ScalingStacks

All the definitions and constructions of §5.1–5.4 extend from the setting of differential algebras to that of differential categories. We will describe this explicitly.

We view the monoidal category ${\mathcal{U}}$ as a $2$-category with one object $\ast$.

We say that $\Upsilon$ is a bimodule $2$-representation on $\Upsilon(\ast)$.

Bimodule $2$-representations form a differential $2$-category.

Let ${\mathcal{C}}$ be a differential category. There are equivalences of differential $2$-categories between

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  the $2$-category of bimodule $2$-representations $\Upsilon$ on ${\mathcal{C}}$

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  the $2$-category with objects differential functors $M:{\mathcal{C}}\times{\mathcal{C}}^{\operatorname{opp}\nolimits}\times{\mathcal{U}}\to k\operatorname{\!-diff}\nolimits$ together with

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    isomorphisms $\mu_{m,n}:M(c,-,e^{m})\otimes_{{\mathcal{C}}}M(-,c^{\prime},e^{n})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}M(c,c^{\prime},e^{n+m})$ functorial in $c$ and $c^{\prime}$, compatible with the canonical morphism $\operatorname{End}\nolimits(e^{m})\otimes\operatorname{End}\nolimits(e^{n})\to\operatorname{End}\nolimits(e^{n+m})$ and satisfying $\mu_{l,n+m}\circ(\operatorname{id}\nolimits\otimes\mu_{m,n})=\mu_{m+l,n}\circ(\mu_{l,m}\otimes\operatorname{id}\nolimits)$

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    an isomorphism $\mu_{0}:M(-,-,e^{0})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Id}\nolimits$ such that $\mu_{m,0}=\mathrm{mult}\circ(M(c,-,e^{m})\otimes\mu_{0})$ and $\mu_{0,m}=\mathrm{mult}\circ(\mu_{0}\otimes M(-,c,e^{m}))$

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  the $2$-category of pairs $(E,\tau)$ where $E$ is a $({\mathcal{C}},{\mathcal{C}})$-bimodule and $\tau\in\operatorname{End}\nolimits(E^{2})$ satisfies (4.1.1).

The category ${\mathcal{H}}{om}(({\mathcal{C}},E,\tau),({\mathcal{C}}^{\prime},E^{\prime},\tau^{\prime}))$ of $1$-arrows in the third $2$-category above has objects pairs $(P,\varphi)$ where $P$ is a $({\mathcal{C}}^{\prime},{\mathcal{C}})$-bimodule and $\varphi:P\otimes_{\mathcal{C}}E\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}\otimes_{{\mathcal{C}}^{\prime}}P^{\prime}$ is a closed isomorphism of $({\mathcal{C}}^{\prime},{\mathcal{C}})$-bimodules satisfying (5.1.1). We leave it to the reader to describe $1$-arrows in the second $2$-category above. In these $2$-categories, the $2$-arrows are morphisms of (non-differential) bimodules or functors compatible with the additional structure.

The equivalences are given by

$$\Upsilon\mapsto(M:(c_{1},c_{2},e^{n})\mapsto\Upsilon(e^{n})(c_{1},c_{2})),\ M\mapsto(E=M(-,-,e),\tau=M(-,-,\tau))$$

$$E\mapsto(\Upsilon:e^{n}\mapsto E^{n}).$$

We will use the terminology “bimodule $2$-representation” for either one of those three equivalent structures.

Note that a $2$-representation $\Upsilon:{\mathcal{U}}\to\operatorname{End}\nolimits({\mathcal{C}})$ gives rise to a bimodule $2$-representation $M$ on ${\mathcal{C}}$ given by $M(c_{1},c_{2},e^{n})=\operatorname{Hom}\nolimits_{\mathcal{C}}(c_{2},\Upsilon\circ\mathrm{rev}(e^{n})(c_{1}))$ (cf §2.2.3). Note also that a bimodule $2$-representation $M$ on a differential category ${\mathcal{C}}$ gives rise to a $2$-representation $\Upsilon:{\mathcal{U}}\to\operatorname{End}\nolimits({\mathcal{C}}\operatorname{\!-diff}\nolimits)$ given by $\Upsilon(e^{n})=M(-,-,e^{n})\otimes_{\mathcal{C}}-$.

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

$$(E_{0,1}E_{1,0})^{i}(c_{1},c_{2})\xrightarrow{{\mathrm{can}}}E_{i,i}(c_{1},c_{2})\xrightarrow{{\mathrm{can}}}E_{i,i}(c_{1},c_{2})/((T_{r}\otimes 1)x-(1\otimes T_{r})x)_{x\in E_{i,i},\ 1\leq r<i}.$$

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

$$F_{1}^{2}E_{2}^{2}(c_{1},c_{2})\xrightarrow{\tau_{1}E_{2}^{2}-F_{1}^{2}\tau_{2}}F_{1}^{2}E_{2}^{2}(c_{1},c_{2})\xrightarrow{F_{1}\lambda E_{2}}(F_{1}E_{2})^{2}(c_{1},c_{2}).$$

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 $\Delta_{\lambda}{\mathcal{C}}$.