ScalingStacks

We consider a differential category ${\mathcal{W}}$ together with three actions $(E_{i},\tau_{i})$, $1\leq i\leq 3$ of ${\mathcal{U}}$.

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

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

We also assume the analogs of the diagram (4.3.1) for the map $\sigma_{ij}$ commute.

Let $(m,\pi)\in\Delta_{\sigma_{21}}({\mathcal{W}})$. We define $\pi^{\prime}\in Z\operatorname{Hom}\nolimits(E_{2}E_{3}(m),E_{1}E_{3}(m))$ as the composition $E_{2}E_{3}(m)\xrightarrow{\sigma_{23}}E_{3}E_{2}(m)\xrightarrow{E_{3}(\pi)}E_{3}E_{1}(m)\xrightarrow{\sigma_{31}}E_{1}E_{3}(m)$.

We have a commutative diagram

It follows that $(E_{3}(m),\pi^{\prime})$ defines an object of $\Delta_{\sigma_{21}}({\mathcal{W}})$ and we put ${\tilde{E}}_{3}(m,\pi)=(E_{3}(m),\pi^{\prime})$. Given $f$ a map in $\Delta_{\sigma_{21}}({\mathcal{W}})$, the map $E_{3}(f)$ is actually in $\Delta_{\sigma_{21}}({\mathcal{W}})$ and this defines ${\tilde{E}}_{3}(f)$.

We have defined an endofunctor ${\tilde{E}}_{3}$ of $\Delta_{\sigma_{21}}({\mathcal{W}})$.

There is a commutative diagram

It follows that $\tau_{3}$ defines an endomorphism of ${\tilde{E}}_{3}^{2}$. So, $({\tilde{E}}_{3},\tau_{3})$ defines a $2$-representation on $\Delta_{\sigma_{21}}({\mathcal{W}})$.

Let $E_{21}$ denote the functor ${\tilde{E}}$ of $\Delta_{\sigma_{21}}({\mathcal{W}})$. We have

where

and

where

We have commutative diagrams

So, $\left(\begin{matrix}\sigma_{23}&0\\ 0&\sigma_{13}\end{matrix}\right)$ defines an isomorphism $E_{21}{\tilde{E}}_{3}(m,\pi)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\tilde{E}}_{3}E_{21}(m,\pi)$. It provides an isomorphism of functors $\sigma_{21,3}:E_{21}{\tilde{E}}_{3}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\tilde{E}}_{3}E_{21}$.

Replacing $1$ by $2$, $2$ by $3$ and $3$ by $1$, the construction above provides a $2$-representation $({\tilde{E}}_{1},\tau_{1})$ of $\Delta_{\sigma_{32}}({\mathcal{W}})$ and we denote by $E_{32}$ the endofunctor ${\tilde{E}}$ of $\Delta_{32}({\mathcal{W}})$. We have an isomorphism $\sigma_{32,1}:E_{32}{\tilde{E}}_{1}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\tilde{E}}_{1}E_{32}$.

Let us now define another $2$-representation. The justifications for the constructions below will be given in the proof of Proposition 4.3.12.

We define a differential category $\Delta_{123}({\mathcal{W}})$. Its objects are quadruples $(m,\pi_{21},\pi_{31},\pi_{32})$ where $m\in\overline{{\mathcal{W}}}^{i}$, $\pi_{ij}:E_{i}(m)\to E_{j}(m)$ satisfy

and given $i,j,k,l\in\{1,2,3\}$ with $j-l\geq i-k>0$, we have an equality between maps $E_{i}E_{j}(m)\to E_{k}E_{l}(m)$:

where we put $\sigma_{rr}=\tau_{r}$.

Note that the equality for $(i,j,k,l)=(3,2,2,1)$ is equivalent to the one for $(i,j,k,l)=(2,3,1,2)$.

We define $\operatorname{Hom}\nolimits_{\Delta_{123}({\mathcal{W}})}((m,\pi_{21},\pi_{31},\pi_{32}),(m^{\prime},\pi^{\prime}_{21},\pi^{\prime}_{31},\pi^{\prime}_{32}))$ to be the differential submodule of $\operatorname{Hom}\nolimits_{\mathcal{W}}(m,m^{\prime})$ of maps $f$ such that $E_{j}f\circ\pi_{ij}=\pi^{\prime}_{ij}\circ E_{i}f$ for all $i>j$.

We define a differential endofunctor $E$ of $\Delta_{123}({\mathcal{W}})$ by $E(m,\pi_{21},\pi_{31},\pi_{32})=(m^{\prime},\pi^{\prime}_{21},\pi^{\prime}_{31},\pi^{\prime}_{32})$ where

We define an endomorphism $\tau$ of $E^{2}$ as follows. We have $E^{2}(m,\pi_{21},\pi_{31},\pi_{32})=(m^{\prime\prime},\pi^{\prime\prime}_{21},\pi^{\prime\prime}_{31},\pi^{\prime\prime}_{32})$ where (ignoring differentials)

We define the endomorphism $\tau$ of $m^{\prime\prime}$ by