Definition 14.2.1. For any integer $n\geq 1$, and any $0\leq j \leq n$ we let $\delta ^ n_ j : [n-1] \to [n]$ denote the injective order preserving map skipping $j$. For any integer $n\geq 0$, and any $0\leq j \leq n$ we denote $\sigma ^ n_ j : [n + 1] \to [n]$ the surjective order preserving map with $(\sigma ^ n_ j)^{-1}(\{ j\} ) = \{ j, j + 1\}$.

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