Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 1.2.5.4 (Coproducts of Kan Complexes). Let $\{ S_{\alpha } \} _{\alpha \in A}$ be a collection of simplicial sets parametrized by a set $A$, and let $S = \coprod _{\alpha \in A} S_{\alpha }$ be their coproduct. For every pair of integers $0 \leq i \leq n$ with $n > 0$, the restriction map

\[ \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^ n_ i, S ) \]

can be identified with the coproduct (formed in the arrow category $\operatorname{Fun}( [1], \operatorname{Set})$) of restriction maps $\theta _{\alpha }: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S_{\alpha }) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^ n_ i, S_{\alpha } )$; this follows from the connectedness of the simplicial sets $\Delta ^{n}$ and $\Lambda ^{n}_{i}$ (see Example 1.2.1.7 and Remark 1.2.4.5). It follows that $\theta $ is surjective if and only if each $\theta _{\alpha }$ is surjective. Allowing $n$ and $i$ to vary, we conclude that $S$ is a Kan complex if and only if each summand $S_{\alpha }$ is a Kan complex.