Kerodon

Example 7.6.1.17 (Products in $\operatorname{\mathcal{QC}}$). Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a collection of $\infty $-categories and let $\operatorname{\mathcal{C}}= {\prod }_{i \in I } \operatorname{\mathcal{C}}_ i$ denote their product, formed in the ordinary category of simplicial sets. For each $i \in I$, let $q_ i: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_ i$ denote the projection map. Applying Example 7.6.1.15 to the simplicial category $\operatorname{QCat}$ (see Construction 5.5.4.1), we deduce that the morphisms $q_ i$ also exhibit $\operatorname{\mathcal{C}}$ as a product of the collection $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ in the $\infty $-category $\operatorname{\mathcal{QC}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$ (this is a special case of the diffraction criterion of Theorem 7.4.4.6). Similarly, if $\operatorname{\mathcal{C}}' = {\coprod }_{i \in I} \operatorname{\mathcal{C}}_ i$ is the coproduct of the collection $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ in the ordinary category of simplicial sets, then the inclusion maps $\operatorname{\mathcal{C}}_ i \hookrightarrow \operatorname{\mathcal{C}}'$ exhibit $\operatorname{\mathcal{C}}'$ as a coproduct of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ in the $\infty $-category $\operatorname{QCat}$ (this is a special case of the refraction criterion of Theorem 7.4.5.11).