ScalingStacks

Consider two actions of ${\mathcal{U}}$ given by $(F_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ on ${\mathcal{W}}$ and a closed morphism of functors $\lambda:F_{1}E_{2}\to E_{2}F_{1}$ such that diagrams (4.2.1) commute. As in §4.2.1, we have maps $\mu_{i,j}=\mu_{(i,i),(j,j)}:E_{2}^{i}F_{1}^{i}E_{2}^{j}F_{1}^{j}\to E_{2}^{i+j}F_{1}^{i+j}$.

We define a differential category $\Delta_{\lambda}{\mathcal{W}}$. Its objects are pairs $(m,\varsigma)$ where $m\in\overline{{\mathcal{W}}}^{i}$ and $\varsigma=(\varsigma_{i})_{i\geq 1}$, $\varsigma_{i}\in Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(E_{2}^{i}F_{1}^{i}(m),m)$, satisfies that

• •

  for all $i,j\geq 1$, we have $\varsigma_{i}\circ E_{2}^{i}F_{1}^{i}\varsigma_{j}=\varsigma_{i+j}\circ\mu_{i,j}$

• •

  $\varsigma_{i}\circ T_{r}F_{1}^{i}=\varsigma_{i}\circ E_{2}^{i}T_{r}$ for all $1\leq r<i$.

We define $\operatorname{Hom}\nolimits_{\Delta_{\lambda}{\mathcal{W}}}((m,\varsigma),(m^{\prime},\varsigma^{\prime}))$ to be the differential submodule of $\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(m,m^{\prime})$ of elements $f$ such that for all $i\geq 1$, the following diagram commutes

The composition of maps is defined to be that of $\overline{{\mathcal{W}}}^{i}$.