ScalingStacks

Definition 1 Let $\mathcal{I}$ be a finite set and $\mathcal{B}$ a category. A commutative $\mathcal{I}$-cube $V$ over $\mathcal{B}$ is a collection of objects $V(\mathcal{L})\in Ob(\mathcal{B})$ for each subset $\mathcal{L}$ of $\mathcal{I},$ morphisms

$$\xi^{V}_{a}(\mathcal{L}):V(\mathcal{L})\longrightarrow V(\mathcal{L}a)$$

(27)

for each $(\mathcal{L},a)\in r(\mathcal{L}),$ such that for each triple $(\mathcal{L},a,b),$ where $\mathcal{L}$ is a subset of $\mathcal{I}$ and $a,b,a\not=b$ are two elements of $\mathcal{I}$ that do not lie in $\mathcal{L},$ there is an equality of morphisms

$$\xi^{V}_{b}(\mathcal{L}a)\xi^{V}_{a}(\mathcal{L})=\xi^{V}_{a}(\mathcal{L}b)\xi^{V}_{b}(\mathcal{L}),$$

(28)

i.e., the following diagram is commutative

$$\begin{CD}V(\mathcal{L})@>{\xi^{V}_{a}(\mathcal{L})}>{}>V(\mathcal{L}a)\\ @V{}V{\xi^{V}_{b}(\mathcal{L})}V@V{}V{\xi^{V}_{b}(\mathcal{L}a)}V\\ V(\mathcal{L}b)@>{\xi_{a}^{V}(\mathcal{L}b)}>{}>V(\mathcal{L}ab)\end{CD}$$