Subsection 4.3. (0PFK) · ScalingStacks

4.3. Diagonal action

4.3.1. Category

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:

(4.3.1)

0P5N

Remark 4.3.1. The data of $\sigma$ and the relations can be described graphically as follows:

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

(4.3.2)

$\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.

0P5P

Remark 4.3.2. The structure of objects and maps in ${\mathcal{V}}$ can be described graphically as follows:

0P5Q

Remark 4.3.3. Assume $E_{1}$ admits a left adjoint $F_{1}$ . The data of the map $\sigma:E_{2}E_{1}\to E_{1}E_{2}$ corresponds by adjunction to the data of a map

\lambda:F_{1}E_{2}\xrightarrow{\bullet\eta_{1}}F_{1}E_{2}E_{1}F_{1}\xrightarrow{F_{1}\sigma F_{1}}F_{1}E_{1}E_{2}F_{1}\xrightarrow{\varepsilon_{1}\bullet}E_{2}F_{1}.

The commutativity of the diagrams (4.3.1) is equivalent to the commutativity of the diagrams (4.2.1). Assume the diagrams commute. We obtain a lax bi- $2$ -representation $(E_{i,j})$ on ${\mathcal{W}}$ (cf §4.2.1).

Let $(m,\varsigma)\in\Delta_{E}{\mathcal{W}}$ . We have an adjunction isomorphism

\phi:\operatorname{Hom}\nolimits(E_{2}(m),E_{1}(m))\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits(F_{1}E_{2}(m),m).

Let $\pi=\phi^{-1}(\varsigma)\in Z\operatorname{Hom}\nolimits(E_{2}(m),E_{1}(m))$ . The object $(m,\pi)$ is in $\Delta_{\sigma}{\mathcal{W}}$ and $(m,\varsigma)\mapsto(m,\pi)$ defines a fully faithful functor of differential categories $\Delta_{E}{\mathcal{W}}\to\Delta_{\sigma}{\mathcal{W}}$ .

Assume now $\lambda$ is invertible. The canonical map $f_{i}:(E_{0,1}E_{1,0})^{i}\to E_{i,i}$ is invertible. Let $\varsigma_{i}=b_{i}\circ f_{i}^{-1}$ . Consider $r\in\{1,\ldots,i-1\}$ . We have

	$\displaystyle\varsigma_{i}\circ(T_{r}\otimes 1)$	$\displaystyle=b_{r-1}\circ(E_{01}E_{10})^{r-1}(\varsigma_{2}\circ(T_{1}\otimes 1)\circ f_{2})\circ(E_{01}E_{10})^{r+1}b_{i-r-1}\circ f_{i}^{-1}$
		$\displaystyle=b_{r-1}\circ(E_{01}E_{10})^{r-1}(\varsigma_{2}\circ(1\otimes T_{1})\circ f_{2})\circ(E_{01}E_{10})^{r+1}b_{i-r-1}\circ f_{i}^{-1}$
		$\displaystyle=\varsigma_{i}\circ(1\otimes T_{r})$

As a consequence, the functor above is an isomorphism of differential categories $\Delta_{E}{\mathcal{W}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\sigma}{\mathcal{W}}$ .

4.3.2. $1$ -arrows

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

\pi^{\prime}=\left(\begin{matrix}\sigma\circ E_{2}\pi\circ\tau_{2}&\sigma\\ 0&\tau_{1}\circ E_{1}\pi\circ\sigma\end{matrix}\right):E_{2}(m^{\prime})\to E_{1}(m^{\prime})

0P5R

Remark 4.3.4. The graphical description of $\pi^{\prime}$ is the following:

0P5S

Lemma 4.3.5. $(m^{\prime},\pi^{\prime})$ is an object of ${\mathcal{V}}$ .

0P5T

Proof. Note that $d(\pi^{\prime})=0$ .

Let $a=\tau_{1}\circ E_{1}\pi^{\prime}\circ\sigma(m^{\prime})\circ E_{2}\pi^{\prime}$ and $b=E_{1}\pi^{\prime}\circ\sigma(m^{\prime})\circ E_{2}\pi^{\prime}\circ\tau_{2}$ . We have

	$\displaystyle a_{11}$	$\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}$
		$\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}$
		$\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}$
		$\displaystyle=E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}$
		$\displaystyle=E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}$
		$\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}$
		$\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}$
		$\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}=b_{11},$

	$\displaystyle a_{12}$	$\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma+\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma$
		$\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}+\tau_{1}^{2}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma$
		$\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}$
		$\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}^{2}E_{1}+E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}$
		$\displaystyle=E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}+E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}=b_{12},$

a_{21}=0=b_{21}\text{ and }

	$\displaystyle a_{22}$	$\displaystyle=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma$
		$\displaystyle=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma$
		$\displaystyle=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma$
		$\displaystyle=E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma$
		$\displaystyle=E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma$
		$\displaystyle=E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}$
		$\displaystyle=E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}$
		$\displaystyle=E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}=b_{22}.$

The lemma follows. ∎

0P5U

Remark 4.3.6. The equalities established in the proof of the lemma above have the following graphical description:

We put $E(m,\pi)=(m^{\prime},\pi^{\prime})$ .

$\bullet\$ Given $f\in\operatorname{Hom}\nolimits_{{\mathcal{V}}}((m,\pi),(\tilde{m},\tilde{\pi}))$ , we put $E(f)=\left(\begin{matrix}E_{2}f&0\\ 0&E_{1}f\end{matrix}\right)$ :

0P5V

Lemma 4.3.7. We have $E(f)\in\operatorname{Hom}\nolimits_{{\mathcal{V}}}(E(m,\pi),E(\tilde{m},\tilde{\pi}))$ . The construction makes $E$ into a differential endofunctor of ${\mathcal{V}}$ .

0P5W

Proof. The lemma follows from the commutativity of the following diagram:

∎

4.3.3. $2$ -arrows

We assume in §4.3.3 that $\sigma$ is invertible.

