Construction 2.5.7.9 (The Shuffle Product). Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. It follows from Proposition 2.5.7.8 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
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 $b$.