ScalingStacks

0P63

Proposition 4.3.12. The construction above defines a 22-representation on Ξ”123​(𝒲)\Delta_{123}({\mathcal{W}}).

We have isomorphisms of 22-representations Δσ21,3βˆ’1​Δσ21​(𝒲)β†’βˆΌΞ”123​(𝒲)\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{123}({\mathcal{W}}) and Δσ32,1​Δσ32​(𝒲)β†’βˆΌΞ”123​(𝒲)\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

Δσ21,3βˆ’1​Δσ21​(𝒲)\textstyle{\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}∼\scriptstyle{\sim}Ο‰\scriptstyle{\omega}Ξ”123​(𝒲)\textstyle{\Delta_{123}({\mathcal{W}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο‰\scriptstyle{\omega}Δσ32,1​Δσ32​(𝒲)\textstyle{\Delta_{\sigma_{32,1}}\Delta_{\sigma_{32}}({\mathcal{W}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}∼\scriptstyle{\sim}Ο‰\scriptstyle{\omega}𝒲\textstyle{\mathcal{W}}
0P64

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

The category Δσ21,3βˆ’1​Δσ21​(𝒲)\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}}) has objects ((m,Ο€21),Ο€3)((m,\pi_{21}),\pi_{3}) where mβˆˆπ’²m\in{\mathcal{W}}, Ο€21:E2​(m)β†’E1​(m)\pi_{21}:E_{2}(m)\to E_{1}(m) and Ο€3:E~3​(m,Ο€21)β†’E21​(m,Ο€21)\pi_{3}:{\tilde{E}}_{3}(m,\pi_{21})\to E_{21}(m,\pi_{21}) satisfy

d⁑(Ο€21)=d⁑(Ο€3)=0d(\pi_{21})=d(\pi_{3})=0

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

For i∈{1,2}i\in\{1,2\}, let Ο€3​i\pi_{3i} be the composition of Ο€3\pi_{3} with the projection onto Ei​(m)E_{i}(m). We have d⁑(Ο€32)=0d(\pi_{32})=0 and d⁑(Ο€31)=Ο€21βˆ˜Ο€32d(\pi_{31})=\pi_{21}\circ\pi_{32}.

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

E22​(m)\textstyle{E_{2}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π21\scriptstyle{E_{2}\pi_{21}}Ο„2\scriptstyle{\tau_{2}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ21\scriptstyle{\sigma_{21}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π21\scriptstyle{E_{1}\pi_{21}}E12​(m)\textstyle{E_{1}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„1\scriptstyle{\tau_{1}}E22​(m)\textstyle{E_{2}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π21\scriptstyle{E_{2}\pi_{21}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ21\scriptstyle{\sigma_{21}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π21\scriptstyle{E_{1}\pi_{21}}E12​(m)\textstyle{E_{1}^{2}(m)}

The maps Ο€3​i:E3​(m)β†’Ei​(m)\pi_{3i}:E_{3}(m)\to E_{i}(m) for i∈{1,2}i\in\{1,2\} give rise to a map

(Ο€32Ο€31)∈HomΔσ21​(𝒲)⁑(E~3​(m,Ο€21),E21​(m,Ο€21))\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

E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ23\scriptstyle{\sigma_{23}}E3​E2​(m)\textstyle{E_{3}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π21\scriptstyle{E_{3}\pi_{21}}E3​E1​(m)\textstyle{E_{3}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ31\scriptstyle{\sigma_{31}}E1​E3​(m)\textstyle{E_{1}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π32\scriptstyle{E_{1}\pi_{32}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)}

is equal to the sum of the following two maps:

E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π32\scriptstyle{E_{2}\pi_{32}}E22​(m)\textstyle{E_{2}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„2\scriptstyle{\tau_{2}}E22​(m)\textstyle{E_{2}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π21\scriptstyle{E_{2}\pi_{21}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ21\scriptstyle{\sigma_{21}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)}

and

E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π31\scriptstyle{E_{2}\pi_{31}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ21\scriptstyle{\sigma_{21}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)}

and the following diagram commutes:

