ScalingStacks

Recall that we have a $(\Delta_{\lambda}{\mathcal{S}}_{M}(Z),\Delta_{\lambda}{\mathcal{S}}_{M}(Z))$-bimodule $E$. Its restriction to a $({\mathcal{S}}_{M}(Z),\Delta_{\lambda}{\mathcal{S}}_{M}(Z))$-bimodule is the cone of $\pi:R_{\xi_{2}^{-}}\otimes_{{\mathcal{S}}_{M}(Z)}\operatorname{Id}\nolimits_{\Delta_{\lambda}{\mathcal{S}}_{M}(Z)}\to R_{\xi_{1}^{-}}\otimes_{{\mathcal{S}}_{M}(Z)}\operatorname{Id}\nolimits_{\Delta_{\lambda}{\mathcal{S}}_{M}(Z)}$.

The $({\mathcal{S}}_{M}(Z),{\mathcal{S}}_{M}(Z_{\xi}))$-bimodule $E^{\prime}=E\circ(1\otimes\Xi^{\prime-1})$ is the cone of the map $u$ defined as follows.

Given $\alpha\in R_{\xi_{2}^{-}}^{\bullet}(T,U)$ with $\alpha_{|U}=\operatorname{id}\nolimits$ and given $\beta\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z_{\xi})}(S,U)$, we have

We construct now an isomorphism between $E^{\prime}$ and the restriction of $R_{\xi^{-}}$ to a $({\mathcal{S}}_{M}(Z),{\mathcal{S}}_{M}(Z_{\xi}))$-bimodule.

We define two morphisms of pointed sets

and

Note that we have an isomorphism of pointed sets