We define an endomorphism $\tau$ of $\omega E^{2}$ . Let $(m,\pi)\in{\mathcal{V}}$ . We have $E^{2}(m,\pi)=(m^{\prime\prime},\pi^{\prime\prime})$ where $m^{\prime\prime}=[E_{2}^{2}(m)\oplus E_{2}E_{1}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m),\partial]$ and

\partial=\left(\begin{matrix}0\\ E_{2}\pi&0\\ \sigma\circ E_{2}\pi\circ\tau_{2}&\sigma&0\\ 0&\tau_{1}\circ E_{1}\pi\circ\sigma&E_{1}\pi&0\end{matrix}\right).

We define an endomorphism $\tau$ of $m^{\prime\prime}$ by

(4.3.3)

\tau=\left(\begin{matrix}\tau_{2}&0&0&0\\ 0&0&\sigma^{-1}&0\\ 0&0&0&0\\ 0&0&0&\tau_{1}\end{matrix}\right).

0P5X

Theorem 4.3.8. The endomorphism $\tau$ of $m^{\prime\prime}$ defines an endomorphism of $E^{2}$ . The data $(\Delta_{\sigma}{\mathcal{W}},E,\tau)$ is an idempotent-complete strongly pretriangulated $2$ -representation.

0P5Y

Proof. The non-zero coefficients of $\pi^{\prime\prime}$ are

	$\displaystyle\pi^{\prime\prime}_{11}$	$\displaystyle=\sigma E_{2}\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}$
	$\displaystyle\pi^{\prime\prime}_{22}$	$\displaystyle=\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}$
	$\displaystyle\pi^{\prime\prime}_{33}$	$\displaystyle=\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}$
	$\displaystyle\pi^{\prime\prime}_{44}$	$\displaystyle=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}$

\pi^{\prime\prime}_{12}=\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1},\ \pi^{\prime\prime}_{13}=\sigma E_{2},\ \pi^{\prime\prime}_{24}=\sigma E_{1},\ \pi^{\prime\prime}_{34}=\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}.

Let $a=E_{1}\tau\circ\pi^{\prime\prime}$ and $b=\pi^{\prime\prime}\circ E_{2}\tau$ . We have

	$\displaystyle a_{11}$	$\displaystyle=\sigma E_{2}\circ E_{2}\sigma\circ E_{1}\tau_{2}\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}=\sigma E_{2}\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}$
		$\displaystyle=\sigma E_{2}\circ E_{2}\sigma\circ E_{2}^{2}\pi\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}=b_{11}$

a_{12}=E_{1}\tau_{2}\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}=0=b_{12}

a_{13}=E_{1}\tau_{2}\circ\sigma E_{2}=b_{13}

	$\displaystyle a_{23}$	$\displaystyle=E_{1}\sigma^{-1}\circ\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ E_{1}\tau_{2}\circ\sigma E_{2}$
		$\displaystyle=E_{1}\sigma^{-1}\circ\tau_{1}E_{2}\circ E_{1}\sigma\circ E_{1}E_{2}\pi\circ\sigma E_{2}\circ E_{2}\sigma\circ\tau_{2}E_{1}\circ E_{2}\sigma^{-1}$
		$\displaystyle=E_{1}\sigma^{-1}\circ\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{2}\circ E_{2}E_{1}\pi\circ\circ E_{2}\sigma\circ\tau_{2}E_{1}\circ E_{2}\sigma^{-1}$
		$\displaystyle=\sigma E_{1}\circ E_{2}\tau_{1}\circ E_{2}E_{1}\pi\circ E_{2}\sigma\circ\tau_{2}E_{1}\circ E_{2}\sigma^{-1}=b_{23}$

a_{24}=E_{1}\sigma^{-1}\circ\tau_{1}E_{2}\circ E_{1}\sigma\circ\sigma E_{1}=\sigma E_{1}\circ E_{2}\tau_{1}=b_{24}

	$\displaystyle a_{44}$	$\displaystyle=E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}$
		$\displaystyle=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ E_{1}^{2}\pi\circ E_{1}\sigma\circ\sigma E_{1}\circ E_{2}\tau_{1}=b_{44}$

All the other coefficients of $a$ and $b$ vanish. We deduce that $a=b$ , hence $\tau$ is an endomorphism of $E^{2}(m,\pi)$ . It follows easily that $\tau$ defines an endomorphism of $E^{2}$ .

We have $\tau^{2}=0$ and

	$\displaystyle d(\tau)$	$\displaystyle=\left(\begin{matrix}d(\tau_{2})&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&d(\tau_{1})\end{matrix}\right)+\tau\circ\partial+\partial\circ\tau$
		$\displaystyle=\left(\begin{matrix}\operatorname{id}\nolimits&&&\\ 2E_{2}\pi\circ\tau_{2}&\operatorname{id}\nolimits&&\\ \sigma\circ E_{2}\pi\circ\tau_{2}^{2}&&\operatorname{id}\nolimits\\ &\tau_{1}^{2}\circ E_{1}\pi&2\tau_{1}\circ E_{1}\pi\circ\sigma&\operatorname{id}\nolimits\end{matrix}\right)$
		$\displaystyle=\operatorname{id}\nolimits.$

We have $E^{3}(m,\pi)=([m^{\prime\prime\prime},\delta^{\prime}],\pi^{\prime\prime\prime})$ , where

m^{\prime\prime\prime}=E_{2}^{3}(m)\oplus E_{2}^{2}E_{1}(m)\oplus E_{2}E_{1}E_{2}(m)\oplus E_{2}E_{1}^{2}(m)\oplus E_{1}E_{2}^{2}(m)\oplus E_{1}E_{2}E_{1}(m)\oplus E_{1}^{2}E_{2}(m)\oplus E_{1}^{3}(m).