E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ23\scriptstyle{\sigma_{23}}E2​π31\scriptstyle{E_{2}\pi_{31}}E3​E2​(m)\textstyle{E_{3}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π21\scriptstyle{E_{3}\pi_{21}}E3​E1​(m)\textstyle{E_{3}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ31\scriptstyle{\sigma_{31}}E1​E3​(m)\textstyle{E_{1}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π31\scriptstyle{E_{1}\pi_{31}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ21\scriptstyle{\sigma_{21}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π21\scriptstyle{E_{1}\pi_{21}}E12​(m)\textstyle{E_{1}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„1\scriptstyle{\tau_{1}}E12​(m)\textstyle{E_{1}^{2}(m)}

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

E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π31\scriptstyle{E_{3}\pi_{31}}Ο„3\scriptstyle{\tau_{3}}E3​E1​(m)\textstyle{E_{3}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ31\scriptstyle{\sigma_{31}}E1​E3​(m)\textstyle{E_{1}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π31\scriptstyle{E_{3}\pi_{31}}E12​(m)\textstyle{E_{1}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„1\scriptstyle{\tau_{1}}E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π31\scriptstyle{E_{3}\pi_{31}}E3​E1​(m)\textstyle{E_{3}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ31\scriptstyle{\sigma_{31}}E1​E3​(m)\textstyle{E_{1}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π31\scriptstyle{E_{3}\pi_{31}}E12​(m)\textstyle{E_{1}^{2}(m)}
E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}Ο„3\scriptstyle{\tau_{3}}E3​E2​(m)\textstyle{E_{3}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ32\scriptstyle{\sigma_{32}}E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}E22​(m)\textstyle{E_{2}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„2\scriptstyle{\tau_{2}}E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}E3​E2​(m)\textstyle{E_{3}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ32\scriptstyle{\sigma_{32}}E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}E22​(m)\textstyle{E_{2}^{2}(m)}
E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π31\scriptstyle{E_{3}\pi_{31}}Ο„3\scriptstyle{\tau_{3}}E3​E1​(m)\textstyle{E_{3}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ31\scriptstyle{\sigma_{31}}E1​E3​(m)\textstyle{E_{1}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π32\scriptstyle{E_{1}\pi_{32}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ12\scriptstyle{\sigma_{12}}E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}E3​E2​(m)\textstyle{E_{3}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ32\scriptstyle{\sigma_{32}}E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π31\scriptstyle{E_{2}\pi_{31}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)}

and the vanishing of the following composition:

E32​(m)β†’Ο„3E32​(m)β†’E3​π31E3​E1​(m)β†’Οƒ31E1​E3​(m)β†’E1​π32E1​E2​(m).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 Δσ21,3βˆ’1​Δσ21​(𝒲)\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}}) can be described as quadruples (m,Ο€21,Ο€31,Ο€32)(m,\pi_{21},\pi_{31},\pi_{32}) where mβˆˆπ’²m\in{\mathcal{W}}, Ο€i​j:Ei​(m)β†’Ej​(m)\pi_{ij}:E_{i}(m)\to E_{j}(m) satisfy

d⁑(Ο€21)=d⁑(Ο€32)=0,d⁑(Ο€31)=Ο€21βˆ˜Ο€32d(\pi_{21})=d(\pi_{32})=0,\ d(\pi_{31})=\pi_{21}\circ\pi_{32}

and given i,j,k,l∈{1,2,3}i,j,k,l\in\{1,2,3\} with jβˆ’lβ‰₯iβˆ’k>0j-l\geq i-k>0, we have an equality between maps Ei​Ej​(m)β†’Ek​El​(m)E_{i}E_{j}(m)\to E_{k}E_{l}(m):

Οƒl​k∘El​πi​kβˆ˜Οƒi​l∘Ei​πj​l+Ek​πj​lβˆ˜Οƒj​k∘Ej​πi​kβˆ˜Οƒi​j+Ξ΄j​k​Ej​πi​lβˆ˜Οƒi​j+Ξ΄i​l​σl​k∘Ei​πj​k=0\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 Οƒr​r=Ο„r\sigma_{rr}=\tau_{r}.

This provides an isomorphism of categories Δσ21,3βˆ’1​Δσ21​(𝒲)β†’βˆΌΞ”123​(𝒲)\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 EE on Δσ21,3βˆ’1​Δσ21​(𝒲)\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}}).

