Kerodon

Remark 4.8.1.2. Let $n$ be a positive integer and let $\operatorname{\mathcal{C}}$ be a simplicial set. If $\operatorname{\mathcal{C}}$ is an $(n,1)$-category, then it is an $\infty $-category. Conversely, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then it is an $(n,1)$-category if and only if, for every pair of integers $0 < i < m$ with $m > n$, the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ m, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{m}_{i}, \operatorname{\mathcal{C}})$ is injective.