Notation 2.5.8.1. Let $n$ be a nonnegative integer. For $0 \leq p \leq n$, we define strictly increasing functions
\[ \iota _{\leq p}: [p] \hookrightarrow [n] \quad \quad \iota _{\geq p}: [n-p] \hookrightarrow [n] \]
by the formulae $\iota _{\leq p}(i) = i$ and $\iota _{\geq p}(j) = j+p$. If $A_{\bullet }$ is a simplicial abelian group, we let $\iota _{\leq p}^{\ast }: A_{n} \rightarrow A_{p}$ and $\iota _{\geq p}^{\ast }: A_{n} \rightarrow A_{n-p}$ denote the associated group homomorphisms.