Proposition 3.5.5.10. Let $n$ be a nonnegative integer and let $X$ be a Kan complex. Then:

- $(1)$
If $X$ is weakly $(n-1)$-coskeletal, then it is an $n$-groupoid.

- $(2)$
If $X$ is an $n$-groupoid, then it is weakly $n$-coskeletal.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.5.5.10. Let $n$ be a nonnegative integer and let $X$ be a Kan complex. Then:

- $(1)$
If $X$ is weakly $(n-1)$-coskeletal, then it is an $n$-groupoid.

- $(2)$
If $X$ is an $n$-groupoid, then it is weakly $n$-coskeletal.

**Proof.**
We first prove $(1)$. Assume that $X$ is weakly $(n-1)$-coskeletal. Suppose we are given an integer $m > n$ and a pair of $m$-simplices $\sigma , \tau : \Delta ^{m} \rightarrow X$ which satisfy $\sigma |_{ \Lambda ^{m}_{i} } = \tau |_{ \Lambda ^{m}_{i} }$ for some $0 \leq i \leq m$; we wish to show that $\sigma = \tau $. Note that $d^{m}_{i}(\sigma )$ and $d^{m}_{i}(\tau )$ are $(m-1)$-simplices of $X$ which coincide on the boundary $\operatorname{\partial \Delta }^{m-1}$. Since $X$ is weakly $(n-1)$-coskeletal, it follows that $d^{m}_{i}(\sigma ) = d^{m}_{i}(\tau )$, so that $\sigma $ and $\tau $ coincide on the boundary $\operatorname{\partial \Delta }^{m}$. Invoking our assumption that $X$ is weakly $(n-1)$-coskeletal again, we conclude that $\sigma = \tau $.

We now prove $(2)$. Suppose that $X$ is an $n$-groupoid; we wish to show that it is weakly $n$-coskeletal. By virtue of Exercise 3.5.4.11, it will suffice to show that for every integer $m > n$, the restriction map

\[ \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, X) \]

is injective. This is clear: our assumption that $X$ is an $n$-groupoid guarantees that the composition of $\theta $ with the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{m}_{0}, X)$ is a bijection. $\square$