We have

\tau E=\left(\begin{matrix}\tau_{2}E_{2}&0&0&0&0&0&0&0\\ 0&\tau_{2}E_{1}&0&0&0&0&0&0\\ 0&0&0&0&\sigma^{-1}E_{2}&0&0&0\\ 0&0&0&0&0&\sigma^{-1}E_{1}&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&\tau_{1}E_{2}&0\\ 0&0&0&0&0&0&0&\tau_{1}E_{1}\end{matrix}\right)

and

E\tau=\left(\begin{matrix}E_{2}\tau_{2}&0&0&0&0&0&0&0\\ 0&0&E_{2}\sigma^{-1}&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&E_{2}\tau_{1}&0&0&0&0\\ 0&0&0&0&E_{1}\tau_{2}&0&0&0\\ 0&0&0&0&0&0&E_{1}\sigma^{-1}&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&E_{1}\tau_{1}\end{matrix}\right)

Let $a=(E\tau)\circ(\tau E)\circ(E\tau)$ and $b=(\tau E)\circ(E\tau)\circ(\tau E)$ . We have

a_{11}=E_{2}\tau_{2}\circ\tau_{2}E_{2}\circ E_{2}\tau_{2}=\tau_{2}E_{2}\circ E_{2}\tau_{2}\circ\tau_{2}E_{2}=b_{11}

a_{44}=E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}\tau_{1}=\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ\tau_{1}E_{1}=b_{44}

a_{25}=E_{2}\sigma^{-1}\circ\sigma^{-1}E_{2}\circ E_{1}\tau_{2}=\tau_{2}E_{1}\circ E_{2}\sigma^{-1}\circ\sigma^{-1}E_{2}=b_{25}

a_{47}=E_{2}\tau_{1}\circ\sigma^{-1}E_{1}\circ E_{1}\sigma^{-1}=\sigma^{-1}E_{1}\circ E_{1}\sigma^{-1}\circ\tau_{1}E_{2}=b_{47}

and all the other coefficients of $a$ and $b$ vanish. It follows that $a=b$ . This completes the proof of the theorem. ∎

4.3.4. Functoriality

We consider two differential categories ${\mathcal{W}}$ and ${\mathcal{W}}^{\prime}$ endowed with actions $(E_{i},\tau_{i})$ and $(E^{\prime}_{i},\tau^{\prime}_{i})$ of ${\mathcal{U}}$ for $i\in\{1,2\}$ and closed morphisms of functors $\sigma:E_{2}E_{1}\to E_{1}E_{2}$ and $\sigma^{\prime}:E^{\prime}_{2}E^{\prime}_{1}\to E^{\prime}_{1}E^{\prime}_{2}$ making (4.3.1) and the similar diagram for $\sigma^{\prime}$ commute.

Let $\Phi:{\mathcal{W}}\to{\mathcal{W}}^{\prime}$ be a differential functor and $\varphi_{i}:\Phi E_{i}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}_{i}\Phi$ be closed isomorphisms of functors making $(\Phi,\varphi_{i})$ into morphisms of $2$ -representations for $i\in\{1,2\}$ . Assume

(4.3.4)

(E^{\prime}_{1}\varphi_{2})\circ(\varphi_{1}E_{2})\circ(\Phi\sigma)=(\sigma^{\prime}\Phi)\circ(E^{\prime}_{2}\varphi_{1})\circ(\varphi_{2}E_{1}):\Phi E_{2}E_{1}\to E^{\prime}_{1}E^{\prime}_{2}\Phi.

0P5Z

Proposition 4.3.9. There is a differential functor $\Delta\Phi:\Delta_{\sigma}{\mathcal{W}}\to\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}$ given by $(m,\pi)\mapsto(\Phi(m),\varphi_{1}(m)\circ\Phi(\pi)\circ\varphi_{2}(m)^{-1})$ .

There is a closed isomorphism of functors

\varphi=\left(\begin{matrix}\varphi_{2}\\ &\varphi_{1}\end{matrix}\right):\Delta\Phi E\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}\Delta\Phi.

If $\sigma$ and $\sigma^{\prime}$ are invertible, then $(\Delta\Phi,\varphi)$ defines a morphism of $2$ -representations $\Delta_{\sigma}{\mathcal{W}}\to\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}$ .

0P60

Proof. Let $(m,\pi)$ be an object of $\Delta_{\sigma}{\mathcal{W}}$ . Let $\pi^{\prime}=\varphi_{1}(m)\circ\Phi(\pi)\circ\varphi_{2}^{-1}(m)$ , an element of $Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}^{\prime}}^{i}}(E^{\prime}_{2}\Phi(m),E^{\prime}_{1}\Phi(m))$ .

We have

(E^{\prime}_{1}\pi^{\prime})\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\pi^{\prime})=

		$\displaystyle=(E^{\prime}_{1}\varphi_{1}(m))\circ(E^{\prime}_{1}\Phi\pi)\circ(E^{\prime}_{1}\varphi_{2}^{-1}(m))\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\varphi_{1}(m))\circ(E^{\prime}_{2}\Phi\pi)\circ(E^{\prime}_{2}\varphi_{2}^{-1}(m))$
		$\displaystyle=(E^{\prime}_{1}\varphi_{1}(m))\circ(E^{\prime}_{1}\Phi\pi)\circ(\varphi_{1}(E_{2}(m)))\circ(\Phi\sigma(m))\circ(\varphi_{2}^{-1}(E_{1}(m)))\circ(E^{\prime}_{2}\Phi\pi)\circ(E^{\prime}_{2}\varphi_{2}^{-1}(m))$
		$\displaystyle=(E^{\prime}_{1}\varphi_{1}(m))\circ(\varphi_{1}(E_{1}(m)))\circ\Phi\bigl((E_{1}\pi)\circ\sigma(m)\circ(E_{2}\pi)\bigr)\circ(\varphi_{2}^{-1}(E_{2}(m)))\circ(E^{\prime}_{2}\varphi_{2}^{-1}(m))$

It follows that

(E^{\prime}_{1}\pi^{\prime})\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\pi^{\prime})\circ(\tau^{\prime}_{2}\Phi(m))=(\tau^{\prime}_{1}\Phi(m))\circ(E^{\prime}_{1}\pi^{\prime})\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\pi^{\prime}),

hence $(\Phi(m),\pi^{\prime})$ is an object of $\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}$ . We put $\Delta\Phi(m,\pi)=(\Phi(m),\pi^{\prime})$ .

Let $f\in\operatorname{Hom}\nolimits_{\Delta_{\sigma}{\mathcal{W}}}((m,\pi),(\tilde{m},\tilde{\pi}))$ . We have a commutative diagram

and it follows that $\Phi(f)\in\operatorname{Hom}\nolimits_{\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}}(\Delta\Phi(m,\pi),\Delta\Phi(\tilde{m},\tilde{\pi}))$ . We put $(\Delta\Phi)(f)=\Phi(f)$ . This makes $\Delta\Phi$ into a differential functor $\Delta_{\sigma}{\mathcal{W}}\to\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}$ .

