Corollary 4.8.3.14. Let $X$ be a Kan complex and let $n$ be a nonnegative integer. The following conditions are equivalent:
The Kan complex $X$ is $n$-reduced: that is, it has a single $m$-simplex for each $0 \leq m \leq n$ (Definition 3.5.2.8).
The Kan complex $X$ is $(n+1)$-connective and minimal in dimensions $\leq n$.
Proof. Without loss of generality, we may assume that $X$ is nonempty (otherwise, neither $(a)$ nor $(b)$ is satisfied). In this case, $X$ is $n$-reduced if and only if the restriction map $\theta _{m}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, \operatorname{\mathcal{C}})$ is injective for each $m \leq n$. Corollary 4.8.3.14 now follows by combining Proposition 4.8.3.12 (and Remark 4.8.3.13) with the criterion of Proposition 3.5.1.12. $\square$