Subsection 4.3.2. (0PFM) · ScalingStacks

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 $E(f)=\left(\begin{matrix}E_{2}f&0\\ 0&E_{1}f\end{matrix}\right)$ :

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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:

∎

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2

4.3.2. 11-arrows

4.3.2. $1$ -arrows