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

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

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 $b_{i}$

is equal to

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.

Consider a differential category ${\mathcal{W}}$ endowed with two actions of ${\mathcal{U}}$ given by $(E_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ and a closed morphism of functors $\sigma:E_{2}E_{1}\to E_{1}E_{2}$ such that the following diagrams commute:

We define a differential category ${\mathcal{V}}=\Delta_{\sigma}{\mathcal{W}}$.

$\bullet\$ The objects of ${\mathcal{V}}$ are pairs $(m,\pi)$ where $m\in\overline{{\mathcal{W}}}^{i}$ and $\pi\in Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(E_{2}(m),E_{1}(m))$ such that the following diagram commutes

$\bullet\$ $\operatorname{Hom}\nolimits_{\mathcal{V}}((m,\pi),(m^{\prime},\pi^{\prime}))$ 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 $\omega=\omega_{\sigma}:{\mathcal{V}}\to\overline{{\mathcal{W}}}^{i},\ (m,\pi)\mapsto m$. Note that ${\mathcal{V}}$ is strongly pretriangulated and idempotent-complete.

We define now a differential functor $E:{\mathcal{V}}\to{\mathcal{V}}$.

$\bullet\$ Let $(m,\pi)\in{\mathcal{V}}$. Let $m^{\prime}={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 35.11345pt\hbox{{\hbox{\kern-35.11345pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{E_{2}(m)\oplus E_{1}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-6.76079pt\raise 19.04272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 19.91682pt\raise 8.53578pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}}$. We define