We have E⁑((m,Ο€21),Ο€3)=(mβ€²,Ο€β€²)E((m,\pi_{21}),\pi_{3})=(m^{\prime},\pi^{\prime}) where

mβ€²=Β Β Β Β E~3​(m,Ο€21)βŠ•E21​(m,Ο€21)Β Β Β Ο€3Β Β Β Β Β Β Β Β Β 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}}}}}
=(Β Β Β Β E3​(m)βŠ•E2​(m)βŠ•E1​(m)Β Β Β Ο€31Β Β Β Β Β Β Β Β Ο€32Β Β Β Β Β Β Β Β Ο€21Β Β Β Β Β Β Β Β Β ,Β Β Β Β E2​E3​(m)βŠ•E22​(m)βŠ•E2​E1​(m)Β Β Β E1​E3​(m)βŠ•E1​E2​(m)βŠ•E12​(m)Β Β Β Β Οƒ31∘E3​π21βˆ˜Οƒ23Β Β Β Β Β Β Β Β Β Β Οƒ21∘E2​π21βˆ˜Ο„2Β Β Β Β Β Β Β Β Β Β Ο„1∘E1​π21βˆ˜Οƒ21Β Β Β Β Β Β Β Β Β Β Οƒ21Β Β Β Β Β Β Β Β Β Β )=({\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 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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{$\scriptstyle{\tau_{1}\circ 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Ο€β€²:Β Β Β Β E~32​(m,Ο€21)βŠ•E~3​E21​(m,Ο€21)Β Β Β E21​E~3​(m,Ο€21)βŠ•E212​(m,Ο€21)Β Β Β Β Οƒ21,3βˆ’1∘E~3​π3βˆ˜Ο„3Β Β Β Β Β Β Β Β Β Β Ο„E21∘E21​π3βˆ˜Οƒ21,3βˆ’1Β Β Β Β Β Β Β Β Β Β Οƒ21,3βˆ’1Β Β Β Β Β Β Β Β Β Β \pi^{\prime}:{\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 99.85797pt\hbox{{\hbox{\kern-58.43185pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\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}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern-36.98839pt\raise-19.91682pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}}
=Β Β Β Β E32​(m)βŠ•E3​E2​(m)βŠ•E3​E1​(m)Β Β Β E2​E3​(m)βŠ•E1​E3​(m)βŠ•E22​(m)βŠ•E2​E1​(m)βŠ•E1​E2​(m)βŠ•E12​(m)Β Β Β Οƒ32∘E3​π32βˆ˜Ο„3Β Β Β Β Β Β Β Β Οƒ31∘E3​π31βˆ˜Ο„3Β Β Β Β Β Β Β Β Οƒ32Β Β Β Β Β Β Β Β Β Ο„2∘E2​π32βˆ˜Οƒ32Β Β Β Β Β Β Β Β Β Β Οƒ12∘E1​π32βˆ˜Οƒ31Β Β Β Β Β Β Β Β Β Β Οƒ31Β Β Β Β Β Β Β Β Β Β Ο„1∘E1​π21βˆ˜Οƒ31Β Β Β Β Β Β Β Β Β Β ={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 161.73352pt\hbox{{\hbox{\kern-68.17015pt\raise 42.67891pt\hbox{\hbox{\kern 0.0pt\raise-2.82002pt\hbox{$\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 EE on Ξ”123​(𝒲)\Delta_{123}({\mathcal{W}}).

The endomorphism Ο„\tau of E2​((m,Ο€21),Ο€3)E^{2}((m,\pi_{21}),\pi_{3}) is

E~32​(m,Ο€21)βŠ•E~3​E21​(m,Ο€21)βŠ•E21​E~3​(m,Ο€21)βŠ•E212​(m,Ο€21)\textstyle{{\tilde{E}}_{3}^{2}(m,\pi_{21})\oplus{\tilde{E}}_{3}E_{21}(m,\pi_{21})\oplus E_{21}{\tilde{E}}_{3}(m,\pi_{21})\oplus E_{21}^{2}(m,\pi_{21})}E~32​(m,Ο€21)βŠ•E~3​E21​(m,Ο€21)βŠ•E21​E~3​(m,Ο€21)βŠ•E212​(m,Ο€21)\textstyle{{\tilde{E}}_{3}^{2}(m,\pi_{21})\oplus{\tilde{E}}_{3}E_{21}(m,\pi_{21})\oplus E_{21}{\tilde{E}}_{3}(m,\pi_{21})\oplus E_{21}^{2}(m,\pi_{21})}Ο„3\scriptstyle{\tau_{3}}Ο„E21\scriptstyle{\tau_{E_{21}}}Οƒ21,3\scriptstyle{\sigma_{21,3}}
=Β Β Β Β E32​(m)βŠ•E3​E2​(m)βŠ•E3​E1​(m)βŠ•E2​E3​(m)βŠ•E1​E3​(m)βŠ•E22​(m)βŠ•E2​E1​(m)βŠ•E1​E2​(m)βŠ•E12​(m)Β Β Β E32​(m)βŠ•E3​E2​(m)βŠ•E3​E1​(m)βŠ•E2​E3​(m)βŠ•E1​E3​(m)βŠ•E22​(m)βŠ•E2​E1​(m)βŠ•E1​E2​(m)βŠ•E12​(m)Β Β Β Β Ο„3Β Β Β Β Β Β Β Β Β Β Οƒ23Β Β Β Β Β Β Β Β Β Β Οƒ13Β Β Β Β Β Β Β Β Β Β Ο„2Β Β Β Β Β Β Β Β Β Β Ο„1Β Β Β Β Β Β Β Β Β Β Οƒ12Β Β Β Β Β Β Β Β Β Β ={\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 Ο„\tau of the endofunctor E2E^{2} of Ξ”123​(𝒲)\Delta_{123}({\mathcal{W}}).

