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Construction (The Shuffle Product). Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. It follows from Proposition that for every pair of integers $p,q \in \operatorname{\mathbf{Z}}$, there is a unique bilinear map $\triangledown : \mathrm{N}_{p}(A) \times \mathrm{N}_{q}(B) \rightarrow \mathrm{N}_{p+q}(A \otimes B)$ for which the diagram

\[ \xymatrix { \mathrm{C}_{p}(A) \times \mathrm{C}_{q}(B) \ar [r]^-{ \bar{\triangledown } } \ar@ {->>}[d] & \mathrm{C}_{p+q}(A \otimes B) \ar@ {->>}[d] \\ \mathrm{N}_{p}(A) \times \mathrm{N}_{q}(B) \ar [r]^-{ \triangledown } & \mathrm{N}_{p+q}(A \otimes B) } \]

commutes. We will refer to $\triangledown : \mathrm{N}_{p}(A) \times \mathrm{N}_{q}(B) \rightarrow \mathrm{N}_{p+q}(A \otimes B)$ as the shuffle product map. Given elements $a \in \mathrm{N}_{p}(A)$ and $b \in \mathrm{N}_{q}(B)$, we will write $a \triangledown b$ for the image of the pair $(a,b)$ under the shuffle product map, which we refer to as the shuffle product of $a$ and $b$.