Let ${\mathcal{L}}$ be a model of $K_{X}$ determined on a proper dlt model ${\mathcal{X}}$. We denote by $\Delta({\mathcal{L}})$ the subcomplex of $\Delta({\mathcal{X}})$ such that a face $\sigma$ of $\Delta({\mathcal{X}})$ is in $\Delta({\mathcal{L}})$ if and only if each vertex of $\sigma$ achieves $\min_{i}\kappa(v_{i})$ with $\kappa=A_{X}-\phi_{\mathcal{L}}$.