The category Δσ32,1​Δσ32​(𝒲)\Delta_{\sigma_{32,1}}\Delta_{\sigma_{32}}({\mathcal{W}}) has objects pairs ((m,Ο€32),Ο€1)((m,\pi_{32}),\pi_{1}) where mβˆˆπ’²m\in{\mathcal{W}}, Ο€32:E3​(m)β†’E2​(m)\pi_{32}:E_{3}(m)\to E_{2}(m) and Ο€1:E32​(m,Ο€32)β†’E~1​(m,Ο€32)\pi_{1}:E_{32}(m,\pi_{32})\to{\tilde{E}}_{1}(m,\pi_{32}) satisfy

d⁑(Ο€32)=d⁑(Ο€1)=0d(\pi_{32})=d(\pi_{1})=0

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

For i∈{2,3}i\in\{2,3\}, let Ο€i​1\pi_{i1} be the composition of the inclusion Ei​(m)β†’E32​(m)E_{i}(m)\to E_{32}(m) with Ο€1\pi_{1}. We have d⁑(Ο€21)=0d(\pi_{21})=0 and d⁑(Ο€31)=Ο€21βˆ˜Ο€32d(\pi_{31})=\pi_{21}\circ\pi_{32}.

As in the case of the category Δσ21,3βˆ’1​Δσ21​(𝒲)\Delta_{\sigma_{21,3}^{-1}}\Delta_{\sigma_{21}}({\mathcal{W}}), the objects of Δσ32,1​Δσ32​(𝒲)\Delta_{\sigma_{32,1}}\Delta_{\sigma_{32}}({\mathcal{W}}) can be described as quadruples (m,Ο€21,Ο€31,Ο€32)(m,\pi_{21},\pi_{31},\pi_{32}) where mβˆˆπ’²m\in{\mathcal{W}}, Ο€i​j:Ei​(m)β†’Ej​(m)\pi_{ij}:E_{i}(m)\to E_{j}(m) satisfy

d⁑(Ο€21)=d⁑(Ο€32)=0,d⁑(Ο€31)=Ο€21βˆ˜Ο€32,d(\pi_{21})=d(\pi_{32})=0,\ d(\pi_{31})=\pi_{21}\circ\pi_{32},

the composition

E3​E2​(m)β†’E3​π21E3​E1​(m)β†’Οƒ31E1​E3​(m)β†’E1​π32E1​E2​(m)β†’Οƒ12E2​E1​(m)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

E3​E2​(m)β†’Οƒ32E2​E3​(m)β†’E2​π32E22​(m)β†’Ο„2E22​(m)β†’E2​π21E2​E1​(m)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

E3​E2​(m)β†’Οƒ32E2​E3​(m)β†’E2​π31E2​E1​(m),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

