# Kerodon

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Example 2.5.5.4. Let $S_{\bullet } = \Delta ^{0}$ be the standard $0$-simplex. Then $S_{\bullet }$ is a simplicial set having a single simplex of each dimension. Consequently, the chain complex $\mathrm{C}_{\ast }( S; \operatorname{\mathbf{Z}})$ is given by $\operatorname{\mathbf{Z}}$ in each nonnegative degree. For $n > 0$, the differential $\operatorname{\mathbf{Z}}\simeq \mathrm{C}_{n}(S; \operatorname{\mathbf{Z}}) \xrightarrow {\partial } \mathrm{C}_{n-1}(S;\operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}$ is given by multiplication by the integer

$\sum _{i = 0}^{n} (-1)^{i} = \begin{cases} 0 & \text{ if n is odd } \\ 1 & \text{ if n is even, } \end{cases}$

as indicated in the diagram

$\cdots \rightarrow \operatorname{\mathbf{Z}}\xrightarrow {0} \operatorname{\mathbf{Z}}\xrightarrow {1} \operatorname{\mathbf{Z}}\xrightarrow {0} \operatorname{\mathbf{Z}}\xrightarrow {1} \operatorname{\mathbf{Z}}\xrightarrow {0} \operatorname{\mathbf{Z}}\xrightarrow {1} \operatorname{\mathbf{Z}}\xrightarrow {0} \operatorname{\mathbf{Z}}.$

It follows that the homology groups of $S_{\bullet }$ are given by

$\mathrm{H}_{n}( S; \operatorname{\mathbf{Z}}) = \begin{cases} \operatorname{\mathbf{Z}}& \text{ if n = 0 } \\ 0 & \text{ otherwise.} \end{cases}$