multinet as well as some interesting known results, and it is meant to give the reader an idea of how the combinatorics of ${\overline{{\mathcal{A}}}}$ can affect $\Delta_{{\overline{{\mathcal{A}}}}}$ as well as of how Theorem \[thm:AP\] fits in the picture; it closes with some sparse facts that we shall need in Section 4, which are placed here for lack of a better alternative. Lastly, Section 4 constitutes the proof of Theorem \[thm:AP\].

Cubical hyperresolutions and de Rham cohomology
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1.  The $n$*-semisimplicial category* is the category ${\triangle}_n$ with objects the sets $[m]:=\{0,\ldots,m\}$ for $0\leq m\leq n$ and with morphisms the strictly increasing maps $[m]\rightarrow [m']$. The $n$*-cubical category* is the category ${\square}_n$ with objects the subsets of $[n-1]$ and with $\text{Hom}(I,J)$ consisting of a single element if $I\subset J$ and empty otherwise. We denote by ${\square}_n^*$ the full subcategory of ${\square}_n$ whose objects are the non-empty subsets of $[n-1]$

2.  If ${\mathcal{C}}$ is any category, an $n$*-cubical ${\mathcal{C}}$-object* is a contravariant functor $K_{{\square}}:{\square}_n\rightarrow{\mathcal{C}}$, and morphisms between such objects are morphisms of the corresponding functors; $K^*_{{\square}}$ denotes the restriction of $K_{{\square}}$ to ${\square}_n^*$. Similarly, we can define $n$*-semisimplicial ${\mathcal{C}}$-objects* $K_{{\bullet}}$ and morphisms thereof. We will use the notations $K_I:=K_{{\square}}(I)$ and $K_m:=K_{{\bullet}}([m])$.

3.  If 