncountable cardinal. We let $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}^{< \kappa }$ denote the full simplicial subset of $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ spanned by those pairs $(\operatorname{\mathcal{C}}, C)$ where the $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-small, and we define $\operatorname{\mathcal{QC}}^{<\kappa }_{\operatorname{Obj}} = \operatorname{Pith}( \operatorname{ \pmb {\mathcal{QC}} }^{< \kappa }_{\operatorname{Obj}})$ similarly. The projection map $\operatorname{\mathcal{QC}}^{< \kappa }_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}^{< \kappa }$ is then a cocartesian fibration of $\infty $-categories, whose fibers are $\kappa $-small.