Exercise 1.2.1.16 (Classification of Summands). Let $S_{}$ be a simplicial set. Show that a simplicial subset $S'_{} \subseteq S_{}$ is a summand if and only if it can be written as a union of connected components of $S_{}$. Consequently, we have a canonical bijection

\[ \{ \text{Subsets of $\pi _0(S_{})$} \} \simeq \{ \text{Summands of $S_{}$} \} . \]