ScalingStacks

We assume given $\sigma_{ij}:E_{i}E_{j}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E_{j}E_{i}$ for $i\neq j$ such that

(4.3.5)

$$\sigma_{ij}\sigma_{ji}=\operatorname{id}\nolimits\text{ for all }i\neq j\text{ and }E_{3}\sigma_{12}\circ\sigma_{13}E_{2}\circ E_{1}\sigma_{23}=\sigma_{23}E_{1}\circ E_{2}\sigma_{13}\circ\sigma_{12}E_{3}.$$

This ensures that by composing $\sigma$’s, we obtain a transitive system of isomorphisms between $E_{i}E_{j}E_{k}$’s for $\{i,j,k\}=\{1,2,3\}$.