$S$ is any object in ${\mathcal{C}}$, the *constant* $n$*-cubical ${\mathcal{C}}$-object* $S$ is the contravariant functor $S_{{\square}}:{\square}_n\rightarrow{\mathcal{C}}$ such that $S_I=S$ for all $I\in{\square}_n$, with all morphisms $S_I\rightarrow S_J$ given by the identity of $S$. An *augmentation* of an $n$-cubical ${\mathcal{C}}$-object $K_{{\square}}$ to $S$ is a morphism of $n$-cubical ${\mathcal{C}}$-objects $K_{{\square}}\rightarrow S$. If we replace ${\square}_n$ by ${\triangle}_n$, we obtain *constant $n$-semisimplicial ${\mathcal{C}}$-objects* and augmentations thereof.

The next observations will be useful in what follows:

\[rmk:cubobj\]

1.  If $X_{{\square}}$ is an $n$-cubical ${\mathcal{C}}$-object we can associate to it the augmented $n$-cubical ${\mathcal{C}}$-object ${\varepsilon}:X_{{\square}}\rightarrow X_{\emptyset}$; sometimes we will call this augmentation the *natural augmentation*.

2.  Any $(n+1)$-cubical ${\mathcal{C}}$-object $X_{{\square}}$ can be considered as a morphism $Y_{{\square}}\rightarrow Z_{{\square}}$ of $n$-cubical ${\mathcal{C}}$-objects by setting $Z_I:=X_I$ and $Y_I:=X_{I\cup\{n\}}$ for $I\in{\square}_n$; in particular, a $1$-cubical ${\mathcal{C}}$-object is the datum of two objects $X,Y\in{\mathcal{C}}$ and a morphism $f:X\rightarrow Y$ between them.

3.  To any $(n+1)$-cubical ${\mathcal{C}}$-object $X_{{\square}}$ we can associate functorially an $n$-semisimplicial ${\mathcal{C}}$-object $X_{{\bullet}}$ and an augmentation ${\varepsilon}:X_{{\bullet}}\rightarrow X_{\emptyset}$. We set:

    $$X_k:=\coprod_{|I|=k+1}X_I\hspace{0.5cm}\text{for }k=0,\ldots,n.$$

    Let ${\beta}:[s]\rightarrow [r]$ be a strictly increasing map (in particular $r\geq s$). If $I\in{\square}_{n+1}$ has cardinality $r+1$ we can write it as $I=\{i_0,\ldots,i_r\}$ with $i_0<\dots<i_r$; the set $J:={\beta}(I):=\{i_{{\beta}(0)},\ldots,i_{{\beta}(s)}\}$ is contained in $I$, so we 