Subsection 4.4.4. (0PFV) · ScalingStacks

4.4.4. $1$ -arrows

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

\pi:E_{2}(m)\xrightarrow{E_{2}\eta_{1}}E_{2}E_{1}F_{1}(m)\xrightarrow{\sigma F_{1}}E_{1}E_{2}F_{1}(m)\xrightarrow{E_{1}\varsigma_{1}}E_{1}(m).

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

\pi:E_{2}(m)\xrightarrow{\eta_{1}E_{2}}E_{1}F_{1}E_{2}(m)\xrightarrow{E_{1}\lambda}E_{1}E_{2}F_{1}(m)\xrightarrow{E_{1}\varsigma_{1}}E_{1}(m)

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 $\Gamma:\Delta_{\lambda}{\mathcal{W}}\to\Delta_{\sigma}{\mathcal{W}},\ (m,\varsigma)\mapsto(m,\pi)$ .

0P6G

Remark 4.4.9. The construction of $\pi$ from $\varsigma_{1}$ is illustrated below.

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

\varsigma^{\prime}_{i}=\left(\begin{matrix}E_{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+1)}&\sum_{r=1}^{i}E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\\ 0&E_{1}\varsigma_{i}\circ\lambda_{(1\cdots 2i+1)}\end{matrix}\right):E_{2}^{i}F_{1}^{i}(m^{\prime})\to m^{\prime}

0P6H

Lemma 4.4.10. $(m^{\prime},\varsigma^{\prime})$ is an object of $\Delta_{\lambda}{\mathcal{W}}$ .

0P6I

Proof. We have

	$\displaystyle d((\varsigma^{\prime}_{i})_{11})$	$\displaystyle=E_{2}\varsigma_{i}\circ d(\tau_{2}E_{2}^{i-1}\circ\cdots\circ E_{2}^{i-1}\tau_{2})F_{1}^{i}\circ\lambda_{(i+1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i}\circ\lambda_{(1\cdots r)(r+1\cdots i+1)}\circ\lambda_{(i+1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i}\circ\lambda_{(1\cdots r)(r+1\cdots i+1)}\circ\lambda_{(i+1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i}\circ\lambda_{(1\cdots r)(2i+1\cdots i+r+1)}\circ\lambda_{(i+1\cdots 2i+1)}$
		$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(1\cdots i)}$

(\varsigma^{\prime}_{i})_{12}\circ E_{2}^{i}F_{1}^{i}\pi=

		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varsigma_{1}\circ\lambda_{(2i,2i+1)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}F_{1}E_{2}\circ E_{2}^{i}F_{1}^{i}\eta_{1}E_{2}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}\varsigma_{i}\circ\lambda_{(i+1\cdots 2i)}\circ\lambda_{(2i,2i+1)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(1\cdots i)}$
		$\displaystyle=d((\varsigma^{\prime}_{i})_{11}).$

	$\displaystyle d((\varsigma^{\prime}_{i})_{22})$	$\displaystyle=E_{1}\varsigma_{i}\circ\lambda_{(1\cdots i+1)}\circ E_{2}^{i}d(\rho F_{1}^{i-1}\circ F_{1}\rho F_{1}^{i-2}\circ\cdots\circ F_{1}^{i-1}\rho)$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(1\cdots i+r)}\circ E_{2}^{i}F_{1}^{r-1}\eta_{1}F_{1}^{i-r}\circ E_{2}^{i}F_{1}^{r-1}\varepsilon_{1}F_{1}^{i-r}\circ\lambda_{(i+r+1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(i+r+1\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(i+r+1\cdots i+2)}\circ\lambda_{(i+2\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(i+2\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=E_{1}\varsigma_{i}\circ\lambda_{(i+2\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+1)}$

	$\displaystyle\pi\circ(\varsigma^{\prime}_{i})_{12}$	$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{1}\circ\lambda_{(12)}\circ E_{2}\eta_{1}\circ E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{1}\circ\lambda_{(23)}\circ\eta_{1}E_{2}\circ E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{1}\circ E_{1}E_{2}F_{1}\varsigma_{i-1}\circ\lambda_{23}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}\varsigma_{i}\circ\lambda_{(i+2\cdots 3)}\circ\lambda_{23}\circ\lambda_{(3\cdots r+2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=E_{1}\varsigma_{i}\circ\lambda_{(i+2\cdots 2)}\circ\eta_{1}E_{2}^{i}F_{1}^{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2i\cdots i+r)}=d((\varsigma^{\prime}_{i})_{22}).$

