Subsection 4.4.5. (0PFW) · ScalingStacks

4.4.5. $2$ -arrows

We assume in §4.4.5 that $\sigma$ is invertible.

Given $(m,\varsigma)\in\Delta_{\lambda}{\mathcal{W}}$ , write $E^{2}(m,\varsigma)=(m^{\prime\prime},\varsigma^{\prime\prime})$ . The formula (4.3.3) defines an endomorphism $\tau$ of $m^{\prime\prime}$ .

0P6P

Lemma 4.4.14. Given $i\geq 1$ , we have $\tau\circ\varsigma^{\prime\prime}_{i}=\varsigma^{\prime\prime}_{i}\circ E_{2}^{i}F_{1}^{i}\tau$ .

0P6Q

Proof. Let $A=\tau\circ\varsigma^{\prime\prime}_{i}$ and $B=\varsigma^{\prime\prime}_{i}\circ E_{2}^{i}F_{1}^{i}\tau$ .

We have

a_{21}=a_{22}=a_{31}=a_{32}=a_{33}=a_{34}=a_{41}=a_{42}=a_{43}=0

	$\displaystyle a_{11}$	$\displaystyle=\lambda_{(12)}\circ E_{2}^{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}$
		$\displaystyle=E_{2}^{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}$

	$\displaystyle a_{12}$	$\displaystyle=\sum_{r=1}^{i}\lambda_{(12)}\circ E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(12)}\circ\lambda_{(1\cdots i+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(i+1\cdots 2i+1)}$
		$\displaystyle=0$

	$\displaystyle a_{13}$	$\displaystyle=\sum_{r=1}^{i}\lambda_{(12)}\circ E_{2}^{2}\varsigma_{i-1}\circ\lambda_{(2\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{2}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ\lambda_{(1\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{2}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(2i+1,2i+2)}\circ E_{2}^{i}F_{1}^{i-1}\sigma^{-1}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ\lambda_{(1\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}E_{2}\varepsilon_{1}\circ\lambda_{(2i,2i+1)}\circ\lambda_{(2i\cdots i+r)}\circ\lambda_{(1\cdots r)}\circ E_{2}^{i}F_{1}^{i-1}\sigma^{-1}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i\cdots i+r)}\circ\lambda_{(1\cdots r)}\circ E_{2}^{i}F_{1}^{i-1}\sigma^{-1}$

	$\displaystyle a_{14}$	$\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}\lambda_{(12)}\circ E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ\lambda_{(2\cdots s+1)(2i-1\cdots i+s)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(1\cdots s+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i-1\cdots i+s)}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=\sum_{1\leq r\leq s<i}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(1\cdots s+1)}\circ\lambda_{(2i\cdots i+r)}\circ\lambda_{(2i\cdots i+s+1)}$

	$\displaystyle a_{23}$	$\displaystyle=\sigma^{-1}\circ E_{1}E_{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}$
		$\displaystyle=\sigma^{-1}\circ E_{1}E_{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1,2i+2)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1}$
		$\displaystyle=\sigma^{-1}\circ E_{1}E_{2}\varsigma_{i}\circ\lambda_{(12)}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1}$
		$\displaystyle=E_{1}E_{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1}$

	$\displaystyle a_{24}$	$\displaystyle=\sum_{r=1}^{i}\sigma^{-1}\circ E_{1}E_{2}\varsigma_{i-1}\circ E_{1}E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(1\cdots 2i+1)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ E_{2}E_{1}E_{2}^{i-1}F_{1}^{i-1}\varepsilon_{1}\circ\sigma^{-1}E_{2}^{i-1}F_{1}^{i}E_{1}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ E_{2}E_{1}E_{2}^{i-1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots 2i+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)}$

	$\displaystyle a_{44}$	$\displaystyle=\lambda_{(12)}\circ E_{1}^{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}$
		$\displaystyle=E_{1}^{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}$

We have

b_{21}=b_{12}=b_{22}=b_{31}=b_{32}=b_{33}=b_{41}=b_{42}=b_{43}=0

	$\displaystyle b_{11}$	$\displaystyle=E_{2}^{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1,2i+2)}$
		$\displaystyle=E_{2}^{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}$

	$\displaystyle b_{13}$	$\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i\cdots i+r)}\circ\lambda_{(1\cdots r)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1}$

	$\displaystyle b_{14}$	$\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ\lambda_{(2\cdots s+1)(2i-1\cdots i+s)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(2i+1,2i+2)}$
		$\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2i+1,2i+2)}\circ\lambda_{(2\cdots s+1)(2i-1\cdots i+s)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2i-1,2i)}\circ\lambda_{(2\cdots s+1)(2i-1\cdots i+s)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{1\leq s<r\leq i}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2\cdots s+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+s)}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=\sum_{1\leq r^{\prime}\leq s^{\prime}\leq i}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2\cdots r^{\prime}+1)}\circ\lambda_{(1\cdots s^{\prime}+1)}\circ\lambda_{(2i\cdots i+r^{\prime})}\circ\lambda_{(2i\cdots i+s^{\prime}+1)}$

b_{23}=E_{2}E_{1}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1}

	$\displaystyle b_{24}$	$\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ\lambda_{(2\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(2i+1,2i+2)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ\lambda_{(2\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}E_{1}\varepsilon_{1}\circ\lambda_{(2i,2i+1)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}E_{1}\varepsilon_{1}\circ\lambda_{(2\cdots 2i+1)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}$

	$\displaystyle b_{34}$	$\displaystyle=\sum_{r=1}^{i}E_{1}E_{2}\varsigma_{i-1}\circ E_{1}E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1,2i+2)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}E_{2}\varsigma_{i-1}\circ E_{1}E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}E_{2}\varsigma_{i-1}\circ\lambda_{(1\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}E_{1}\varepsilon_{1}\circ\lambda_{(2i,2i+1)}\circ\lambda_{(2i+1,2i+2)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=\sum_{r=1}^{i}E_{1}E_{2}\varsigma_{i-1}\circ\lambda_{(1\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2i+1,2i+2)}\circ\lambda_{(2i+1,2i+2)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)}$
		$\displaystyle=0$

	$\displaystyle b_{44}$	$\displaystyle=E_{1}^{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1,2i+2)}$
		$\displaystyle=E_{1}^{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}$

We deduce that $A=B$ and the lemma follows. ∎

Lemma 4.4.14 shows that $\tau$ defines an endomorphism of $E^{2}(m,\varsigma)$ for all $(m,\varsigma)\in\Delta_{\lambda}{\mathcal{W}}$ . The functor $\Gamma$ is faithful, $\Gamma E^{2}=E^{2}\Gamma$ (Lemma 4.4.13) and $\tau$ commutes with $\Gamma$ . It follows that $\tau$ is functorial.

Theorem 4.3.8 has the following consequence.

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Theorem 4.4.15. The data $(\Delta_{\lambda}{\mathcal{W}},E,\tau)$ is an idempotent-complete strongly pretriangulated $2$ -representation.

The following proposition is a consequence of Lemma 4.4.13 and the construction of $\tau$ .

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Proposition 4.4.16. The functor $\Gamma:\Delta_{\lambda}{\mathcal{W}}\to\Delta_{\sigma}{\mathcal{W}}$ induces a morphism of $2$ -representations.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2

4.4.5. 22-arrows

4.4.5. $2$ -arrows