ScalingStacks

A bimodule lax bi-$2$-representation is a lax differential $2$-functor $\Upsilon:{\mathcal{U}}\otimes{\mathcal{U}}\to\mathrm{Bimod}$. We say it is a bimodule lax bi-$2$-representation on $\Upsilon(\ast\otimes\ast)$.

A bimodule lax bi-$2$-representation on ${\mathcal{C}}$ is the same as the data of

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  $({\mathcal{C}},{\mathcal{C}})$-bimodules $E_{i,j}$ for $i,j\geq 0$

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  morphisms of differential algebras $H_{i}\otimes H_{j}\to\operatorname{End}\nolimits(E_{i,j})$

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  morphisms $\mu_{(i,j),(i^{\prime},j^{\prime})}:E_{i,j}E_{i^{\prime},j^{\prime}}\to E_{i+i^{\prime},j+j^{\prime}}$ satisfying properties (1) and (2) of §4.2.1.

We define the differential category $\Delta_{E}({\mathcal{C}})$ as the additive category quotient of $T_{{\mathcal{C}}}(E_{0,1}E_{1,0})$ by the ideal of maps generated by the kernels of the compositions

Assume now ${\mathcal{C}}$ is a differential category endowed with two structures $(F_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ of bimodule $2$-representations together with a closed morphism $\lambda:F_{1}E_{2}\to E_{2}F_{1}$ such that the diagrams (4.2.1) commute.

We define the differential category $\Delta^{\prime}_{\lambda}({\mathcal{C}})$ as the additive category quotient of $T_{\mathcal{C}}(F_{1}E_{2})$ by the ideal of maps generated by the image of the composition

We have a differential category ${\mathcal{C}}^{\prime}=\bigoplus_{i\geq 0}E_{2}^{i}F_{1}^{i}$. Its objects are those of ${\mathcal{C}}$ and $\operatorname{Hom}\nolimits_{{\mathcal{C}}^{\prime}}(c_{1},c_{2})=\bigoplus_{i\geq 0}E_{2}^{i}F_{1}^{i}(c_{1},c_{2})$. The multiplication is induced by the maps $\mu_{i,j}$. We define the differential category $\Delta_{\lambda}({\mathcal{C}})$ as the additive category quotient of $\bigoplus_{i\geq 0}E_{2}^{i}F_{1}^{i}$ by the ideal of maps generated by the images of $T_{r}\otimes 1-1\otimes T_{r}:E_{2}^{i}F_{1}^{i}\to E_{2}^{i}F_{1}^{i}$ for $1\leq r<i$.

Assume now ${\mathcal{C}}$ is a differential category endowed with two structures $(E_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ of bimodule $2$-representations, the first of which is right finite. Consider $\sigma:E_{2}E_{1}\to E_{1}E_{2}$ closed such that the diagrams (4.3.1) commute. We define $\lambda:E_{1}^{\vee}E_{2}\to E_{2}E_{1}^{\vee}$ as in (5.3.1).

$\bullet\$We put $\Delta_{\sigma}({\mathcal{C}})=\Delta^{\prime}_{\lambda}({\mathcal{C}})$. As in §5.3.3, we define a $(\Delta_{\sigma}{\mathcal{C}},{\mathcal{C}})$-bimodule $E$ and extend it to a $(\Delta_{\sigma}{\mathcal{C}},\Delta_{\sigma}{\mathcal{C}})$-bimodule. Assume finally that $\sigma$ is invertible. We construct in addition an endomorphism $\tau$ of $E^{2}$. We obtain a bimodule $2$-representation on $\Delta_{\sigma}{\mathcal{C}}$ and an isomorphism of $2$-representations $\Delta_{\sigma}({\mathcal{C}}\operatorname{\!-diff}\nolimits)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\sigma}({\mathcal{C}})\operatorname{\!-diff}\nolimits$. The $2$-representation is right finite if $E_{2}$ is right finite.

As in §5.3.4, we have a monoidal structure on the differential $2$-category of right finite bimodule $2$-representations.

$\bullet\$We drop now the assumption that $\sigma$ is invertible. We define as in §5.4.2 a $(\Delta_{\lambda}{\mathcal{C}},\Delta_{\lambda}{\mathcal{C}})$-bimodule $E$. Assume $\sigma$ is invertible. We obtain an endomorphism $\tau$ of $E^{2}$ and a bimodule $2$-representation on $\Delta_{\lambda}{\mathcal{C}}$.