ScalingStacks

Remark 5.3.7. Consider three $2$-representations $(E_{i},\tau_{i})_{1\leq i\leq 3}$ on a differential algebra $B$ together with closed morphisms $\sigma_{ij}:E_{i}E_{j}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E_{j}E_{i}$ for $i\neq j$ satisfying (4.3.5). Assume $E_{1}$ and $E_{2}$ are right finite. We will construct a triple tensor product $2$-representation.

Define

$$\lambda_{ij}:E_{i}^{\vee}E_{j}\xrightarrow{E_{i}^{\vee}E_{j}\eta_{i}}E_{i}^{\vee}E_{j}E_{i}E_{i}^{\vee}\xrightarrow{E_{i}^{\vee}\sigma_{ji}E_{i}^{\vee}}E_{i}^{\vee}E_{i}E_{j}E_{i}^{\vee}\xrightarrow{\varepsilon_{i}E_{j}E_{i}^{\vee}}E_{j}E_{i}^{\vee}$$

and denote by $\sigma_{ij}^{\vee}:E_{i}^{\vee}E_{j}^{\vee}\to E_{j}^{\vee}E_{i}^{\vee}$ the map adjoint to $\sigma_{ij}$.

Let $A^{\prime}=T_{B}(E_{1}^{\vee}E_{2}\oplus E_{2}^{\vee}E_{3}\oplus E_{1}^{\vee}E_{3})$. There is a derivation $\partial$ of $A^{\prime}$ whose restriction to $B\oplus E_{1}^{\vee}E_{2}\oplus E_{2}^{\vee}E_{3}$ is $0$ and whose restriction to $E_{1}^{\vee}E_{3}$ is

$$\partial:E_{1}^{\vee}E_{3}\xrightarrow{E_{1}^{\vee}\eta_{2}E_{3}}(E_{1}^{\vee}E_{2})(E_{2}^{\vee}E_{3}).$$

Define $A^{\prime\prime}$ to be the differential algebra with underlying algebra $A^{\prime}$ and with differential $\partial+d_{A^{\prime}}$.

Let $E$ be the set of quadruples $(i,j,k,l)$ with $i,j,k,l\in\{1,2,3\}$, $j-l\geq i-k>0$ and $(i,j,k,l)\neq(2,3,1,2)$. Given such a quadruple, we define

$$f_{ijkl}:E_{l}^{\vee}E_{k}^{\vee}E_{i}E_{j}\xrightarrow{\sigma_{lk}^{\vee}E_{i}E_{j}}E_{k}^{\vee}E_{l}^{\vee}E_{i}E_{j}\xrightarrow{E_{k}\lambda_{li}E_{j}}(E_{k}^{\vee}E_{i})(E_{l}^{\vee}E_{j})$$

$$g_{ijkl}:E_{l}^{\vee}E_{k}^{\vee}E_{i}E_{j}\xrightarrow{E_{l}^{\vee}E_{k}^{\vee}\sigma_{ij}}E_{l}^{\vee}E_{k}^{\vee}E_{j}E_{i}\xrightarrow{E_{l}\lambda_{kj}E_{i}}(E_{l}^{\vee}E_{j})(E_{k}^{\vee}E_{i})$$

$$h_{3221}:E_{1}^{\vee}E_{2}^{\vee}E_{3}E_{2}\xrightarrow{E_{1}^{\vee}\lambda_{23}E_{2}}E_{1}^{\vee}E_{3}E_{2}^{\vee}E_{2}\xrightarrow{E_{1}^{\vee}E_{3}\varepsilon_{2}}E_{1}^{\vee}E_{3}$$

where we put $\sigma_{rr}=\tau_{r}$ and $\sigma_{rr}^{\vee}=\tau_{r}$. We define $I^{\prime\prime}$ to be the two-sided ideal generated by the images of $f_{ijkl}+g_{ijkl}+\delta_{jk}h_{3221}$ for $(i,j,k,l)\in E$. We put $A=A^{\prime\prime}/I^{\prime\prime}$.

As in §5.3.2, we have an isomorphism of differential categories $\Delta_{123}(B\operatorname{\!-diff}\nolimits)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}A\operatorname{\!-diff}\nolimits$ (cf §4.3.5).

We obtain a bimodule $2$-representation on $A$ as in §4.3.5. We define the differential $(B,A)$-bimodule

$$E={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 71.13152pt\hbox{{\hbox{\kern-64.13002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.66666pt\hbox{$\textstyle{E_{3}\otimes_{B}A\oplus E_{2}\otimes_{B}A\oplus E_{1}\otimes_{B}A}$}}}}}\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 71.13152pt\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-42.33748pt\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 5.69052pt\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 25.32281pt\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 68.28625pt\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

$$\pi_{ij}:E_{i}\otimes_{B}A\xrightarrow{\eta_{j}\operatorname{id}\nolimits}E_{j}E_{j}^{\vee}E_{i}\otimes_{B}A\xrightarrow{E_{j}\mathrm{mult}}E_{j}\otimes_{B}A.$$

We extend the left action of $B$ to an action of $A$ by letting the action maps

$$E_{i}^{\vee}E_{j}\otimes_{B}E\to E$$

for $i<j$ be given by

Finally, we define the endomorphism $\tau$ of $E^{2}$ as in (4.3.6).