E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„3\scriptstyle{\tau_{3}}E3​π31\scriptstyle{E_{3}\pi_{31}}E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}E3​E2​(m)\textstyle{E_{3}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ32\scriptstyle{\sigma_{32}}E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π31\scriptstyle{E_{2}\pi_{31}}E3​E1​(m)\textstyle{E_{3}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ31\scriptstyle{\sigma_{31}}E1​E3​(m)\textstyle{E_{1}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π32\scriptstyle{E_{1}\pi_{32}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ12\scriptstyle{\sigma_{12}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)}
E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}Ο„3\scriptstyle{\tau_{3}}E3​E2​(m)\textstyle{E_{3}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ32\scriptstyle{\sigma_{32}}E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}E22​(m)\textstyle{E_{2}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„2\scriptstyle{\tau_{2}}E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}E3​E2​(m)\textstyle{E_{3}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ32\scriptstyle{\sigma_{32}}E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π32\scriptstyle{E_{3}\pi_{32}}E22​(m)\textstyle{E_{2}^{2}(m)}
E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π31\scriptstyle{E_{3}\pi_{31}}Ο„3\scriptstyle{\tau_{3}}E3​E1​(m)\textstyle{E_{3}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ31\scriptstyle{\sigma_{31}}E1​E3​(m)\textstyle{E_{1}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π31\scriptstyle{E_{3}\pi_{31}}E12​(m)\textstyle{E_{1}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„1\scriptstyle{\tau_{1}}E32​(m)\textstyle{E_{3}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π31\scriptstyle{E_{3}\pi_{31}}E3​E1​(m)\textstyle{E_{3}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ31\scriptstyle{\sigma_{31}}E1​E3​(m)\textstyle{E_{1}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π31\scriptstyle{E_{3}\pi_{31}}E12​(m)\textstyle{E_{1}^{2}(m)}
E22​(m)\textstyle{E_{2}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π21\scriptstyle{E_{2}\pi_{21}}Ο„2\scriptstyle{\tau_{2}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ21\scriptstyle{\sigma_{21}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π21\scriptstyle{E_{1}\pi_{21}}E12​(m)\textstyle{E_{1}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„1\scriptstyle{\tau_{1}}E22​(m)\textstyle{E_{2}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π21\scriptstyle{E_{2}\pi_{21}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ21\scriptstyle{\sigma_{21}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π21\scriptstyle{E_{1}\pi_{21}}E12​(m)\textstyle{E_{1}^{2}(m)}
E2​E3​(m)\textstyle{E_{2}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​π31\scriptstyle{E_{2}\pi_{31}}Οƒ23\scriptstyle{\sigma_{23}}E2​E1​(m)\textstyle{E_{2}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ21\scriptstyle{\sigma_{21}}E1​E2​(m)\textstyle{E_{1}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π21\scriptstyle{E_{1}\pi_{21}}E12​(m)\textstyle{E_{1}^{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„1\scriptstyle{\tau_{1}}E3​E2​(m)\textstyle{E_{3}E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E3​π21\scriptstyle{E_{3}\pi_{21}}E3​E1​(m)\textstyle{E_{3}E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Οƒ31\scriptstyle{\sigma_{31}}E1​E3​(m)\textstyle{E_{1}E_{3}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​π31\scriptstyle{E_{1}\pi_{31}}E12​(m)\textstyle{E_{1}^{2}(m)}

and the following composition vanishes:

E3​E2​(m)β†’E3​π21E3​E1​(m)β†’Οƒ31E1​E3​(m)β†’E1​π31E12​(m)β†’Ο„1E12​(m).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 Δσ32,1​Δσ32​(𝒲)β†’βˆΌΞ”123​(𝒲)\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 EE on Δσ32,1​Δσ32​(𝒲)\Delta_{\sigma_{32,1}}\Delta_{\sigma_{32}}({\mathcal{W}}).

We have E⁑((m,Ο€32),Ο€1)=(mβ€²,Ο€β€²)E((m,\pi_{32}),\pi_{1})=(m^{\prime},\pi^{\prime}) where

