Kerodon

Example 2.5.5.13. Let $S_{\bullet }$ be a simplicial set. It follows from Proposition 2.5.5.11 that the quotient map $\mathrm{C}_{\ast }( S; \operatorname{\mathbf{Z}}) \twoheadrightarrow \mathrm{N}_{\ast }( S; \operatorname{\mathbf{Z}})$ induces an isomorphism on homology. In particular, the homology groups $\mathrm{H}_{\ast }(S; \operatorname{\mathbf{Z}})$ of the simplicial set $S_{\bullet }$ (in the sense of Definition 2.5.5.2) can be computed by means of the normalized chain complex $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$. This has various practical advantages. For example, if $S_{\bullet }$ is a simplicial set of dimension $\leq d$, then the chain complex $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ is concentrated in degrees $\leq d$. It follows that the homology groups $\mathrm{H}_{\ast }(S; \operatorname{\mathbf{Z}})$ are also concentrated in degrees $\leq d$, which is not immediately obvious from the definition (note that the chain complex $\mathrm{C}_{\ast }(S; \operatorname{\mathbf{Z}})$ is never concentrated in degrees $\leq d$, except in the trivial case where $S_{\bullet }$ is empty).