Example 4.3.4.10. Let $X = \Delta ^{p}$ and $Y = \Delta ^{q}$ be standard simplices. Then the join comparison map $T_{X,Y}: | \Delta ^{p} \star \Delta ^{q} | \rightarrow |\Delta ^{p}| \star | \Delta ^{q} |$ fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ | \Delta ^{p} \star \Delta ^{q} | \ar [dr]^{ | \rho | } \ar [rr]^{ T_{X,Y} } & & | \Delta ^{p} | \star | \Delta ^{q} | \ar [dl]_{ H_{p,q} } \\ & | \Delta ^{p+1+q} |, & } \]
where $\rho : \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}$ denotes the isomorphism of simplicial sets appearing in Example 4.3.3.23 and $H_{p,q}$ is the homeomorphism of Proposition 4.3.4.8. In particular, $T_{X,Y}$ is a homeomorphism.