ScalingStacks

Consider two actions of ${\mathcal{U}}$ given by $(F_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ on ${\mathcal{W}}$ and a closed morphism of functors $\lambda:F_{1}E_{2}\to E_{2}F_{1}$ such that diagrams (4.2.1) commute. As in §4.2.1, we have maps $\mu_{i,j}=\mu_{(i,i),(j,j)}:E_{2}^{i}F_{1}^{i}E_{2}^{j}F_{1}^{j}\to E_{2}^{i+j}F_{1}^{i+j}$.

We define a differential category $\Delta_{\lambda}{\mathcal{W}}$. Its objects are pairs $(m,\varsigma)$ where $m\in\overline{{\mathcal{W}}}^{i}$ and $\varsigma=(\varsigma_{i})_{i\geq 1}$, $\varsigma_{i}\in Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(E_{2}^{i}F_{1}^{i}(m),m)$, satisfies that

• •

  for all $i,j\geq 1$, we have $\varsigma_{i}\circ E_{2}^{i}F_{1}^{i}\varsigma_{j}=\varsigma_{i+j}\circ\mu_{i,j}$

• •

  $\varsigma_{i}\circ T_{r}F_{1}^{i}=\varsigma_{i}\circ E_{2}^{i}T_{r}$ for all $1\leq r<i$.

We define $\operatorname{Hom}\nolimits_{\Delta_{\lambda}{\mathcal{W}}}((m,\varsigma),(m^{\prime},\varsigma^{\prime}))$ to be the differential submodule of $\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}}^{i}}(m,m^{\prime})$ of elements $f$ such that for all $i\geq 1$, the following diagram commutes

The composition of maps is defined to be that of $\overline{{\mathcal{W}}}^{i}$.

We assume $F_{1}$ has a right adjoint $E_{1}$ and denote by $\varepsilon_{1}$ and $\eta_{1}$ the counit and unit of the adjunction. We denote by $\tau_{1}$ the endomorphism of $E_{1}^{2}$ corresponding by adjunction to the endomorphism $\tau_{1}$ of $F_{1}^{2}$. The pair $(E_{1},\tau_{1})$ provides an action of ${\mathcal{U}}$ on ${\mathcal{W}}$.

We denote by $\sigma$ the composition

and by $\rho$ the composition

The diagram (4.3.1) is commutative.

Let ${\mathcal{M}}$ be the strict monoidal pointed category generated by objects $a_{l}$ for $1\leq l\leq 3$ and maps $\lambda_{lm}:a_{l}a_{m}\to a_{m}a_{l}$ for $l\leq m$ with relations $\lambda_{ll}^{2}=0$ and

Given $l_{1},\ldots,l_{r},m_{1},\ldots,m_{r}\in\{1,2,3\}$ and $w\in{\mathfrak{S}}_{r}$ satisfying the assumptions of Lemma 4.4.6, we put $\lambda_{w}=H^{-1}(T_{w})$.

We denote by ${\mathcal{M}}^{\prime}$ the strict monoidal $k$-linear category obtained from $k[{\mathcal{M}}]$ by adding maps $\varepsilon:a_{1}a_{3}\to 1$ and $\eta:1\to a_{3}a_{1}$ and relations

There is a monoidal duality, i.e. a monoidal equivalence ${\mathcal{M}}^{\prime{\operatorname{opp}\nolimits}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{M}}^{\prime}$ given by

Lemmas 4.4.5 and 4.4.6 show that there is a $k$-linear monoidal functor $R:{\mathcal{M}}^{\prime}\to{\mathcal{W}}$

Given $l_{1},\ldots,l_{r},m_{1},\ldots,m_{r}\in\{1,2,3\}$ and $w\in{\mathfrak{S}}_{r}$ satisfying the assumptions of Lemma 4.4.6, we still denote by $\lambda_{w}$ the element $R(\lambda_{w})$.

Lemma 4.4.7 has the following consequence.

Let $(m,\varsigma)\in\Delta_{\lambda}{\mathcal{W}}$. Let $\pi=\pi(\varsigma)$ be the composition

Note that $\pi$ is also equal to the composition

since $E_{1}\lambda E_{1}F_{1}\circ\eta_{1}E_{2}E_{1}F_{1}\circ E_{2}\eta_{1}=E_{1}E_{2}F_{1}\eta_{1}\circ E_{1}\lambda\circ\eta_{1}E_{2}$ and $E_{1}E_{2}F_{1}\eta_{1}\circ E_{1}E_{2}F_{1}\varepsilon_{1}=\operatorname{id}\nolimits_{E_{1}E_{2}F_{1}}$.

The pair $(m,\pi)$ defines an object of $\Delta_{\sigma}{\mathcal{W}}$. We obtain a faithful differential functor $\Gamma:\Delta_{\lambda}{\mathcal{W}}\to\Delta_{\sigma}{\mathcal{W}},\ (m,\varsigma)\mapsto(m,\pi)$.

We define now a differential functor $E:\Delta_{\lambda}{\mathcal{W}}\to\Delta_{\lambda}{\mathcal{W}}$.

Let $(m,\varsigma)\in\Delta_{\lambda}{\mathcal{W}}$. 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}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 19.91682pt\raise 8.53578pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}}$ where $\pi=\pi(\varsigma_{1})$. Given $i\geq 1$, we define