Definition 1.2.5.1. Let $S$ be a simplicial set. We will say that $S$ is a Kan complex if it satisfies the following condition:
- $(\ast )$
For every pair of integers $0 \leq i \leq n$ with $n > 0$, every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow S$ can be extended to a map $\sigma : \Delta ^{n} \rightarrow S$. Here $\Lambda ^{n}_{i} \subseteq \Delta ^ n$ denotes the $i$th horn (see Construction 1.2.4.1).