Construction 2.5.7.3 (The Unnormalized Shuffle Product). Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups, and suppose we are given elements $a \in A_{p}$ and $b \in B_{q}$. We let $a \bar{\triangledown } b$ denote the sum

\[ \sum _{\sigma } (-1)^{\sigma } \sigma _{-}^{\ast }(a) \otimes \sigma _{+}^{\ast }(b) \in (A \otimes B)_{p+q} \]

Here the sum is taken over all $(p,q)$-shuffles $\sigma = (\sigma _{-}, \sigma _{+})$ (Notation 2.5.7.2), and we write $\sigma _{-}^{\ast }: A_{p} \rightarrow A_{p+q}$ and $\sigma _{+}^{\ast }: B_{q} \rightarrow B_{p+q}$ for the structure morphisms of the simplicial abelian groups $A_{\bullet }$ and $B_{\bullet }$, respectively. We will refer to $a \bar{\triangledown } b$ as the *unnormalized shuffle product* of $a$ and $b$.