ScalingStacks

A lax bi-$2$-representation on ${\mathcal{V}}$ is a lax monoidal differential functor $E:{\mathcal{U}}\otimes{\mathcal{U}}\to\operatorname{End}\nolimits({\mathcal{V}})$. It corresponds to the data of

• •

  differential endofunctors $E_{i,j}=E(e^{i}\otimes e^{j})$ of ${\mathcal{V}}$

• •

  morphisms of differential algebras $H_{i}\otimes H_{j}\to\operatorname{End}\nolimits(E_{i,j})$

• •

  morphisms of differential functors $\mu_{(i,j),(i^{\prime},j^{\prime})}:E_{i,j}E_{i^{\prime},j^{\prime}}\to E_{i+i^{\prime},j+j^{\prime}}$

such that

1. (1)

   $\mu_{(i,j),(i^{\prime},j^{\prime})}$ is equivariant for the action of $(H_{i}\otimes H_{j})\otimes(H_{i^{\prime}}\otimes H_{j^{\prime}})$, where the action on $E_{i+i^{\prime},j+j^{\prime}}$ is the restriction of the action of $H_{i+i^{\prime}}\otimes H_{j+j^{\prime}}$ via the morphism $(a\otimes b)\otimes(a^{\prime}\otimes b^{\prime})\mapsto af_{i}(a^{\prime})\otimes bf_{j}(b^{\prime})$

2. (2)

   $\mu_{(i+i^{\prime},j+j^{\prime}),(i^{\prime\prime},j^{\prime\prime})}\circ(\mu_{(i,j),(i^{\prime},j^{\prime})}E_{i^{\prime\prime},j^{\prime\prime}})=\mu_{(i,j),(i^{\prime}+i^{\prime\prime},j^{\prime}+j^{\prime\prime})}\circ(E_{i,j}\mu_{(i^{\prime},j^{\prime}),(i^{\prime\prime},j^{\prime\prime})})$.

Consider two actions of ${\mathcal{U}}$ given by $(F_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ on ${\mathcal{V}}$ and a closed morphism of functors $\lambda:F_{1}E_{2}\to E_{2}F_{1}$ such that the following diagrams commute:

(4.2.1)

Define morphisms

$$\lambda_{i,1}=(\lambda F_{1}^{i-1})\circ\cdots\circ(F_{1}^{i-2}\lambda F_{1})\circ(F_{1}^{i-1}\lambda):F_{1}^{i}E_{2}\to E_{2}F_{1}^{i}$$

and

$$\lambda_{i,j}=(E_{2}^{j-1}\lambda_{i,1})\circ\cdots\circ(E_{2}\lambda_{i,1}E_{2}^{j-2})\circ(\lambda_{i,1}E_{2}^{j-1}):F_{1}^{i}E_{2}^{j}\to E_{2}^{j}F_{1}^{i}.$$

We define a lax bi-$2$-representation on ${\mathcal{V}}$ by $E_{i,j}=E_{2}^{i}F_{1}^{j}$. The actions of $H_{i}$ on $E_{2}^{i}$ and $H_{j}$ on $F_{1}^{j}$ provide an action of $H_{i}\otimes H_{j}$ on $E_{i,j}$ and $\mu_{(i,j),(i^{\prime},j^{\prime})}=E_{2}^{i}\lambda_{j,i^{\prime}}F_{1}^{j^{\prime}}$:

$$\mu_{(i,j),(i^{\prime},j^{\prime})}:E_{2}^{i}F_{1}^{j}E_{2}^{i^{\prime}}F_{1}^{j^{\prime}}\xrightarrow{E_{2}^{i}\lambda_{j,i^{\prime}}F_{1}^{j^{\prime}}}E_{2}^{i}E_{2}^{i^{\prime}}F_{1}^{j}F_{1}^{j^{\prime}}=E_{2}^{i+i^{\prime}}F_{1}^{j+j^{\prime}}.$$

Let ${\mathcal{W}}$ be a differential category endowed with a lax action $(E_{i,j})$ of ${\mathcal{U}}^{2}$.

We define a differential category $\Delta_{E}{\mathcal{W}}$.

$\bullet\$ The objects of $\Delta_{E}{\mathcal{W}}$ are pairs $(m,\varsigma)$ where $m\in\overline{{\mathcal{W}}}^{i}$ and $\varsigma\in Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(E_{0,1}E_{1,0}(m),m)$ such that for all $i\geq 1$, there exists $\varsigma_{i}\in Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(E_{i,i}(m),m)$ such that the composition $b_{i}$

(4.2.2)

$$b_{i}:(E_{0,1}E_{1,0})^{i}(m)\xrightarrow{(E_{0,1}E_{1,0})^{i-1}\varsigma}(E_{0,1}E_{1,0})^{i-1}(m)\xrightarrow{(E_{0,1}E_{1,0})^{i-2}\varsigma}\cdots\to E_{0,1}E_{1,0}(m)\xrightarrow{\varsigma}m$$

is equal to

$$(E_{0,1}E_{1,0})^{i}(m)\xrightarrow{{\mathrm{can}}}E_{i,i}(m)\xrightarrow{\varsigma_{i}}m$$

and $\varsigma_{i}\circ(T_{r}\otimes 1)=\varsigma_{i}\circ(1\otimes T_{r})$ for $1\leq r<i$.

$\bullet\$ $\operatorname{Hom}\nolimits_{\Delta_{E}{\mathcal{W}}}((m,\varsigma),(m^{\prime},\varsigma^{\prime}(m))$ is the differential submodule of $\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(m,m^{\prime})$ of elements $f$ such that the following diagram commutes

The composition of maps is defined by restricting that of $\overline{{\mathcal{W}}}^{i}$. So, we have a faithful forgetful functor $\omega:\Delta_{E}{\mathcal{W}}\to\overline{{\mathcal{W}}}^{i},\ (m,\varsigma)\mapsto m$. Note that $\Delta_{E}{\mathcal{W}}$ is strongly pretriangulated and idempotent-complete.