# Kerodon

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Remark 2.5.9.7. Let $n$ be a nonnegative integer. Then the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{n} ]$ of Construction 2.5.9.6 is given by the iterated shuffle product

$[ \Delta ^1] \triangledown [ \Delta ^1 ] \triangledown \cdots \triangledown [\Delta ^1] \in \mathrm{N}_{n}( \Delta ^1 \times \Delta ^1 \times \cdots \times \Delta ^1; \operatorname{\mathbf{Z}}) \simeq \mathrm{N}_{n}( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}})$

(see §2.5.7); here $[ \Delta ^1 ]$ denotes the generator of the group $\mathrm{N}_{1}( \Delta ^1; \operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}$ (which is also the fundamental chain of the $1$-dimensional cube $\operatorname{\raise {0.1ex}{\square }}^1$).