Kerodon

Remark 2.5.7.5 (Commutativity of the Shuffle Product). Let $\sigma : [p+q] \rightarrow [p] \times [q]$ be a $(p,q)$-shuffle, and let $\sigma ': [p+q] \rightarrow [q] \times [p]$ denote the composition of $\sigma $ with the isomorphism $[p] \times [q] \simeq [q] \times [p]$ given by permuting the factors. Then $\sigma '$ is a $(q,p)$-shuffle, whose sign is given by $(-1)^{\sigma '} = (-1)^{pq} \cdot (-1)^{\sigma }$. Consequently, if $A_{\bullet }$ and $B_{\bullet }$ are simplicial abelian groups containing simplices $a \in A_{p}$ and $b \in B_{q}$, then the canonical isomorphism $(A \otimes B)_{p+q} \simeq (B \otimes A)_{p+q}$ carries $a \bar{\triangledown } b$ to $(-1)^{pq} ( b \bar{\triangledown } a)$.