We have

(\Delta\Phi)(E(m,\pi))=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 50.11348pt\hbox{{\hbox{\kern-50.11348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{\Phi(E_{2}(m))\oplus\Phi(E_{1}(m))}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-11.31735pt\raise 21.03578pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\Phi(\pi)}$}}}\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}}}}},\beta),

\beta=\left(\begin{matrix}\varphi_{1}E_{2}\circ\Phi\sigma\circ\Phi E_{2}\pi\circ\Phi\tau_{2}\circ\varphi_{2}^{-1}E_{2}&\varphi_{1}E_{2}\circ\Phi\sigma\circ\varphi_{2}^{-1}E_{1}\\ 0&\varphi_{1}E_{1}\circ\Phi\tau_{1}\circ\Phi E_{1}\pi\circ\Phi\sigma\circ\varphi_{2}^{-1}E_{1}\end{matrix}\right)

and

E^{\prime}((\Delta\Phi)(m,\pi))=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 50.11348pt\hbox{{\hbox{\kern-50.11348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.50891pt\hbox{$\textstyle{E^{\prime}_{2}(\Phi(m))\oplus E^{\prime}_{1}(\Phi(m))}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-41.0553pt\raise 21.53079pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.24501pt\hbox{$\scriptstyle{\varphi_{1}(m)\circ\Phi(\pi)\circ\varphi_{2}(m)^{-1}}$}}}\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}}}}},\beta^{\prime}),

\beta^{\prime}=\left(\begin{matrix}\sigma^{\prime}\Phi\circ E^{\prime}_{2}(\varphi_{1}\circ\Phi\pi\circ\varphi_{2}^{-1})\circ\tau^{\prime}_{2}\Phi&\sigma^{\prime}\Phi\\ 0&\tau^{\prime}_{1}\Phi\circ E^{\prime}_{1}(\varphi_{1}\circ\Phi\pi\circ\varphi_{2}^{-1})\circ\sigma^{\prime}\Phi\end{matrix}\right))

We have

\beta^{\prime}\left(\begin{matrix}E^{\prime}_{2}\varphi_{2}&0\\ 0&E^{\prime}_{2}\varphi_{1}\end{matrix}\right)=\left(\begin{matrix}E^{\prime}_{1}\varphi_{2}&0\\ 0&E^{\prime}_{1}\varphi_{1}\end{matrix}\right)\beta,

hence $\left(\begin{matrix}\varphi_{2}(m)\\ &\varphi_{1}(m)\end{matrix}\right)$ defines a closed isomorphism $\Delta\Phi(E(m,\pi))\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}(\Delta\Phi(m,\pi))$ . The naturality of $\varphi_{1}$ and $\varphi_{2}$ implies immediately that of $\varphi$ .

We have $\tau^{\prime}_{i}\Phi\circ E^{\prime}_{i}\varphi_{i}\circ\varphi_{i}E_{i}=E^{\prime}_{i}\varphi_{i}\circ\varphi_{i}E_{i}\circ\Phi\tau_{i}$ for $i\in\{1,2\}$ . Together with (4.3.4), it follows that $\tau^{\prime}(\Delta\Phi)\circ E^{\prime}\varphi\circ\varphi E=E^{\prime}\varphi\circ\varphi E\circ(\Delta\Phi)\tau$ , hence $(\Delta\Phi,\varphi)$ defines a morphism of $2$ -representations. ∎

0P61

Remark 4.3.10. The data of $\varphi_{1}$ and $\varphi_{2}$ , the relations they are required to satisfy, and the map $\pi^{\prime}$ in the proof of the proposition are described graphically as:

The following proposition is immediate.

0P62

Proposition 4.3.11. If $\Phi$ is faithful, then $\Delta\Phi$ is faithful.

4.3.5. Associativity

We consider a differential category ${\mathcal{W}}$ together with three actions $(E_{i},\tau_{i})$ , $1\leq i\leq 3$ of ${\mathcal{U}}$ .

We assume given $\sigma_{ij}:E_{i}E_{j}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E_{j}E_{i}$ for $i\neq j$ such that

(4.3.5)

\sigma_{ij}\sigma_{ji}=\operatorname{id}\nolimits\text{ for all }i\neq j\text{ and }E_{3}\sigma_{12}\circ\sigma_{13}E_{2}\circ E_{1}\sigma_{23}=\sigma_{23}E_{1}\circ E_{2}\sigma_{13}\circ\sigma_{12}E_{3}.

This ensures that by composing $\sigma$ ’s, we obtain a transitive system of isomorphisms between $E_{i}E_{j}E_{k}$ ’s for $\{i,j,k\}=\{1,2,3\}$ .

We also assume the analogs of the diagram (4.3.1) for the map $\sigma_{ij}$ commute.

Let $(m,\pi)\in\Delta_{\sigma_{21}}({\mathcal{W}})$ . We define $\pi^{\prime}\in Z\operatorname{Hom}\nolimits(E_{2}E_{3}(m),E_{1}E_{3}(m))$ as the composition $E_{2}E_{3}(m)\xrightarrow{\sigma_{23}}E_{3}E_{2}(m)\xrightarrow{E_{3}(\pi)}E_{3}E_{1}(m)\xrightarrow{\sigma_{31}}E_{1}E_{3}(m)$ .

We have a commutative diagram

It follows that $(E_{3}(m),\pi^{\prime})$ defines an object of $\Delta_{\sigma_{21}}({\mathcal{W}})$ and we put ${\tilde{E}}_{3}(m,\pi)=(E_{3}(m),\pi^{\prime})$ . Given $f$ a map in $\Delta_{\sigma_{21}}({\mathcal{W}})$ , the map $E_{3}(f)$ is actually in $\Delta_{\sigma_{21}}({\mathcal{W}})$ and this defines ${\tilde{E}}_{3}(f)$ .

We have defined an endofunctor ${\tilde{E}}_{3}$ of $\Delta_{\sigma_{21}}({\mathcal{W}})$ .

There is a commutative diagram

It follows that $\tau_{3}$ defines an endomorphism of ${\tilde{E}}_{3}^{2}$ . So, $({\tilde{E}}_{3},\tau_{3})$ defines a $2$ -representation on $\Delta_{\sigma_{21}}({\mathcal{W}})$ .

