Variant 1.3.2.12. Let $M$ be a nonunital monoid. We let $B_{\bullet } M$ denote the semisimplicial set which assigns to each object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}$ the collection of tuples $\{ \alpha _{j,i} \in M \} _{0 \leq i < j \leq n}$ which satisfy the identity $\alpha _{k,i} = \alpha _{k,j} \alpha _{j,i}$ for $0 \leq i < j < k \leq n$. As in Remark 1.3.2.6, the construction
\[ \{ \alpha _{j,i} \} _{0 \leq i < j \leq n} \mapsto ( \alpha _{n,n-1}, \alpha _{n-1,n-2}, \cdots , \alpha _{1,0} ) \]
induces an identification $B_{n} M \simeq M^ n$. Under this identification, the face operators of $B_{\bullet } M$ are given by the formula
\[ d^{n}_ i( x_ n, x_{n-1}, \ldots , x_1) = \begin{cases} (x_ n, x_{n-1}, \ldots , x_2) & \text{ if } i = 0 \\ (x_ n, \ldots , x_{i+2}, x_{i+1} x_ i, x_{i-1}, \ldots , x_1) & \text{ if } 0 < i < n \\ ( x_{n-1}, x_{n-2}, \ldots , x_1) & \text{ if } i = n. \end{cases} \]