ScalingStacks

Given two $\mathcal{I}$-cubes $V,W$ over a category $\mathcal{B},$ an $\mathcal{I}$-cube map $\psi:V\longrightarrow W$ is a collection of maps

$$\psi(\mathcal{L}):V(\mathcal{L})\longrightarrow W(\mathcal{L}),\hskip 21.68121pt\mbox{ for all }\mathcal{L}\subset\mathcal{I}$$

that make diagrams

$$\begin{CD}V(\mathcal{L})@>{\psi(\mathcal{L})}>{}>W(\mathcal{L})\\ @V{}V{\xi^{V}_{a}(\mathcal{L})}V@V{}V{\xi^{W}_{a}(\mathcal{L})}V\\ V(\mathcal{L}a)@>{\psi(\mathcal{L}a)}>{}>W(\mathcal{L}a)\end{CD}$$

(29)

commutative for all $(\mathcal{L},a)\in r(\mathcal{I}).$ The map $\psi$ is called an isomorphism if $\psi(\mathcal{L})$ is an isomorphism for all $\mathcal{L}\subset\mathcal{I}.$ The map $\psi$ of $\mathcal{I}$-cubes over an abelian category $\mathcal{B}$ is called injective/surjective if $\psi(\mathcal{L})$ is injective/surjective for all $\mathcal{L}\subset\mathcal{I}.$