Remark 1.3.2.6 (04G1)

Remark 1.3.2.6. Let $M$ be a monoid with unit element $e$ and let $B_{\bullet }M$ denote its classifying simplicial set. By definition, $n$-simplices of the simplicial set $B_{\bullet }M$ are functors from the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ to the category $BM$. Such a functor can be identified with a collection of elements $\{ \alpha _{j,i} \in M \} _{0 \leq i \leq j \leq n}$ (where $\alpha _{j,i}$ denotes the image in $BM$ of the unique element of $\operatorname{Hom}_{[n]}(i,j)$) which are required to satisfy the identities

\[ \alpha _{i,i} = e \quad \quad \alpha _{k,i} = \alpha _{k,j} \alpha _{j,i} \text{ for $0 \leq i \leq j \leq k \leq n$.} \]

For each $n \geq 0$, the construction

\[ \{ \alpha _{j,i} \} _{0 \leq i \leq j \leq n} \mapsto ( \alpha _{n,n-1}, \alpha _{n-1,n-2}, \cdots , \alpha _{1,0} ) \]

induces a bijection $B_{n} M \simeq M^ n$. Under the resulting identification, the face and degeneracy operators of $B_{\bullet }M$ are given concretely by the formulae

\[ d^{n}_ i( x_ n, x_{n-1}, \ldots , x_1) = \begin{cases} (x_ n, x_{n-1}, \ldots , x_2) & \text{ if } i = 0 \\ (x_ n, \ldots , x_{i+2}, x_{i+1} x_ i, x_{i-1}, \ldots , x_1) & \text{ if } 0 < i < n \\ ( x_{n-1}, x_{n-2}, \ldots , x_1) & \text{ if } i = n \end{cases} \]

\[ s^{n}_ i( x_ n, x_{n-1}, \ldots , x_1 ) = (x_ n, \ldots , x_{i+1}, e, x_{i}, \ldots , x_1) \]

(see Remarks 1.3.1.7 and 1.3.1.8).