Variant 5.5.4.12. Let $\kappa $ be an uncountable cardinal. We let $\operatorname{\mathcal{S}}^{< \kappa }$ denote the full subcategory of $\operatorname{\mathcal{S}}$ spanned by the $\kappa $-small Kan complexes, which we also regard as a full subcategory of $\operatorname{\mathcal{QC}}^{< \kappa }$. Similarly, we let $\operatorname{\mathcal{S}}^{< \kappa }_{\ast }$ denote the full subcategory of $\operatorname{\mathcal{S}}_{\ast }$ spanned by those pointed Kan complexes $(X,x)$ where $X$ is $\kappa $-small.
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