# Kerodon

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

Example 9.1.0.1 (Simplicial Sets of Dimension $\leq 0$). The contents of this tag have been expanded to subsection §1.1.4.

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

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