Kerodon

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

Corollary 4.3.4.12. Let $X$ be a simplicial set. Then the join comparison maps $T_{\Delta ^0,X}$ and $T_{X, \Delta ^0}$ supply homeomorphisms of topological spaces

\[ | X^{\triangleleft } | \simeq |X|^{\triangleleft } \quad \quad | X^{\triangleright } | \simeq | X |^{\triangleright }. \]

Here $X^{\triangleleft }$ and $X^{\triangleright }$ denote the left and right cones on $X$ in the category of simplicial sets (Construction 4.3.3.22), while $|X|^{\triangleleft }$ and $|X|^{\triangleright }$ denote the cone $|X|$ in the category of topological spaces (Example 4.3.4.5).