Kerodon

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

Remark 3.2.2.9. Let $n$ be a nonnegative integer. By virtue of Corollary 3.1.6.2, there exists an anodyne morphism $f: \Delta ^ n / \operatorname{\partial \Delta }^ n \rightarrow Q$, where $Q$ is a Kan complex. Let $q \in Q$ denote the image of the base point $q_0$ of $\Delta ^ n / \operatorname{\partial \Delta }^ n$. If $(X,x)$ is a pointed Kan complex, then precomposition with $f$ induces a trivial Kan fibration $\operatorname{Fun}( Q, X ) \rightarrow \operatorname{Fun}( \Delta ^ n / \operatorname{\partial \Delta }^ n, X)$ (Theorem 3.1.3.5), hence also a trivial Kan fibration

\[ \operatorname{Fun}(Q,X) \times _{ \operatorname{Fun}( \{ q\} , X) } \{ x\} \rightarrow \operatorname{Fun}( \Delta ^ n / \operatorname{\partial \Delta }^ n ) \times _{ \operatorname{Fun}( \{ q_0 \} , X) } \{ x\} . \]

Passing to connected components, we see that $f$ induces a bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }}( Q, X ) \simeq \pi _{n}(X,x)$. In other words, the functor $(X,x) \mapsto \pi _{n}(X,x)$ is corepresentable (in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$) by the pointed Kan complex $(Q,q)$ (which can be regarded as a combinatorial incarnation of the $n$-sphere).