Proposition 7.6.6.27 (06BJ)

Proposition 7.6.6.27. Let $\kappa $ be an infinite cardinal, let $\lambda $ be an uncountable cardinal of exponential cofinality $\geq \kappa $, and let $\mathbb {K}$ be any collection of simplicial sets. Then:

$(1)$: The $\infty $-categories $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{comp} }_{< \lambda }$ and $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{ccomp} }_{< \lambda }$ are $\kappa $-complete.
$(2)$: The inclusion functors

\[ \operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{comp} }_{< \lambda } \hookrightarrow \operatorname{\mathcal{QC}}_{< \lambda } \hookleftarrow \operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{ccomp} }_{< \lambda } \]

are $\kappa $-continuous.

Proof. Our assumption that $\lambda $ has exponential cofinality $\geq \kappa $ guarantees that the $\infty $-category $\operatorname{\mathcal{QC}}_{< \lambda }$ is $\kappa $-complete (Remark 7.4.4.13). The assertions regarding $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cont}}_{< \lambda }$ are a reformulation of Proposition 7.6.6.24, and the dual assertions for $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda }$ are a formal consequence (see §8.6.7). $\square$