degeneracy map

Let \(\Delta\) be the simplex category and \(\Delta_n = [n],\Delta_{n-1} = [n-1]\) be simplices. Then the \(k\)-th degeneracy map is defined as map

\begin{align*} \sigma_k: [n] \rightarrow& [n-1] \\ j \mapsto& \begin{cases} j & \mbox{if } j \leq k \\ j - 1 & \mbox{else } \\ \end{cases} \end{align*}