Proposition 3.6.5.2 (0149)

Proposition 3.6.5.2. Let $X$ and $Y$ be simplicial sets. If $Y$ is a Kan complex, then the comparison map

\[ \theta : \operatorname{Fun}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{Top}}( |X|, |Y| )_{\bullet } \]

of Construction 3.6.5.1 is a homotopy equivalence of Kan complexes.

Proof. Using Example 3.6.2.5, we can identify $\theta $ with the morphism

\[ \operatorname{Fun}(X,Y) \rightarrow \operatorname{Fun}( X, \operatorname{Sing}_{\bullet }(|Y| ) ) \]

given by postcomposition with the unit map $u_{Y}: Y \rightarrow \operatorname{Sing}_{\bullet }(|Y|)$. By virtue of Theorem 3.6.4.1, the map $u_{Y}$ is a weak homotopy equivalence. Since $Y$ and $\operatorname{Sing}_{\bullet }(|Y|)$ are Kan complexes, we conclude that $u_{Y}$ is a homotopy equivalence (Proposition 3.1.6.13). It follows that $\theta $ is also a homotopy equivalence (it admits a homotopy inverse, given by postcomposition with any homotopy inverse to $u_{Y}$). $\square$