Definition 1.2.1.8 (Connected Components). Let $S_{}$ be a simplicial set. We will say that a simplicial subset $S'_{} \subseteq S_{}$ is a connected component of $S_{}$ if $S'_{}$ is a summand of $S_{}$ (Definition 1.2.1.1) and $S'_{}$ is connected (Definition 1.2.1.6). We let $\pi _0( S_{} )$ denote the set of all connected components of $S_{}$.
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