$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 2.5.9.16. Let $I$ be a finite linearly ordered set which is a union of disjoint subsets $I_{-}, I_{+} \subseteq I$ satisfying $i_{-} < i_{+}$ for each $i_{-} \in I_{-}$ and $i_{+} \in I_{+}$. Then the Alexander-Whitney homomorphism $\mathrm{AW}: \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I}; Z) \rightarrow \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{-}}; \operatorname{\mathbf{Z}}) \times \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{+}}; \operatorname{\mathbf{Z}})$ satisfies
\[ \mathrm{AW}( [ \operatorname{\raise {0.1ex}{\square }}^{I} ]) = [ \operatorname{\raise {0.1ex}{\square }}^{I_{-}} ] \boxtimes [ \operatorname{\raise {0.1ex}{\square }}^{I_{+}} ]. \]
Proof.
Using Remark 2.5.9.7 (and the graded-commutativity of the shuffle product; see Proposition 2.5.7.10), we observe that the shuffle product map
\[ \triangledown : \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{-}}; \operatorname{\mathbf{Z}}) \times \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{+}}; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{-}} \times \operatorname{\raise {0.1ex}{\square }}^{I_{+}}; \operatorname{\mathbf{Z}}) \simeq \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}}) \]
satisfies $[ \operatorname{\raise {0.1ex}{\square }}^{I} ] = [ \operatorname{\raise {0.1ex}{\square }}^{I_{-}} ] \triangledown [ \operatorname{\raise {0.1ex}{\square }}^{I_{+}}] $. Applying the Alexander-Whitney homomorphism and invoking Proposition 2.5.8.9, we obtain the identity
\[ \mathrm{AW}( [ \operatorname{\raise {0.1ex}{\square }}^{I} ] ) = \mathrm{AW}([ \operatorname{\raise {0.1ex}{\square }}^{I_{-}} ] \triangledown [ \operatorname{\raise {0.1ex}{\square }}^{I_{+}}] ) = [ \operatorname{\raise {0.1ex}{\square }}^{I_{-}} ] \boxtimes [ \operatorname{\raise {0.1ex}{\square }}^{I_{+}} ]. \]
$\square$