Example 2.5.5.14 (00QN)

Example 2.5.5.14. Let $S_{\bullet } = \operatorname{N}_{\bullet }(Q)$ be the nerve of a partially ordered set $Q$. Suppose that $Q$ has a least element $e$, which determines a map of simplicial sets $i: \Delta ^{0} \rightarrow S_{\bullet }$ which is right inverse to the projection map $q: S_{\bullet } \rightarrow \Delta ^{0}$. Passing to normalized chain complexes, we obtain chain maps

\[ \widehat{i}: \operatorname{\mathbf{Z}}[0] \simeq \mathrm{N}_{\ast }(\Delta ^0; \operatorname{\mathbf{Z}}) \hookrightarrow \mathrm{N}_{\ast }( S_{\bullet }; \operatorname{\mathbf{Z}}) \quad \quad \widehat{q}: \mathrm{N}_{\ast }(S_{\bullet }; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }( \Delta ^{0}; \operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}[0]. \]

We claim that $\widehat{i}$ and $\widehat{q}$ are chain homotopy inverse to one another. In one direction, this is clear: the composition $\widehat{q} \circ \widehat{i}$ is equal to the identity. We complete the proof by constructing a chain homotopy from the composite map $\widehat{i} \circ \widehat{q}$ to the identity $\operatorname{id}$ on $\mathrm{N}_{\ast }(S_{\bullet }; \operatorname{\mathbf{Z}})$. This chain homotopy is given by a collection of maps $h_{m}: \mathrm{N}_{m}( S; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{m+1}( S; \operatorname{\mathbf{Z}})$, given on nondegenerate simplices by the construction

\[ ( q_0 < q_1 < \cdots < q_ m ) \mapsto \begin{cases} 0 & \text{ if } q_0 = e \\ ( e < q_0 < q_1 < \cdots < q_ m ) & \text{ otherwise. } \end{cases} \]

In particular, if $Q$ is a partially ordered set with a least element, then the homology groups of the nerve $S_{\bullet } = \operatorname{N}_{\bullet }(Q)$ are given by

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