We have

d((\varsigma^{\prime}_{i})_{12})=A+B

where

	$\displaystyle A$	$\displaystyle=\sum_{1\leq s<r\leq i}E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots s)(s+1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{1\leq s<r\leq i}E_{2}\varsigma_{i-1}\circ\lambda_{(s+1\cdots r)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots s)(2i\cdots i+r)}$
		$\displaystyle=\sum_{1\leq s<r\leq i}E_{2}\varsigma_{i-1}\circ\lambda_{(i+r-1\cdots s+i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots s)(2i\cdots i+r)}$
		$\displaystyle=\sum_{1\leq s<r\leq i}E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots s)(2i\cdots i+r)(i+r-1\cdots i+s)}$

and

B=\sum_{\begin{subarray}{c}1\leq r^{\prime}\leq i\\ 1\leq s^{\prime}\leq i-r^{\prime}\end{subarray}}E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r^{\prime})(2i\cdots i+r^{\prime}+s^{\prime})(i+r^{\prime}+s^{\prime}-1\cdots i+r^{\prime})}

So $A=B$ and $d((\varsigma^{\prime}_{i})_{12})=0$ .

We have shown that $d(\varsigma^{\prime}_{i})=0$ ,

Fix $r\in\{1,\ldots,i\}$ . We put $b_{r}=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}:E_{2}^{i}F_{1}^{i}E_{1}(m)\to E_{2}(m)$ .

Consider $s\in\{1,\ldots,i-1\}$ .

If $s>r$ , we have

	$\displaystyle b_{r}(T_{s}\otimes 1)$	$\displaystyle=E_{2}\varsigma_{i-1}\circ\lambda_{(s,s+1)}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ\lambda_{(i+s-1,i+s)}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(i+s-1,i+s)}\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\lambda_{(i+s,i+s+1)}$
		$\displaystyle=b_{r}(1\otimes T_{s}).$

If $s<r-1$ , we have

	$\displaystyle b_{r}(1\otimes T_{s})$	$\displaystyle=E_{2}\varsigma_{i-1}\circ\lambda_{(i+s,i+s+1)}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ\lambda_{(s+1,s+2)}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(s+1,s+2)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(s,s+1)}$
		$\displaystyle=b_{r}(T_{s}\otimes 1).$

We have

b_{r}(T_{r-1}\otimes 1)=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(r-1,r)}=0

b_{r}(1\otimes T_{r})=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(i+r,i+r+1)}=0

b_{r}(1\otimes T_{r-1})=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{i-1}\circ\lambda_{(1\cdots r)(2i\cdots i+r-1)}=b_{r-1}(T_{r-1}\otimes 1).

We have shown that $(\varsigma_{i})_{12}(1\otimes T_{s})=(\varsigma_{i})_{12}(T_{s}\otimes 1)$ .

We have

	$\displaystyle(\varsigma^{\prime}_{i})_{11}(T_{s}\otimes 1)$	$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(s+1,s+2)}\circ\lambda_{(1\cdots 2i+1)}$
		$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(i+s+1,i+s+2)}\circ\lambda_{(1\cdots 2i+1)}$
		$\displaystyle=E_{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+1)}\lambda_{(i+s,i+s+1)}$
		$\displaystyle=(\varsigma^{\prime}_{i})_{11}(1\otimes T_{s}).$

Similarly,

(\varsigma^{\prime}_{i})_{22}(T_{s}\otimes 1)=(\varsigma^{\prime}_{i})_{11}(1\otimes T_{s}).

So $\varsigma_{i}(1\otimes T_{s})=\varsigma_{i}(T_{s}\otimes 1)$ .

Let $l\in\{1,2\}$ . We have

(\varsigma^{\prime}_{i+j})_{ll}\circ\mu_{ij}=E_{l}\varsigma_{i+j}\circ\lambda_{w}

where $w(r)=r$ and $w(i+r)=i+r+j+1$ for $1\leq r\leq i$ , $w(2i+r)=i+r$ and $w(2i+j+r)=2i+j+r+1$ for $1\leq r\leq j$ and $w(2i+2j+1)=i+j+1$ .

