Let $\mathscr{X}$ be a dlt model of $X$. To each stratum $Y$ of $\mathscr{X}_{k}$ which is a connected component of $D_{J}$, we associate a simplex:

$$\tau_{Y}=\{w\in\mathbb{R}_{\geqslant 0}^{|J|}\,|\sum_{j\in J}a_{j}w_{j}=1\}.$$

We define the cell complex $\mathcal{D}(\mathscr{X}_{k})$ by the following incidence relations: $\tau_{Y}$ is a face of $\tau_{Y^{\prime}}$ if and only if $Y^{\prime}\subset Y$.