Let ${\mathscr{C}}$ be a polyhedral complex in $N_{\mathbb{R}}$. A superform on ${\mathscr{C}}$ is the restriction of a superform on (an open subset of) $N_{\mathbb{R}}$ to $|{\mathscr{C}}|$. This means that two superforms agree if their restrictions to any polyhedron of $|{\mathscr{C}}|$ agree. Let $A({\mathscr{C}})$ be the space of superforms on ${\mathscr{C}}$. It is an alternating algebra with respect to the induced wedge product. We have also differential operators $d$, $d^{\prime}$ and $d^{\prime\prime}$ on $A({\mathscr{C}})$ given by restriction of the corresponding operators on $A(N_{\mathbb{R}})$. Let $A^{p,q}({\mathscr{C}})$ be the space of $(p,q)$-superforms on ${\mathscr{C}}$. The support of $\alpha\in A({\mathscr{C}})$ is the complement of $\{\omega\in|{\mathscr{C}}|\mid\text{$\alpha$ vanishes identically in a neighbourhood of $\omega$}\}$ in $|{\mathscr{C}}|$. We denote by $A_{c}^{p,q}({\mathscr{C}})$ the subspace of $A^{p,q}({\mathscr{C}})$ of superforms of compact support.

Let $N^{\prime}$ be a free abelian group of rank $r^{\prime}$ and let $F:N^{\prime}_{\mathbb{R}}\rightarrow N_{\mathbb{R}}$ be an affine map. Suppose that ${\mathscr{C}}^{\prime}$ is a polyhedral complex of $N^{\prime}_{\mathbb{R}}$ with $F(|{\mathscr{C}}^{\prime}|)\subset|{\mathscr{C}}|$, then the pull-back in 2.3 induces a pull-back $F^{*}:A^{p,q}({\mathscr{C}})\rightarrow A^{p,q}({\mathscr{C}}^{\prime})$.