Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Definition 1.1.9.1. Let $S_{\bullet }$ be a simplicial set. We will say that $S_{\bullet }$ is a Kan complex if it satisfies the following condition:

$(\ast )$

For $n > 0$ and $0 \leq i \leq n$, any map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow S_{\bullet }$ can be extended to a map $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$. Here $\Lambda ^{n}_{i} \subseteq \Delta ^ n$ denotes the $i$th horn (see Construction 1.1.2.9).