Let $E_{21}$ denote the functor ${\tilde{E}}$ of $\Delta_{\sigma_{21}}({\mathcal{W}})$ . We have

{\tilde{E}}_{3}E_{21}(m,\pi)=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 47.55789pt\hbox{{\hbox{\kern-47.55789pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{E_{3}E_{2}(m)\oplus E_{3}E_{1}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-13.54237pt\raise 21.03578pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{E_{3}(\pi)}$}}}\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}}}}},\pi^{\prime}),

where

\pi^{\prime}=\left(\begin{matrix}\sigma_{31}E_{2}\circ E_{3}(\sigma_{21}\circ E_{2}\pi\circ\tau_{2})\circ\sigma_{23}E_{2}&\sigma_{31}E_{2}\circ E_{3}\sigma_{21}\circ\sigma_{23}E_{1}\\ 0&\sigma_{31}E_{1}\circ E_{3}(\tau_{1}\circ E_{1}\pi\circ\sigma_{21})\circ\sigma_{23}E_{1}\end{matrix}\right)

and

E_{21}{\tilde{E}}_{3}(m,\pi)=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 47.55789pt\hbox{{\hbox{\kern-47.55789pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{E_{2}E_{3}(m)\oplus E_{1}E_{3}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-29.57921pt\raise 21.03578pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\sigma_{31}\circ E_{3}(\pi)\circ\sigma_{23}}$}}}\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}}}}},\pi^{\prime\prime})

where

\pi^{\prime\prime}=\left(\begin{matrix}\sigma_{21}E_{3}\circ E_{2}(\sigma_{31}\circ E_{3}(\pi)\circ\sigma_{23})\circ\tau_{2}E_{3}&\sigma_{21}E_{3}\\ 0&\tau_{1}E_{3}\circ\sigma_{13}E_{1}\circ E_{1}E_{3}\pi\circ E_{1}\sigma_{23}\circ\sigma_{21}E_{3}\end{matrix}\right)\bigl).

We have commutative diagrams

So, $\left(\begin{matrix}\sigma_{23}&0\\ 0&\sigma_{13}\end{matrix}\right)$ defines an isomorphism $E_{21}{\tilde{E}}_{3}(m,\pi)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\tilde{E}}_{3}E_{21}(m,\pi)$ . It provides an isomorphism of functors $\sigma_{21,3}:E_{21}{\tilde{E}}_{3}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\tilde{E}}_{3}E_{21}$ .

Replacing $1$ by $2$ , $2$ by $3$ and $3$ by $1$ , the construction above provides a $2$ -representation $({\tilde{E}}_{1},\tau_{1})$ of $\Delta_{\sigma_{32}}({\mathcal{W}})$ and we denote by $E_{32}$ the endofunctor ${\tilde{E}}$ of $\Delta_{32}({\mathcal{W}})$ . We have an isomorphism $\sigma_{32,1}:E_{32}{\tilde{E}}_{1}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\tilde{E}}_{1}E_{32}$ .

Let us now define another $2$ -representation. The justifications for the constructions below will be given in the proof of Proposition 4.3.12.

We define a differential category $\Delta_{123}({\mathcal{W}})$ . Its objects are quadruples $(m,\pi_{21},\pi_{31},\pi_{32})$ where $m\in\overline{{\mathcal{W}}}^{i}$ , $\pi_{ij}:E_{i}(m)\to E_{j}(m)$ satisfy

d(\pi_{21})=d(\pi_{32})=0,\ d(\pi_{31})=\pi_{21}\circ\pi_{32}

and given $i,j,k,l\in\{1,2,3\}$ with $j-l\geq i-k>0$ , we have an equality between maps $E_{i}E_{j}(m)\to E_{k}E_{l}(m)$ :

\sigma_{lk}\circ E_{l}\pi_{ik}\circ\sigma_{il}\circ E_{i}\pi_{jl}+E_{k}\pi_{jl}\circ\sigma_{jk}\circ E_{j}\pi_{ik}\circ\sigma_{ij}+\delta_{jk}E_{j}\pi_{il}\circ\sigma_{ij}+\delta_{il}\sigma_{lk}\circ E_{i}\pi_{jk}=0

where we put $\sigma_{rr}=\tau_{r}$ .

Note that the equality for $(i,j,k,l)=(3,2,2,1)$ is equivalent to the one for $(i,j,k,l)=(2,3,1,2)$ .

We define $\operatorname{Hom}\nolimits_{\Delta_{123}({\mathcal{W}})}((m,\pi_{21},\pi_{31},\pi_{32}),(m^{\prime},\pi^{\prime}_{21},\pi^{\prime}_{31},\pi^{\prime}_{32}))$ to be the differential submodule of $\operatorname{Hom}\nolimits_{\mathcal{W}}(m,m^{\prime})$ of maps $f$ such that $E_{j}f\circ\pi_{ij}=\pi^{\prime}_{ij}\circ E_{i}f$ for all $i>j$ .

We define a differential endofunctor $E$ of $\Delta_{123}({\mathcal{W}})$ by $E(m,\pi_{21},\pi_{31},\pi_{32})=(m^{\prime},\pi^{\prime}_{21},\pi^{\prime}_{31},\pi^{\prime}_{32})$ where

m^{\prime}={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 55.72571pt\hbox{{\hbox{\kern-55.72571pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{E_{3}(m)\oplus E_{2}(m)\oplus E_{1}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-8.99098pt\raise 37.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 48.36943pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-35.22432pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\ \pi_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern-2.84526pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 13.7711pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{21}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 45.52417pt\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}}}}}

\pi^{\prime}_{31}:{\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 118.46088pt\hbox{{\hbox{\kern-68.17015pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{3}E_{1}(m)}$}}}}}{\hbox{\kern-68.17015pt\raise-28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{1}E_{3}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-118.46088pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{31}\circ E_{3}\pi_{31}\circ\tau_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-62.59573pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-12.10573pt\raise 5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-56.90521pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 62.59573pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\tau_{1}\circ E_{1}\pi_{31}\circ\sigma_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 62.59573pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}}

