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Notation Let $S$ be a set. We will often abuse notation by identifying $S$ with the constant simplicial set $\underline{S}_{\bullet }$ of Construction (by virtue of Corollary, this is mostly harmless). This abuse will occur most frequently in the special case where $S = \{ v \} $ consists of a single vertex $v$ of some other simplicial set $X_{\bullet }$: in this case, we view $\{ v\} $ as a simplicial subset of $X_{\bullet }$ which is abstractly isomorphic to $\Delta ^0$ (see Example