A polyhedral complex ${\mathscr{D}}$ subdivides the polyhedral complex ${\mathscr{C}}$ if they have the same support and if every polyhedron $\Delta$ of ${\mathscr{D}}$ is contained in a polyhedron of ${\mathscr{C}}$. In this case, we say that ${\mathscr{D}}$ is a subdivision of ${\mathscr{C}}$. All our constructions here will be compatible with subdivisions. This is no problem for the definition of superforms on ${\mathscr{C}}$ as they depend only on the support $|{\mathscr{C}}|$.

A weight on a pure dimensional polyhedral complex ${\mathscr{C}}$ is a function $m$ which assigns to every maximal polyhedron $\sigma\in{\mathscr{C}}$ a number $m_{\sigma}\in{\mathbb{Z}}$. Then we get a canonical weight on every subdivision of ${\mathscr{C}}$. For a weighted polyhedral complex $({\mathscr{C}},m)$, only the polyhedra $\Delta\in{\mathscr{C}}$ which are contained in a maximal dimensional $\sigma\in{\mathscr{C}}$ with $m_{\sigma}\neq 0$ are of interest. They form a subcomplex ${\mathscr{D}}$ of ${\mathscr{C}}$ and we define the support of $({\mathscr{C}},m)$ as the support of ${\mathscr{D}}$. The polyhedra of ${\mathscr{C}}\setminus{\mathscr{D}}$ will usually be neglected.