\pi^{\prime}_{32}:{\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 118.46088pt\hbox{{\hbox{\kern-68.17015pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{3}E_{1}(m)}$}}}}}{\hbox{\kern-68.17015pt\raise-28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{2}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{2}E_{1}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-118.46088pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{32}\circ E_{3}\pi_{32}\circ\tau_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-62.59573pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-50.9615pt\raise 5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-56.90521pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 62.59573pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{12}\circ E_{1}\pi_{32}\circ\sigma_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 62.59573pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-5.69052pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\tau_{2}\circ E_{2}\pi_{32}\circ\sigma_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-5.69052pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}}

\pi^{\prime}_{21}:{\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 122.80978pt\hbox{{\hbox{\kern-68.17015pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{2}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{2}E_{1}(m)}$}}}}}{\hbox{\kern-68.17015pt\raise-28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{1}E_{3}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-122.80978pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{31}\circ E_{3}\pi_{21}\circ\sigma_{23}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-62.59573pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 62.59573pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\tau_{1}\circ E_{1}\pi_{21}\circ\sigma_{21}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 62.59573pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 13.19344pt\raise 5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{21}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-60.389pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{21}\circ E_{2}\pi_{21}\circ\tau_{2}\!}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-5.69052pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}}

We define an endomorphism $\tau$ of $E^{2}$ as follows. We have $E^{2}(m,\pi_{21},\pi_{31},\pi_{32})=(m^{\prime\prime},\pi^{\prime\prime}_{21},\pi^{\prime\prime}_{31},\pi^{\prime\prime}_{32})$ where (ignoring differentials)

m^{\prime\prime}=E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{3}E_{1}(m)\oplus E_{2}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{2}E_{1}(m)\oplus E_{1}E_{3}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m).

We define the endomorphism $\tau$ of $m^{\prime\prime}$ by

(4.3.6)

\tau=\left(\begin{matrix}\tau_{3}\\ &&&\sigma_{23}\\ &&&&&&\sigma_{13}\\ \\ &&&&\tau_{2}\\ &&&&&&&\sigma_{12}\\ \\ \\ &&&&&&&&\tau_{1}\end{matrix}\right).

0P63

Proposition 4.3.12. The construction above defines a $2$ -representation on $\Delta_{123}({\mathcal{W}})$ .

We have isomorphisms of $2$ -representations $\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{123}({\mathcal{W}})$ and $\Delta_{\sigma_{32,1}}\Delta_{\sigma_{32}}({\mathcal{W}})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{123}({\mathcal{W}})$ whose underlying functors make the following diagram commutative

0P64

Proof. Replacing ${\mathcal{W}}$ by $\overline{{\mathcal{W}}}^{i}$ , we can assume it is strongly pretriangulated and idempotent-complete.

The category $\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}})$ has objects $((m,\pi_{21}),\pi_{3})$ where $m\in{\mathcal{W}}$ , $\pi_{21}:E_{2}(m)\to E_{1}(m)$ and $\pi_{3}:{\tilde{E}}_{3}(m,\pi_{21})\to E_{21}(m,\pi_{21})$ satisfy

d(\pi_{21})=d(\pi_{3})=0

and the diagram (4.3.2) commutes for $\pi_{21}$ and for $\pi_{3}$ .

For $i\in\{1,2\}$ , let $\pi_{3i}$ be the composition of $\pi_{3}$ with the projection onto $E_{i}(m)$ . We have $d(\pi_{32})=0$ and $d(\pi_{31})=\pi_{21}\circ\pi_{32}$ .

The commutativity of (4.3.2) for $\pi_{21}$ is the commutativity of

The maps $\pi_{3i}:E_{3}(m)\to E_{i}(m)$ for $i\in\{1,2\}$ give rise to a map

\left(\begin{matrix}\pi_{32}\\ \pi_{31}\end{matrix}\right)\in\operatorname{Hom}\nolimits_{\Delta_{\sigma_{21}}({\mathcal{W}})}({\tilde{E}}_{3}(m,\pi_{21}),E_{21}(m,\pi_{21}))

if and only if the composition

is equal to the sum of the following two maps:

and

and the following diagram commutes:

The commutativity of (4.3.2) for $\pi_{3}$ is equivalent to the commutativity of the following diagrams:

and the vanishing of the following composition:

E_{3}^{2}(m)\xrightarrow{\tau_{3}}E_{3}^{2}(m)\xrightarrow{E_{3}\pi_{31}}E_{3}E_{1}(m)\xrightarrow{\sigma_{31}}E_{1}E_{3}(m)\xrightarrow{E_{1}\pi_{32}}E_{1}E_{2}(m).

Note that the vanishing of that composition follows from the commutativity of the diagram immediately above.

We deduce that the objects of $\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}})$ can be described as quadruples $(m,\pi_{21},\pi_{31},\pi_{32})$ where $m\in{\mathcal{W}}$ , $\pi_{ij}:E_{i}(m)\to E_{j}(m)$ satisfy

d(\pi_{21})=d(\pi_{32})=0,\ d(\pi_{31})=\pi_{21}\circ\pi_{32}

and given $i,j,k,l\in\{1,2,3\}$ with $j-l\geq i-k>0$ , we have an equality between maps $E_{i}E_{j}(m)\to E_{k}E_{l}(m)$ :

\sigma_{lk}\circ E_{l}\pi_{ik}\circ\sigma_{il}\circ E_{i}\pi_{jl}+E_{k}\pi_{jl}\circ\sigma_{jk}\circ E_{j}\pi_{ik}\circ\sigma_{ij}+\delta_{jk}E_{j}\pi_{il}\circ\sigma_{ij}+\delta_{il}\sigma_{lk}\circ E_{i}\pi_{jk}=0

where we put $\sigma_{rr}=\tau_{r}$ .

This provides an isomorphism of categories $\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{123}({\mathcal{W}})$ .

Let us now describe the action of $E$ on $\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}})$ .

