ScalingStacks

Let ${\mathcal{C}}$ and ${\mathcal{C}}^{\prime}$ be differential categories. A $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule is a differential functor ${\mathcal{C}}\otimes{\mathcal{C}}^{\prime{\operatorname{opp}\nolimits}}\to k\operatorname{\!-diff}\nolimits$. There is a $2$-category $\mathrm{Bimod}$ of differential categories and bimodules. Its objects are differential categories and ${{\mathcal{H}}om}_{\mathrm{Bimod}}({\mathcal{C}},{\mathcal{C}}^{\prime})$ is the differential category of $({\mathcal{C}}^{\prime},{\mathcal{C}})$-bimodules. Composition is given by tensor product: given ${\mathcal{C}}^{\prime\prime}$ a differential category, $M$ a $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule and $N$ a $({\mathcal{C}}^{\prime},{\mathcal{C}}^{\prime\prime})$-bimodule, we put

There is an equivalence of $2$-categories $\mathrm{Bimod}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\mathrm{Bimod}^{\mathrm{rev}}$ sending a differential category ${\mathcal{C}}$ to ${\mathcal{C}}^{\operatorname{opp}\nolimits}$ and a $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule to the same functor, viewed as a $({\mathcal{C}}^{\prime{\operatorname{opp}\nolimits}},{\mathcal{C}}^{\operatorname{opp}\nolimits})$-bimodule.

The bimodule $\operatorname{Hom}\nolimits:{\mathcal{C}}\otimes{\mathcal{C}}^{\operatorname{opp}\nolimits}\to k\operatorname{\!-diff}\nolimits,\ (c_{1},c_{2})\mapsto\operatorname{Hom}\nolimits_{\mathcal{C}}(c_{2},c_{1})$ is an identity for the tensor product. The canonical isomorphism of $({\mathcal{C}},{\mathcal{C}})$-bimodules $\operatorname{Hom}\nolimits\otimes_{\mathcal{C}}\operatorname{Hom}\nolimits\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits$ is given by

Let $M$ be a $({\mathcal{C}}^{\prime},{\mathcal{C}})$-bimodule. We define the $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule $M^{\vee}$ by

There is a morphism of $({\mathcal{C}},{\mathcal{C}})$-bimodules $\varepsilon_{M}:M^{\vee}\otimes_{{\mathcal{C}}^{\prime}}M\to\operatorname{Hom}\nolimits$ given by

Given $L\in{\mathcal{C}}\operatorname{\!-diff}\nolimits$ and $L^{\prime}\in{\mathcal{C}}^{\prime}\operatorname{\!-diff}\nolimits$, we have a morphism functorial in $L$ and $L^{\prime}$

We say that $M$ is right finite if the morphism above is an isomorphism for all $L$ and $L^{\prime}$. When this holds, the functor $M^{\vee}\otimes_{{\mathcal{C}}^{\prime}}-$ is left adjoint to $M\otimes_{{\mathcal{C}}}-$ and $M^{\vee}$ is left dual to $M$ . We also write ${{}^{\vee}N}=M$ where $N=M^{\vee}$. We say that $M$ is left finite if it is a right finite $({\mathcal{C}}^{\prime{\operatorname{opp}\nolimits}},{\mathcal{C}}^{\operatorname{opp}\nolimits})$-bimodule.

Let $M$ be a $({\mathcal{C}},{\mathcal{C}})$-bimodule. We define the differential category $T_{{\mathcal{C}}}(M)$. Its objects are those of ${\mathcal{C}}$ and