ScalingStacks

Definition 2 Let $\mathcal{I}$ be a finite set and $\mathcal{B}$ an additive category. A skew-commutative $\mathcal{I}$-cube $V$ over $\mathcal{B}$ is a collection of objects $V(\mathcal{L})\in Ob(\mathcal{B})$ for $\mathcal{L}\subset\mathcal{I}$ and morphisms

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

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

$$\xi^{V}_{b}(\mathcal{L}a)\xi^{V}_{a}(\mathcal{L})+\xi^{V}_{a}(\mathcal{L}b)\xi^{V}_{b}(\mathcal{L})=0.$$