Example 1.2.3.2. For each $n \geq 0$, the identity map $\operatorname{id}: | \Delta ^{n} | \simeq | \Delta ^{n} |$ determines an $n$-simplex of the simplicial set $\operatorname{Sing}_{\bullet }( | \Delta ^{n} |)$, which we can identify with a morphism of simplicial sets $u: \Delta ^{n} \rightarrow \operatorname{Sing}_{\bullet }( | \Delta ^ n | )$. It follows from Proposition 1.1.0.12 that $u$ exhibits the topological space $| \Delta ^{n} |$ as a geometric realization of the simplicial set $\Delta ^{n}$.
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