Definition 1.4.0.1. An $\infty $-category is a simplicial set $S$ which satisfies the following condition:
- $(\ast '')$
For $0 < i < n$, every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow S$ can be extended to an $n$-simplex $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$.