Kerodon

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Warning 2.5.8.8. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. Then we have a canonical isomorphism of simplicial abelian groups $A_{\bullet } \otimes B_{\bullet } \simeq B_{\bullet } \otimes A_{\bullet }$, given degreewise by the construction $a \otimes b \mapsto b \otimes a$. Likewise, there is a canonical isomorphism of chain complexes $\mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \simeq \mathrm{N}_{\ast }(B) \boxtimes \mathrm{N}_{\ast }(A)$ given by the Koszul sign rule (see Warning 2.5.1.14). Beware that these isomorphisms are not compatible with the Alexander-Whitney construction: that is, the diagram

\[ \xymatrix { \mathrm{N}_{\ast }(A \otimes B) \ar [d]^{ \mathrm{AW} } \ar [r] & \mathrm{N}_{\ast }(B \otimes A) \ar [d]^{\mathrm{AW}} \\ \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \ar [r] & \mathrm{N}_{\ast }(B) \boxtimes \mathrm{N}_{\ast }(A) } \]

usually does not commute. Instead, the composite map

\[ \mathrm{N}_{\ast }(A \otimes B) \simeq \mathrm{N}_{\ast }(B \otimes A) \xrightarrow { \mathrm{AW} } \mathrm{N}_{\ast }(B) \boxtimes \mathrm{N}_{\ast }(A) \simeq \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \]

can be identified with the Alexander-Whitney homomorphism associated to the opposite simplicial abelian groups $A_{\bullet }^{\operatorname{op}}$ and $B_{\bullet }^{\operatorname{op}}$. In other words, the colax monoidal structure of Proposition 2.5.8.7 is not a colax symmetric monoidal structure (see Definition ). The same remark applies to the unnormalized Alexander-Whitney construction $\overline{\mathrm{AW}}$ of Construction 2.5.8.2.