Construction 4.3.3.25. Let $X$ be a simplicial set. We will denote the join $\Delta ^{0} \star X$ by $X^{\triangleleft }$ and refer to it as the *left cone of $X$*. Similarly, we denote the join $X \star \Delta ^0$ by $X^{\triangleright }$ and refer to it as the *right cone of $X$*. We will often abuse notation by using Remark 4.3.3.14 to identify $X$ with its image in the cones $X^{\triangleleft }$ and $X^{\triangleright }$. Moreover, Remark 4.3.3.14 also supplies morphisms of simplicial sets $X^{\triangleleft } \hookleftarrow \Delta ^0 \hookrightarrow X^{\triangleright }$, which we can identify with vertices which we refer to as the *cone points* of $X^{\triangleleft }$ and $X^{\triangleright }$, respectively.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$