Definition 4.8.1.1. Let $n$ be a positive integer. We say that a simplicial set $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if it satisfies the following condition for every pair of integers $0 < i < m$:
- $(\ast )$
Every morphism of simplicial sets $\sigma _0: \Lambda ^{m}_{i} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $m$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$. Moreover, if $m > n$, then $\sigma $ is unique.