ScalingStacks

Let ${\mathcal{U}}$ be the differential strict monoidal category generated by an object $e$ and a map $\tau:e^{2}\to e^{2}$ subject to the relations

There are isomorphisms of differential monoidal categories ${\operatorname{opp}\nolimits}:{\mathcal{U}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{U}}^{\operatorname{opp}\nolimits}$ and $\mathrm{rev}:{\mathcal{U}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{U}}^{\mathrm{rev}}$ given on generators by $e\mapsto e$ and $\tau\mapsto\tau$.

The following result is clear.

The isomorphism ${\operatorname{opp}\nolimits}:{\mathcal{U}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{U}}^{\operatorname{opp}\nolimits}$ gives rise to the isomorphism of differential algebras

The isomorphism $\mathrm{rev}:{\mathcal{U}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{U}}^{\mathrm{rev}}$ gives rise to the isomorphism of differential algebras

The functor $-\otimes E^{n}$ induces an injective morphism of differential algebras $H_{r}=\operatorname{End}\nolimits(E^{r})\to H_{r+n}=\operatorname{End}\nolimits(E^{r+n}),\ T_{i}\mapsto T_{i}$ and we will identify $H_{r}$ with a subalgebra of $H_{r+n}$ via this morphism.

The functor $E^{n}\otimes-$ induces a morphism of differential algebras

Note that $H_{n}$ commutes with $f_{n}(H_{r})$ and that $f_{n}=\iota_{n+r}\circ\iota_{r}$.

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

The data of a $2$-representation on ${\mathcal{V}}$ is the same as the data of a differential endofunctor $E$ of ${\mathcal{V}}$ and of $\tau=\tau_{E}\in\operatorname{End}\nolimits(E^{2})$ satisfying (4.1.1).

Note that a $2$-representation on ${\mathcal{V}}$ extends to a $2$-representation on $\bar{{\mathcal{V}}}$ and on ${\mathcal{V}}^{i}$ (uniquely up to an equivalence unique up to isomorphism).

A morphism of $2$-representations $({\mathcal{V}},E,\tau)\to({\mathcal{V}}^{\prime},E^{\prime},\tau)$ is the data of a differential functor $\Phi:{\mathcal{V}}\to{\mathcal{V}}^{\prime}$ and of an isomorphism of functors $\varphi:\Phi E\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}\Phi$ (with $d(\varphi)=0$) such that $\tau^{\prime}\Phi\circ E^{\prime}\varphi\circ\varphi E=E^{\prime}\varphi\circ\varphi E\circ\Phi\tau:\Phi E^{2}\to E^{\prime 2}\Phi$.

Let ${\mathcal{V}}$ be a $2$-representation. The opposite $2$-representation is $({\mathcal{V}}\operatorname{\!-diff}\nolimits,E^{\prime},\tau^{\prime})$, where $E^{\prime}(\zeta)=\zeta E$ and $\tau^{\prime}(\zeta)=\zeta\tau\in\operatorname{End}\nolimits(E^{\prime 2}(\zeta))$ for $\zeta\in{\mathcal{V}}\operatorname{\!-diff}\nolimits$. Note that the canonical functor ${\mathcal{V}}\to({\mathcal{V}}\operatorname{\!-diff}\nolimits)\operatorname{\!-diff}\nolimits,\ v\mapsto(\zeta\mapsto\zeta(v))$ is a fully faithful morphism of $2$-representations.

Assume $E$ has a left adjoint $E^{\vee}$. We still denote by $\tau$ the endomorphism of $(E^{\vee})^{2}$ corresponding to $\tau$ (cf §2.1.1). The pair $(E^{\vee},\tau)$ defines the left dual $2$-representation of $(E,\tau)$. Similarly, if $E$ has a right adjoint ${{}^{\vee}E}$, we obtain a right dual $2$-representation $({{}^{\vee}E},\tau)$ of $(E,\tau)$.

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 $k[{\mathcal{V}}]$.