Kerodon

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

Proposition 3.2.5.4. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes and let $n \geq 0$ be an integer. Then the sequence of pointed sets $\pi _{n+1}(S,s) \xrightarrow { \partial } \pi _{n}(X_ s, x) \rightarrow \pi _{n}(X,x)$ is exact, where $\partial $ is the connecting homomorphism of Construction 3.2.4.3.

Proof of Proposition 3.2.5.4. Fix an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X_ s$ such that $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ is the constant map carrying $\operatorname{\partial \Delta }^{n}$ to the base point $x \in X_ s$. By construction, the homotopy class $[ \sigma ] \in \pi _{n}(X_ s, x)$ belongs to the image of the connecting homomorphism $\partial : \pi _{n+1}(S,s) \rightarrow \pi _{n}(X_ s, x)$ if and only if there exists an $(n+1)$-simplex $\tau : \Delta ^{n+1} \rightarrow S$ such that $\tau |_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $s$ and $\sigma $ is incident to $\tau $, in the sense of Definition 3.2.4.1. This condition is equivalent to the existence of an $(n+1)$-simplex $\widetilde{\tau }: \Delta ^{n+1} \rightarrow X$ satisfying $d_0( \widetilde{\tau } ) = \sigma $ and $d_ i( \widetilde{\tau } )$ is equal to the constant map $e: \Delta ^{n} \rightarrow \{ x\} \subseteq X$ for $1 \leq i \leq n+1$. In other words, it is equivalent to the assertion that the tuple of $n$-simplices of $X$ $( \sigma , e, e, \ldots , e)$ bounds, in the sense of Notation 3.2.3.1. For $n \geq 1$, this is equivalent to the vanishing of the image of $[\sigma ]$ in the homotopy group $\pi _{n}(X,x)$ (Theorem 3.2.2.10). When $n=0$, it is equivalent to the equality $[\sigma ] = [x]$ in $\pi _0(X)$ by virtue of Remark 1.3.6.13. $\square$