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:

Define morphisms

and

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