We have $E((m,\pi_{21}),\pi_{3})=(m^{\prime},\pi^{\prime})$ where

m^{\prime}={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 54.52208pt\hbox{{\hbox{\kern-54.52208pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{{\tilde{E}}_{3}(m,\pi_{21})\oplus E_{21}(m,\pi_{21})}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-7.28957pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 34.14313pt\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}}}}}

=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 55.72571pt\hbox{{\hbox{\kern-55.72571pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{E_{3}(m)\oplus E_{2}(m)\oplus E_{1}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-8.99098pt\raise 37.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 48.36943pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-37.27293pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\ \ \pi_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern-2.84526pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 13.7711pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{21}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 45.52417pt\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}}}}},{\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 117.11926pt\hbox{{\hbox{\kern-68.17015pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{2}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{2}E_{1}(m)}$}}}}}{\hbox{\kern-68.17015pt\raise-28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{1}E_{3}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-117.11926pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{31}\circ E_{3}\pi_{21}\circ\sigma_{23}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-56.90521pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-53.53183pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{21}\circ E_{2}\pi_{21}\circ\tau_{2}\!\!}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 56.90521pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\tau_{1}\circ E_{1}\pi_{21}\circ\sigma_{21}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 56.90521pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 27.33139pt\raise-5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{21}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 5.69052pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}})

\pi^{\prime}:{\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 99.85797pt\hbox{{\hbox{\kern-58.43185pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{{\tilde{E}}_{3}^{2}(m,\pi_{21})\oplus{\tilde{E}}_{3}E_{21}(m,\pi_{21})}$}}}}}{\hbox{\kern-62.73737pt\raise-28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{21}{\tilde{E}}_{3}(m,\pi_{21})\oplus E_{21}^{2}(m,\pi_{21})}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-99.85797pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.62502pt\hbox{$\scriptstyle{\sigma_{21,3}^{-1}\circ{\tilde{E}}_{3}\pi_{3}\circ\tau_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-42.67891pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 42.67891pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.86725pt\hbox{$\scriptstyle{\tau_{E_{21}}\circ E_{21}\pi_{3}\circ\sigma_{21,3}^{-1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 42.67891pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-17.97343pt\raise 7.10612pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.13391pt\hbox{$\scriptstyle{\sigma_{21,3}^{-1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-36.98839pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}}

={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 161.73352pt\hbox{{\hbox{\kern-68.17015pt\raise 42.67891pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{3}E_{1}(m)}$}}}}}{\hbox{\kern-142.45137pt\raise-42.67891pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{2}E_{3}(m)\oplus E_{1}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{2}E_{1}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-161.73352pt\raise 14.86165pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{32}\circ E_{3}\pi_{32}\circ\tau_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern-139.41777pt\raise-34.14313pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\ignorespaces\ignorespaces{\hbox{\kern-79.89424pt\raise 7.94972pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\!\!\!\sigma_{31}\circ E_{3}\pi_{31}\circ\tau_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern-88.20308pt\raise-34.14313pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\ignorespaces\ignorespaces{\hbox{\kern-106.93022pt\raise-5.6085pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\!\!\!\!\sigma_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern-133.72725pt\raise-34.14313pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-25.03835pt\raise-21.959pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\!\!\tau_{2}\circ E_{2}\pi_{32}\circ\sigma_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-28.4526pt\raise-34.14313pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 36.41922pt\raise-8.54361pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\!\!\sigma_{12}\circ E_{1}\pi_{32}\circ\sigma_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 28.4526pt\raise-34.14313pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 16.79288pt\raise 15.47916pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\!\!\sigma_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-71.13152pt\raise-34.14313pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 91.83302pt\raise 5.89168pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\tau_{1}\circ E_{1}\pi_{21}\circ\sigma_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 142.26303pt\raise-34.14313pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}}

Via the isomorphism of categories above, this corresponds to the functor $E$ on $\Delta_{123}({\mathcal{W}})$ .

The endomorphism $\tau$ of $E^{2}((m,\pi_{21}),\pi_{3})$ is

={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 246.90207pt\hbox{{\hbox{\kern-216.73259pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{3}E_{1}(m)\oplus E_{2}E_{3}(m)\oplus E_{1}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{2}E_{1}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}$}}}}}{\hbox{\kern-216.73259pt\raise-28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{3}E_{1}(m)\oplus E_{2}E_{3}(m)\oplus E_{1}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{2}E_{1}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-246.90207pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\tau_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-233.31137pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-131.60667pt\raise 5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{23}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-176.40616pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-68.4296pt\raise-5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{13}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-119.50095pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 46.15977pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\tau_{2}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 59.75047pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 224.77559pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\tau_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 224.77559pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 121.31334pt\raise 5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{12}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 108.1199pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}}

This coincides with the endomorphism $\tau$ of the endofunctor $E^{2}$ of $\Delta_{123}({\mathcal{W}})$ .

The category $\Delta_{\sigma_{32,1}}\Delta_{\sigma_{32}}({\mathcal{W}})$ has objects pairs $((m,\pi_{32}),\pi_{1})$ where $m\in{\mathcal{W}}$ , $\pi_{32}:E_{3}(m)\to E_{2}(m)$ and $\pi_{1}:E_{32}(m,\pi_{32})\to{\tilde{E}}_{1}(m,\pi_{32})$ satisfy

d(\pi_{32})=d(\pi_{1})=0

and the diagram (4.3.2) commutes for $\pi_{32}$ and for $\pi_{1}$ .

For $i\in\{2,3\}$ , let $\pi_{i1}$ be the composition of the inclusion $E_{i}(m)\to E_{32}(m)$ with $\pi_{1}$ . We have $d(\pi_{21})=0$ and $d(\pi_{31})=\pi_{21}\circ\pi_{32}$ .

As in the case of the category $\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}})$ , the objects of $\Delta_{\sigma_{32,1}}\Delta_{\sigma_{32}}({\mathcal{W}})$ can be described as quadruples $(m,\pi_{21},\pi_{31},\pi_{32})$ where $m\in{\mathcal{W}}$ , $\pi_{ij}:E_{i}(m)\to E_{j}(m)$ satisfy

d(\pi_{21})=d(\pi_{32})=0,\ d(\pi_{31})=\pi_{21}\circ\pi_{32},

the composition

E_{3}E_{2}(m)\xrightarrow{E_{3}\pi_{21}}E_{3}E_{1}(m)\xrightarrow{\sigma_{31}}E_{1}E_{3}(m)\xrightarrow{E_{1}\pi_{32}}E_{1}E_{2}(m)\xrightarrow{\sigma_{12}}E_{2}E_{1}(m)

is equal to the sum of the following two maps

E_{3}E_{2}(m)\xrightarrow{\sigma_{32}}E_{2}E_{3}(m)\xrightarrow{E_{2}\pi_{32}}E_{2}^{2}(m)\xrightarrow{\tau_{2}}E_{2}^{2}(m)\xrightarrow{E_{2}\pi_{21}}E_{2}E_{1}(m)

and

E_{3}E_{2}(m)\xrightarrow{\sigma_{32}}E_{2}E_{3}(m)\xrightarrow{E_{2}\pi_{31}}E_{2}E_{1}(m),

the following diagrams commute

and the following composition vanishes:

E_{3}E_{2}(m)\xrightarrow{E_{3}\pi_{21}}E_{3}E_{1}(m)\xrightarrow{\sigma_{31}}E_{1}E_{3}(m)\xrightarrow{E_{1}\pi_{31}}E_{1}^{2}(m)\xrightarrow{\tau_{1}}E_{1}^{2}(m).

