Kerodon

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

Definition 5.3.6.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. We will say that $W$ has the two-out-of-six property if it satisfies the following condition:

  • Let $\sigma $ be a $3$-simplex of $\operatorname{\mathcal{C}}$ and, for every pair of integers $0 \leq i < j \leq 3$, let $\sigma _{ji}$ denote the edge of $\operatorname{\mathcal{C}}$ given by $\sigma |_{ \operatorname{N}_{\bullet }( \{ i < j \} )}$. If the edges $\sigma _{20}$ and $\sigma _{31}$ belong to $W$, then the edges $\sigma _{10}$, $\sigma _{21}$, $\sigma _{32}$, and $\sigma _{30}$ also belong to $W$.