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Definition (Homology of Simplicial Sets). Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$ denote the simplicial abelian group freely generated by $S_{\bullet }$. We let $\mathrm{C}_{\ast }( S; \operatorname{\mathbf{Z}})$ denote the Moore complex of $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$. We will refer to $\mathrm{C}_{\ast }(S; \operatorname{\mathbf{Z}})$ as the chain complex of $S_{\bullet }$. For each integer $n$, we denote the $n$th homology group of $\mathrm{C}_{\ast }(S; \operatorname{\mathbf{Z}})$ by $\mathrm{H}_{n}(S; \operatorname{\mathbf{Z}})$ and refer to it as the $n$th homology group of $X$ (with coefficients in $\operatorname{\mathbf{Z}}$).