Kerodon

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Construction 2.5.5.9 (The Normalized Chain Complex of a Simplicial Set). Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$ be the simplicial abelian group freely generated by $S_{\bullet }$. We let $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ denote the normalized Moore complex of $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$. This chain complex can be described more concretely as follows:

  • For each integer $n \geq 0$, we can identify $\mathrm{N}_{n}(S)$ with the free abelian group generated by the set $S_{n}^{\mathrm{nd}}$ of nondegenerate $n$-simplices of $S_{\bullet }$.

  • The boundary map $\partial : \mathrm{N}_{n}(S) \rightarrow \mathrm{N}_{n-1}(S)$ is given by the formula

    \[ \partial (\sigma ) = \sum _{i=0}^{n} (-1)^{i} \begin{cases} d_ i(\sigma ) & \text{ if $d_ i(\sigma )$ is nondegenerate } \\ 0 & \text{otherwise.} \end{cases} \]

We will refer to $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ as the normalized chain complex of the simplicial set $S_{\bullet }$.