Example 3.3.3.5. Let $n$ be a nonnegative integer. Then the identity map
\[ \operatorname{id}: \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) \]
determines a map of simplicial sets $u: \Delta ^ n \rightarrow \operatorname{Ex}( \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) )$, which exhibits $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ as the subdivision of $\Delta ^ n$. In particular, the subdivision $\operatorname{Sd}(\Delta ^2)$ is the $2$-dimensional simplicial set indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & & \{ 1\} \ar [ddl] \ar [ddr] \ar [ddd] & & \\ & & & & \\ & \{ 0,1\} \ar [dr] & & \{ 1,2\} \ar [dl] & \\ & & \{ 0,1,2\} & & \\ \{ 0\} \ar [urr] \ar [rr] \ar [uur] & & \{ 0,2 \} \ar [u] & & \{ 2\} . \ar [ll] \ar [ull] \ar [uul] } \]