Remark 3.5.5.2. In the situation of Definition 3.5.5.1, the assumption that $X$ is a Kan complex already guarantees that the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ m, X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{m}_{i}, X )$ is surjective. Consequently, $X$ is an $n$-groupoid if and only if, whenever $\sigma $ and $\tau $ are simplices of $X$ having dimension $m > n$ which satisfy $\sigma |_{ \Lambda ^{m}_{i} } = \tau |_{ \Lambda ^{m}_{i} }$ for some $0 \leq i \leq m$, we have $\sigma = \tau $.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$