ScalingStacks

We denote by ${\mathcal{U}}^{\bullet}$ the strict monoidal differential pointed category generated by an object $e$ and a map $\tau\in\operatorname{End}\nolimits(e^{2})$ subject to the relations (4.1.1). Its objects are the $e^{n}$, $n\geq 0$, $\operatorname{Hom}\nolimits(e^{n},e^{m})=0$ for $m\neq n$ and $\operatorname{End}\nolimits(e^{n})=H_{n}^{\bullet}$.

Let ${\mathcal{V}}$ be a differential pointed category.

A $2$-representation on ${\mathcal{V}}$ is the data of a strict monoidal differential pointed functor ${\mathcal{U}}^{\bullet}\to\operatorname{End}\nolimits({\mathcal{V}})$. This is equivalent to the data of an endofunctor $E$ of the differential pointed category ${\mathcal{V}}$ and $\tau\in\operatorname{End}\nolimits(E^{2})$ such that $(E,\tau)$ induce a $2$-representation on $k[{\mathcal{V}}]$.