The vanishing of that composition follows from the commutativity of the diagram immediately above.

This description of objects provides an isomorphism of categories $\Delta_{\sigma_{32,1}}\Delta_{\sigma_{32}}({\mathcal{W}})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{123}({\mathcal{W}})$ .

Let us now describe the action of $E$ on $\Delta_{\sigma_{32,1}}\Delta_{\sigma_{32}}({\mathcal{W}})$ .

We have $E((m,\pi_{32}),\pi_{1})=(m^{\prime},\pi^{\prime})$ where

m^{\prime}={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 53.411pt\hbox{{\hbox{\kern-53.411pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{E_{32}(m,\pi_{32})\oplus{\tilde{E}}_{1}(m,\pi_{32})}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-7.28957pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 34.14313pt\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}}}}}

=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 55.72571pt\hbox{{\hbox{\kern-55.72571pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{E_{3}(m)\oplus E_{2}(m)\oplus E_{1}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-8.99098pt\raise 37.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 48.36943pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-37.27293pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\ \ \pi_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern-2.84526pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 13.7711pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{21}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 45.52417pt\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}}}}},{\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 112.77036pt\hbox{{\hbox{\kern-68.17015pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{3}E_{1}(m)}$}}}}}{\hbox{\kern-68.17015pt\raise-28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{2}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{2}E_{1}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-112.77036pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{32}\circ E_{3}\pi_{32}\circ\tau_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-56.90521pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\!\!\tau_{2}\circ E_{2}\pi_{32}\circ\sigma_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 56.90521pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{12}\circ E_{1}\pi_{32}\circ\sigma_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 56.90521pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-45.27098pt\raise 5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-51.21469pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}})

\pi^{\prime}:{\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 113.15536pt\hbox{{\hbox{\kern-62.73737pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{32}^{2}(m,\pi_{32})\oplus E_{32}{\tilde{E}}_{1}(m,\pi_{32})}$}}}}}{\hbox{\kern-58.43185pt\raise-28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{{\tilde{E}}_{1}E_{32}(m,\pi_{32})\oplus{\tilde{E}}_{1}^{2}(m,\pi_{32})}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-113.15536pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.1389pt\hbox{$\scriptstyle{\sigma_{32,1}\circ E_{32}\pi_{1}\circ\tau_{E_{32}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-42.67891pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 42.67891pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.62502pt\hbox{$\scriptstyle{\tau_{1}\circ{\tilde{E}}_{1}\pi_{1}\circ\sigma_{32,1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 42.67891pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-17.97343pt\raise 5.49306pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.52084pt\hbox{$\scriptstyle{\sigma_{32,1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-36.98839pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}}

={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 161.73352pt\hbox{{\hbox{\kern-142.45137pt\raise 42.67891pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{2}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{3}E_{1}(m)\oplus E_{2}E_{1}(m)}$}}}}}{\hbox{\kern-68.17015pt\raise-42.67891pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{1}E_{3}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-161.73352pt\raise-14.86163pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{31}\circ E_{3}\pi_{31}\circ\tau_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern-62.59573pt\raise-34.14313pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-22.92957pt\raise-8.65009pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{31}\!\!\!}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-45.52417pt\raise-34.14313pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\ignorespaces\ignorespaces{\hbox{\kern 19.00433pt\raise-5.78938pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\tau_{1}\circ E_{1}\pi_{31}\circ\sigma_{31}\!\!}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 56.90521pt\raise-34.14313pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\ignorespaces\ignorespaces{\hbox{\kern-108.4182pt\raise 20.73994pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{31}\circ E_{3}\pi_{21}\circ\sigma_{23}\!\!}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern-56.90521pt\raise-34.14313pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-35.32208pt\raise 22.45383pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\sigma_{21}\circ E_{2}\pi_{21}\circ\tau_{2}\!\!}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-34.14313pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\ignorespaces\ignorespaces{\hbox{\kern 93.58415pt\raise 15.21579pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{21}\!\!}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 5.69052pt\raise-34.14313pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 109.4812pt\raise-18.87695pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.89168pt\hbox{$\scriptstyle{\tau_{1}\circ E_{1}\pi_{31}\circ\sigma_{21}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 62.59573pt\raise-34.14313pt\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}}}}}

Via the isomorphism of categories above, this corresponds to the functor $E$ on $\Delta_{123}({\mathcal{W}})$ .

The endomorphism $\tau$ of $E^{2}((m,\pi_{32}),\pi_{1})$ is

={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 246.90207pt\hbox{{\hbox{\kern-216.73259pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{2}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{3}E_{1}(m)\oplus E_{2}E_{1}(m)\oplus E_{1}E_{3}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}$}}}}}{\hbox{\kern-216.73259pt\raise-28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\textstyle{E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{2}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{3}E_{1}(m)\oplus E_{2}E_{1}(m)\oplus E_{1}E_{3}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-246.90207pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\tau_{3}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-233.31137pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-163.21272pt\raise 5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{23}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-176.40616pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 39.10896pt\raise 5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{13}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-5.69052pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-76.18643pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\tau_{2}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-62.59573pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 224.77559pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\tau_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 224.77559pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 96.01418pt\raise 5.00694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\sigma_{12}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 51.21469pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}}

This coincides with the endomorphism $\tau$ of the endofunctor $E^{2}$ of $\Delta_{123}({\mathcal{W}})$ . ∎

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2

4.3. Diagonal action

4.3.1. Category

4.3.2. 11-arrows

4.3.3. 22-arrows

4.3.4. Functoriality

4.3.5. Associativity

4.3.2. $1$ -arrows

4.3.3. $2$ -arrows