We have

	$\displaystyle(\varsigma^{\prime}_{i})_{ll}\circ(\varsigma^{\prime}_{j})_{ll}$	$\displaystyle=E_{l}\varsigma_{i}\circ E_{l}E_{2}^{i}F_{1}^{i}\varsigma_{j}\circ\lambda_{(1\cdot 2i+1)}\circ\lambda_{(2i+1\cdots 2i+2j+1)}$
		$\displaystyle=E_{l}\varsigma_{i+j}\circ\lambda_{w^{\prime}}\circ\lambda_{(1\cdot 2i+1)}\circ\lambda_{(2i+1\cdots 2i+2j+1)}$

where $w^{\prime}(r)=r$ for $1\leq r\leq i+1$ , $w^{\prime}(i+1+r)=i+j+1+r$ for $1\leq r\leq i$ , $w^{\prime}(1+2i+r)=1+i+r$ and $w^{\prime}(1+2i+j+r)=1+2i+j+r$ for $1\leq r\leq j$ .

It follows that $(\varsigma^{\prime}_{i+j})_{ll}\circ\mu_{ij}=(\varsigma^{\prime}_{i})_{ll}\circ(\varsigma^{\prime}_{j})_{ll}$ .

Given $l\leq l^{\prime}\leq 1$ , we put $b_{l^{\prime},l}=E_{2}\varsigma_{l^{\prime}-1}\circ E_{2}^{l^{\prime}}F_{l^{\prime}-1}\varepsilon_{1}\circ\lambda_{(1\cdots l)(2l^{\prime}\cdots l^{\prime}+l)}:E_{2}^{l^{\prime}}F_{1}^{l^{\prime}}E_{1}(m)\to E_{2}(m)$ . We denote by $w_{l_{1},l_{2}}$ the permutation of ${\mathfrak{S}}_{l_{1}+l_{2}}$ given by $s\mapsto s+l_{2}$ for $1\leq s\leq l_{1}$ and $s\mapsto s-l_{1}$ for $l_{1}+1\leq s\leq l_{1}+l_{2}$ .

Consider $r\in\{1,\ldots,i\}$ . We have

	$\displaystyle b_{i,r}\circ(\varsigma^{\prime}_{j})_{22}$	$\displaystyle=E_{2}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}\varsigma_{j}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{2}^{j}F_{1}^{j}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(2i+1\cdots 2i+2j+1)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ\lambda_{w_{i-1,j}}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{2}^{j}F_{1}^{j}\circ\lambda_{(2i+1\cdots 2i+2j+1)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i}\lambda_{w_{i-1,j}}F_{1}^{j}\circ E_{2}^{i}F_{1}^{i-1}E_{2}^{j}F_{1}^{j-1}\varepsilon_{1}\circ\lambda_{(2i+2j\cdots 2i)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i}\lambda_{w_{i-1,j}}F_{1}^{j}\circ E_{2}^{i}F_{1}^{i-1}E_{2}^{j}F_{1}^{j-1}\varepsilon_{1}\circ\lambda_{(2i+2j\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+j}F_{1}^{i+j-1}\varepsilon_{1}\circ E_{2}^{i}\lambda_{w_{i-1,j}}F_{1}^{j+1}E_{1}\circ\lambda_{(2i+2j\cdots i+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+j}F_{1}^{i+j-1}\varepsilon_{1}\circ\lambda_{(2i+2j\cdots i+j+r)}\circ E_{2}^{i}\lambda_{w_{i,j}}F_{1}^{j}E_{1}$
		$\displaystyle=b_{i+j,r}\circ\mu_{i,j}.$

Consider $r\in\{1,\ldots,j\}$ . We have

	$\displaystyle(\varsigma^{\prime}_{i})_{11}\circ b_{j,r}$	$\displaystyle=E_{2}\varsigma_{i}\circ E_{2}^{i+1}F_{1}^{i}\varsigma_{j-1}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}E_{2}^{j}F_{1}^{j-1}\varepsilon_{1}\circ\lambda_{(2i+1\cdots 2i+r)(2i+2j\cdots 2i+j+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+1}\lambda_{w_{i,j-1}}F_{1}^{j-1}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}E_{2}^{j}F_{1}^{j-1}\varepsilon_{1}\circ\lambda_{(2i+1\cdots 2i+r)(2i+2j\cdots 2i+j+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+j}F_{1}^{i+j-1}\varepsilon_{1}\circ E_{2}^{i+1}\lambda_{w_{i,j-1}}F_{1}^{j}E_{1}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1\cdots 2i+r)(2i+2j\cdots 2i+j+r)}$
		$\displaystyle=E_{2}\varsigma_{i+j-1}\circ E_{2}^{i+j}F_{1}^{i+j-1}\varepsilon_{1}\circ\lambda_{(1\cdots i+r)(2i+2j\cdots 2i+j+r)}\circ E_{2}^{i}\lambda_{w_{i,j}}F_{1}^{j}E_{1}$
		$\displaystyle=b_{i+j,j+r}\circ\mu_{i,j}.$

