Kerodon

Example 3.3.2.9. Let $X$ be a topological space and let $\operatorname{Sing}_{\bullet }(X)$ denote the singular simplicial set of $X$. For each nonnegative integer $n$, the $n$-simplices of $\operatorname{Sing}_{\bullet }(X)$ are given by continuous functions $| \Delta ^{n} | \rightarrow X$, and the $n$-simplices of $\operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$ are given by continuous functions $| \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \rightarrow X$. The homeomorphism $| \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \simeq | \Delta ^{n} |$ of Proposition 3.3.2.3 determines a bijection $\operatorname{Sing}_{n}(X) \xrightarrow {\sim } \operatorname{Ex}_{n}( \operatorname{Sing}_{\bullet }(X) )$, and Remark 3.3.2.4 guarantees that these bijections are compatible with the face operators on the simplicial sets $\operatorname{Sing}_{\bullet }(X)$ and $\operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$. In other words, Proposition 3.3.2.3 supplies an isomorphism of semisimplicial sets $\varphi : \operatorname{Sing}_{\bullet }(X) \xrightarrow {\sim } \operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$. Beware that $\varphi $ is generally not an isomorphism of simplicial sets: that is, it usually does not commute with the degeneracy operators on $\operatorname{Sing}_{\bullet }(X)$ and $\operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$.