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Example (Simplicial Sets of Dimension $\leq 0$). The contents of this tag have been expanded to subsection ยง1.1.4.

Let $S_{\bullet }$ be a simplicial set. Then the $0$-skeleton $\operatorname{sk}_0( S_{\bullet } )$ can be identified with the coproduct $\coprod _{v \in S_0} \{ v\} $, indexed by the collection of all vertices of $S_{\bullet }$. In particular, the simplicial set $S_{\bullet }$ has dimension $\leq 0$ if and only if it is isomorphic to a coproduct of copies of $\Delta ^0$. We therefore obtain an equivalence of categories

\[ \xymatrix { \{ \text{Simplicial Sets of Dimension $\leq 0$} \} \simeq \{ \text{Sets} \} . } \]