# Kerodon

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Notation 2.5.7.11 (The Eilenberg-Zilber Homomorphism). Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. It follows from assertion $(4)$ of Proposition 2.5.7.10 that there is a unique chain map

$\mathrm{EZ}: \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \rightarrow \mathrm{N}_{\ast }(A \otimes B)$

satisfying $\mathrm{EZ}( a \boxtimes b) = a \triangledown b$ (see Exercise 2.5.1.15). We will refer to $\mathrm{EZ}$ as the Eilenberg-Zilber homomorphism (see Remark 2.5.7.16). It follows from assertions $(1)$ and $(3)$ of Proposition 2.5.7.10 that the collection of chain maps

$\{ \mathrm{EZ}: \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \rightarrow \mathrm{N}_{\ast }(A \otimes B) \} _{A_{\bullet }, B_{\bullet } \in \operatorname{ Ab }_{\Delta }}$

determine a lax monoidal structure (Definition 2.1.5.8) on the normalized Moore complex functor $\mathrm{N}_{\ast }: \operatorname{ Ab }_{\Delta } \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})$, with unit given by the canonical isomorphism of chain complexes $\operatorname{\mathbf{Z}}[0] \simeq \mathrm{N}_{\ast }( \operatorname{\mathbf{Z}}[\Delta ^0] )$ (in fact, it is even a lax symmetric monoidal structure in the sense of Definition : this follows from assertion $(2)$ of Proposition 2.5.7.10).