The images under $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ of the active simplices in $\Delta({\mathcal{X}}^{\prime})$ form a simplicial ${\mathbb{Z}}$-subdivision of $\Delta({\mathcal{X}})$. As a consequence, there exists a unique, ${\mathbb{Z}}$-PA map $i_{{\mathcal{X}}^{\prime}{\mathcal{X}}}\colon\Delta({\mathcal{X}})\to\Delta({\mathcal{X}}^{\prime})$ such that $i_{{\mathcal{X}}^{\prime}{\mathcal{X}}}(\Delta({\mathcal{X}}))=A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ and $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\circ i_{{\mathcal{X}}^{\prime}{\mathcal{X}}}=\operatorname{id}$.