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

0P6J

Remark 4.4.11. The graphical description of $\varsigma^{\prime}$ is the following:

Given $f\in\operatorname{Hom}\nolimits_{\Delta_{\lambda}{\mathcal{W}}}((m,\varsigma),(\tilde{m},\tilde{\varsigma}))$ , we put $E(f)=\left(\begin{matrix}E_{2}(f)&0\\ 0&E_{1}(f)\end{matrix}\right)$ .

0P6K

Lemma 4.4.12. We have $E(f)\in\operatorname{Hom}\nolimits_{\Delta_{\lambda}{\mathcal{W}}}(E(m,\varsigma),E(\tilde{m},\tilde{\varsigma}))$ . The construction makes $E$ into a differential endofunctor of $\Delta_{\lambda}{\mathcal{W}}$ .

0P6L

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

∎

0P6M

Lemma 4.4.13. We have $E\circ\Gamma=\Gamma\circ E$ .

0P6N

Proof. Let $(m,\varsigma)\in\Delta_{\lambda}{\mathcal{W}}$ . We have $E(m,\pi)=(m^{\prime},\pi^{\prime})$ where $m^{\prime}=\operatorname{cone}\nolimits(\pi)$ and $\pi^{\prime}$ is given in §4.3.2. We have $\Gamma\circ E(m,\varsigma)=(m^{\prime},\pi^{\prime\prime})$ where

\pi^{\prime\prime}_{12}=E_{1}E_{2}\varepsilon_{1}\circ E_{1}\lambda E_{1}\circ\eta_{1}E_{2}E_{1}=\sigma,\ \pi^{\prime\prime}_{21}=0

	$\displaystyle\pi^{\prime\prime}_{11}$	$\displaystyle=E_{1}E_{2}\varsigma_{1}\circ E_{1}E_{2}\lambda\circ E_{1}\lambda E_{2}\circ E_{1}F_{1}\tau_{2}\circ\eta_{1}E_{2}^{2}$
		$\displaystyle=E_{1}E_{2}\varsigma_{1}\circ E_{1}E_{2}\lambda\circ E_{1}\lambda E_{2}\circ\eta_{1}E_{2}^{2}\circ\tau_{2}$
		$\displaystyle=E_{1}E_{2}\varsigma_{1}\circ E_{1}E_{2}\lambda\circ\sigma F_{1}E_{2}\circ E_{2}\eta_{1}E_{2}\circ\tau_{2}$
		$\displaystyle=\sigma\circ E_{2}E_{1}\varsigma_{1}\circ E_{2}E_{1}\lambda\circ E_{2}\eta_{1}E_{2}\circ\tau_{2}$
		$\displaystyle=\pi^{\prime}_{11}$

	$\displaystyle\pi^{\prime\prime}_{22}$	$\displaystyle=E_{1}^{2}\varsigma_{1}\circ E_{1}^{2}\lambda\circ E_{1}\rho E_{2}\circ E_{1}F_{1}\sigma\circ\eta_{1}E_{2}E_{1}$
		$\displaystyle=E_{1}^{2}\varsigma_{1}\circ E_{1}^{2}\lambda\circ E_{1}\rho E_{2}\circ\eta_{1}E_{1}E_{2}\circ\sigma$
		$\displaystyle=E_{1}^{2}\varsigma_{1}\circ E_{1}^{2}\lambda\circ\tau_{1}F_{1}E_{2}\circ E_{1}\eta_{1}E_{2}\circ\sigma$
		$\displaystyle=\tau_{1}\circ E_{1}^{2}\varsigma_{1}\circ E_{1}^{2}\lambda\circ E_{1}\eta_{1}E_{2}\circ\sigma$
		$\displaystyle=\pi^{\prime}_{22}$

It follows that $\pi^{\prime\prime}=\pi^{\prime}$ . ∎

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2

4.4.4. 11-arrows

4.4.4. $1$ -arrows