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

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 $\Upsilon(e^{n})=M(-,-,e^{n})\otimes_{\mathcal{C}}-$.