Construction 4.6.8.11. Let $K$ be a simplicial set. We let $\Sigma (K)$ denote the pushout $(\{ x\} \star K) \coprod _{K} \{ y\} $ (this is a model for the unreduced suspension of $K$). Let $\operatorname{Path}[ \Sigma (K) ]_{\bullet }$ denote the simplicial path category of $\Sigma (K)$ (Notation 2.4.4.2). Then $\operatorname{Path}[ \Sigma (K) ]_{\bullet }$ has exactly two objects, which we denote by $x$ and $y$. We let $\Phi (K)$ denote the simplicial set $\operatorname{Hom}_{ \operatorname{Path}[ \Sigma (K) ]}(x,y)_{\bullet }$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$