We can now state the main result we shall need on cubical hyperresolutions:

\[thm:iperrescub\] Any variety $X$ admits an $(n+1)$-cubical hyperresolution $X_{{\square}}$.

A full proof can be found in [@GNPP Théorème I.2.15] or in [@PeSt Thereom 5.26]; here we are only interested in sketching how such a cubical hyperresolution can be constructed.

-   Take a resolution $\pi:\tilde{X}\rightarrow X$ of $X$ and consider the $2$-cubical variety $X^{(1)}_{{\square}}$ given by the associated resolution square:

    $$X^{(1)}_{\emptyset}:=X,\hspace{0.5cm}X^{(1)}_{\{0\}}:=\tilde{X},\hspace{0.5cm}X^{(1)}_{\{1\}}:=D,\hspace{0.5cm}X^{(1)}_{\{0,1\}}:=\pi_1^{-1}(D).$$

    $X^{(1)}_{{\square}}$ can be seen as a morphism of $1$-cubical varieties $f^{(1)}:Y_{{\square}}^{(1)}\rightarrow Z_{{\square}}^{(1)}$, with $Z_I$ smooth for $I\neq\emptyset$.

-   Consider a resolution $\pi_2:\tilde{Y}_{{\square}}^{(1)}\rightarrow Y_{{\square}}^{(1)}$ and the corresponding resolution square; we obtain the diagram

    $$\xymatrixcolsep{1.5cm}\xymatrixrowsep{1.5cm}\xymatrix{
    E_{\square}^{(1)} \ar[r] \ar[d] & \tilde{Y}_{\square}^{(1)} \ar[d]^{\pi_2} \ar[rd]^{f^{(1)}\circ\pi_2} &\\
    D_{\square}^{(1)} \ar[r]        & Y_{\square}^{(1)} \ar[r]^{f^{(1)}} & Z_{\square}^{(1)}.}$$ The outer commutative square of $1$-cubical varieties can be considered as a $3$-cubical variety $X_{\square}^{(2)}$ i.e. as a morphism of $2$-cubical varieties $f^{(2)}:Y_{{\square}}^{(2)}\rightarrow Z_{{\square}}^{(2)}$, with $Z_I$ smooth for $I\neq\emptyset$.

-   Repeat the previous step enough times.

Observe that if we take for ${\mathcal{C}}$ the category of $n$-cubical varieties and consider $X_{{\square}}\in{\mathcal{C}}$, we can still apply the construction of Theorem \[thm:iperrescub\] to $X_{{\square}}$: at each step we obtain an $m$-cubical variety whose entries are $n$-cubical varieties. More precisely, Theorem \[thm:iperrescub\] 