Kerodon

Example 7.6.1.16 (Products in $\operatorname{\mathcal{S}}$). Let $\{ Y_ i \} _{i \in I}$ be a collection of Kan complexes and let $Y = {\prod }_{i \in I } Y_ i$ denote their product, formed in the ordinary category of simplicial sets. For each $i \in I$, let $q_ i: Y \rightarrow Y_ i$ denote the projection map. Applying Example 7.6.1.15 to the simplicial category category $\operatorname{Kan}$, we deduce that the morphisms $q_ i$ also exhibit $Y$ as a product of the collection $\{ Y_ i \} _{i \in I}$ in the $\infty $-category of spaces $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$. Similarly, if $Y' = {\coprod }_{i \in I} Y_ i$ is the coproduct of the collection $\{ Y_ i \} _{i \in I}$ in the ordinary category of simplicial sets, then the inclusion maps $Y_ i \hookrightarrow Y'$ exhibit $Y'$ as a coproduct of $\{ Y_ i \} _{i \in I}$ in the $\infty $-category $\operatorname{\mathcal{S}}$.