Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 3.5.2.5. Let $X$ be a simplicial set and let $Y$ be a topological space, and let $\operatorname{Hom}_{\operatorname{Top}}( |X|, Y)_{\bullet }$ be the simplicial set defined in Example 2.4.1.5. For each $n \geq 0$, precomposition with the homeomorphism $|X \times \Delta ^ n| \rightarrow |X| \times | \Delta ^ n |$ induces a bijection

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Top}}( |X|, Y)_{n} & = & \operatorname{Hom}_{\operatorname{Top}}( |X| \times | \Delta ^ n |, Y) \\ & \simeq & \operatorname{Hom}_{\operatorname{Top}}( |X \times \Delta ^ n|, Y) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X \times \Delta ^ n, \operatorname{Sing}_{\bullet }(Y) ) \\ & = & \operatorname{Fun}(X, \operatorname{Sing}_{\bullet }(Y) )_{n}. \end{eqnarray*}

These bijections are compatible with face and degeneracy operators, and therefore determine an isomorphism of simplicial sets $\operatorname{Hom}_{\operatorname{Top}}( |X|, Y)_{\bullet } \rightarrow \operatorname{Fun}(X, \operatorname{Sing}_{\bullet }(Y) )$.