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

Construction 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 to identify $X$ with its image in the cones $X^{\triangleleft }$ and $X^{\triangleright }$. Moreover, Remark 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.