mβ€²=Β Β Β Β E32​(m,Ο€32)βŠ•E~1​(m,Ο€32)Β Β Β Ο€1Β Β Β Β Β Β Β Β Β 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}}}}}
=(Β Β Β Β E3​(m)βŠ•E2​(m)βŠ•E1​(m)Β Β Β Ο€31Β Β Β Β Β Β Β Β Ο€32Β Β Β Β Β Β Β Β Ο€21Β Β Β Β Β Β Β Β Β ,Β Β Β Β E32​(m)βŠ•E3​E2​(m)βŠ•E3​E1​(m)Β Β Β E2​E3​(m)βŠ•E22​(m)βŠ•E2​E1​(m)Β Β Β Β Οƒ32∘E3​π32βˆ˜Ο„3Β Β Β Β Β Β Β Β Β Β Ο„2∘E2​π32βˆ˜Οƒ32Β Β Β Β Β Β Β Β Β Β Οƒ12∘E1​π32βˆ˜Οƒ31Β Β Β Β Β Β Β Β Β Β Οƒ32Β Β Β Β Β Β Β Β Β Β )=({\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}}}}})
Ο€β€²:Β Β Β Β E322​(m,Ο€32)βŠ•E32​E~1​(m,Ο€32)Β Β Β E~1​E32​(m,Ο€32)βŠ•E~12​(m,Ο€32)Β Β Β Β Οƒ32,1∘E32​π1βˆ˜Ο„E32Β Β Β Β Β Β Β Β Β Β Ο„1∘E~1​π1βˆ˜Οƒ32,1Β Β Β Β Β Β Β Β Β Β Οƒ32,1Β Β Β Β Β Β Β Β Β Β \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}}}}}
=Β Β Β Β E32​(m)βŠ•E3​E2​(m)βŠ•E2​E3​(m)βŠ•E22​(m)βŠ•E3​E1​(m)βŠ•E2​E1​(m)Β Β Β E1​E3​(m)βŠ•E1​E2​(m)βŠ•E12​(m)Β Β Β Οƒ31∘E3​π31βˆ˜Ο„3Β Β Β Β Β Β Β Β Β Οƒ31Β Β Β Β Β Β Β Β Β Ο„1∘E1​π31βˆ˜Οƒ31Β Β Β Β Β Β Β Β Οƒ31∘E3​π21βˆ˜Οƒ23Β Β Β Β Β Β Β Β Β Οƒ21∘E2​π21βˆ˜Ο„2Β Β Β Β Β Β Β Β Β Οƒ21Β Β Β Β Β Β Β Β Ο„1∘E1​π31βˆ˜Οƒ21Β Β Β Β Β Β Β Β Β ={\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 EE on Ξ”123​(𝒲)\Delta_{123}({\mathcal{W}}).

The endomorphism Ο„\tau of E2​((m,Ο€32),Ο€1)E^{2}((m,\pi_{32}),\pi_{1}) is

E322​(m,Ο€32)βŠ•E32​E~1​(m,Ο€32)βŠ•E~1​E32​(m,Ο€32)βŠ•E~12​(m,Ο€32)\textstyle{E_{32}^{2}(m,\pi_{32})\oplus E_{32}{\tilde{E}}_{1}(m,\pi_{32})\oplus{\tilde{E}}_{1}E_{32}(m,\pi_{32})\oplus{\tilde{E}}_{1}^{2}(m,\pi_{32})}E322​(m,Ο€32)βŠ•E32​E~1​(m,Ο€32)βŠ•E~1​E32​(m,Ο€32)βŠ•E~12​(m,Ο€32)\textstyle{E_{32}^{2}(m,\pi_{32})\oplus E_{32}{\tilde{E}}_{1}(m,\pi_{32})\oplus{\tilde{E}}_{1}E_{32}(m,\pi_{32})\oplus{\tilde{E}}_{1}^{2}(m,\pi_{32})}Ο„E32\scriptstyle{\tau_{E_{32}}}Ο„1\scriptstyle{\tau_{1}}Οƒ32,1βˆ’1\scriptstyle{\sigma_{32,1}^{-1}}
=Β Β Β Β E32​(m)βŠ•E3​E2​(m)βŠ•E2​E3​(m)βŠ•E22​(m)βŠ•E3​E1​(m)βŠ•E2​E1​(m)βŠ•E1​E3​(m)βŠ•E1​E2​(m)βŠ•E12​(m)Β Β Β E32​(m)βŠ•E3​E2​(m)βŠ•E2​E3​(m)βŠ•E22​(m)βŠ•E3​E1​(m)βŠ•E2​E1​(m)βŠ•E1​E3​(m)βŠ•E1​E2​(m)βŠ•E12​(m)Β Β Β Β Ο„3Β Β Β Β Β Β Β Β Β Β Οƒ23Β Β Β Β Β Β Β Β Β Β Οƒ13Β Β Β Β Β Β Β Β Β Β Ο„2Β Β Β Β Β Β Β Β Β Β Ο„1Β Β Β Β Β Β Β Β Β Β Οƒ12Β Β Β Β Β Β Β Β Β Β ={\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 Ο„\tau of the endofunctor E2E^{2} of Ξ”123​(𝒲)\Delta_{123}({\mathcal{W}}). ∎

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2