Kerodon

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Example 1.1.7.8. The construction $[n] \mapsto | \Delta ^{n} |$ determines a functor from the simplex category $\operatorname{{\bf \Delta }}$ to the category $\operatorname{Top}$ of topological spaces, which assigns to each morphism $\alpha : [m] \rightarrow [n]$ the continuous map

\[ | \Delta ^{m} | \rightarrow | \Delta ^{n} | \quad \quad (t_0, \ldots , t_ m) \mapsto ( \sum _{\alpha (i) = 0} t_ i, \ldots , \sum _{\alpha (i)=n} t_ i). \]

We regard this functor as a cosimplicial topological space, which we denote by $| \Delta ^{\bullet } |$. Applying Variant 1.1.7.7 to this cosimplicial space yields a functor $\operatorname{Sing}^{ | \Delta |}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}$, which coincides with the singular simplicial set functor $\operatorname{Sing}_{\bullet }$ of Construction 1.1.7.1.