Remark 2.5.7.6 (Associativity of the Shuffle Product). Let $A_{\bullet }$, $B_{\bullet }$, and $C_{\bullet }$ be simplicial abelian groups containing simplices $a \in A_{p}$, $b \in B_{q}$, and $c \in C_{r}$. Then the canonical isomorphism $( A \otimes (B \otimes C))_{p+q+r} \simeq ( (A \otimes B) \otimes C)_{p+q+r}$ carries $a \bar{\triangledown } (b \bar{\triangledown } c)$ to $( a \bar{\triangledown } b) \bar{\triangledown } c$. Both of these iterated shuffle products can be described concretely as the sum
\[ \sum _{ \sigma } (-1)^{\sigma } \sigma _{-}^{\ast }(a) \otimes \sigma _{0}^{\ast }(b) \otimes \sigma _{+}^{\ast }(c), \]
where the sum is taken over all strictly increasing maps $\sigma = (\sigma _{-}, \sigma _{0}, \sigma _{+}): [p+q+r] \rightarrow [p] \times [q] \times [r]$, and $(-1)^{\sigma }$ denotes the product
\[ \prod _{1 \leq i < j \leq p+q+r} \begin{cases} -1 & \text{ if } \sigma _{-}(j-1) < \sigma _{-}(j) \text{ and } \sigma _{-}(i-1) = \sigma _{-}(i) \\ -1 & \text{ if } \sigma _+(j-1) = \sigma _+(j) \text{ and } \sigma _+(i-1) < \sigma _{+}(i) \\ 1 & \text{ otherwise.} \end{cases} \]