ScalingStacks

Let $\mathrm{Alg}$ be the $2$-category with objects the differential algebras, and $\operatorname{Hom}\nolimits_{\mathrm{Alg}}(A,A^{\prime})$ the category of $(A^{\prime},A)$-bimodules. The composition of $1$-arrows is the tensor product of differential bimodules.

Given $M$ an $(A^{\prime},A)$-bimodule, we put $M^{\vee}=\operatorname{Hom}\nolimits_{A^{\operatorname{opp}\nolimits}}(M,A)$, an $(A,A^{\prime})$-bimodule.

There is a morphism of $(A^{\prime},A)$-bimodules

It is an isomorphism if $M$ is finitely generated and projective as a (non-differential) $A^{\operatorname{opp}\nolimits}$-module.

There is a morphism of functors

It is an isomorphism if $M^{\vee}$ is finitely generated and projective as a (non-differential) $A$-module.

Combining those two morphisms, we obtain a morphism of functors

that is an isomorphism if $M$ is finitely generated and projective as a (non-differential) $A^{\operatorname{opp}\nolimits}$-module. So, when this holds, we have an adjoint pair $(M^{\vee}\otimes_{A^{\prime}}-,M\otimes_{A}-)$, with corresponding unit $\eta:A^{\prime}\to M\otimes_{A}M^{\vee}$ and counit $\varepsilon:M^{\vee}\otimes_{A^{\prime}}M\to A$. In other terms, the bimodule $M^{\vee}$ is a left dual of $M$.

Note conversely that given $M$ such that $(M^{\vee}\otimes_{A^{\prime}}-,M\otimes_{A}-)$ is an adjoint pair, then $M^{\vee}$ is a finitely generated projective $A$-module because $\operatorname{Hom}\nolimits_{A}(M^{\vee},-)$ is exact and commutes with direct sums, hence $M\simeq\operatorname{Hom}\nolimits_{A}(M^{\vee},A)$ is finitely generated and projective as an $A^{\operatorname{opp}\nolimits}$-module.

We say that $M$ is right finite when it is finitely generated and projective as an $A^{\operatorname{opp}\nolimits}$-module. We say that $M$ is left finite when it is finitely generated and projective as an $A^{\prime}$-module.

Consider the $2$-full subcategory $\mathrm{Alg}^{r}$ (resp. $\mathrm{Alg}^{l}$) of $\mathrm{Alg}$ with same objects and $1$-arrows the right (resp. left) finite bimodules. There is an equivalence of $2$-categories $\mathrm{Alg}^{r}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(\mathrm{Alg}^{l})^{\mathrm{rev}{\operatorname{opp}\nolimits}}$. It is the identity on objects and sends a bimodule $M$ to $M^{\vee}$.

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

There is a $2$-functor from $\mathrm{Alg}$ to $\mathrm{Bimod}$: it sends $A$ to the differential category ${\mathcal{C}}_{A}$ with one object $c_{A}$ and $\operatorname{End}\nolimits(c_{A})=A$. It sends an $(A^{\prime},A)$-bimodule $M$ to the $({\mathcal{C}}_{A^{\prime}},{\mathcal{C}}_{A})$-bimodule ${\mathcal{C}}_{M}$ given by ${\mathcal{C}}_{M}(c_{A},c_{A^{\prime}})=M$. This $2$-functor provides isomorphisms of categories $\operatorname{Hom}\nolimits_{\mathrm{Alg}}(A,A^{\prime})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{\mathrm{Bimod}}({\mathcal{C}}_{A},{\mathcal{C}}_{A^{\prime}})$.

There is a $2$-fully faithful $2$-functor from the $2$-category of differential categories to $\mathrm{Bimod}^{\mathrm{rev}}$: it sends ${\mathcal{C}}$ to ${\mathcal{C}}$ and $F:{\mathcal{C}}\to{\mathcal{C}}^{\prime}$ to the $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule $(c,c^{\prime})\mapsto\operatorname{Hom}\nolimits(c^{\prime},F(c))$.

There is a $2$-fully faithful $2$-functor from $\mathrm{Bimod}$ to the $2$-category of differential categories: it sends ${\mathcal{C}}$ to ${\mathcal{C}}\operatorname{\!-diff}\nolimits$ and $M$ a $({\mathcal{C}}^{\prime},{\mathcal{C}})$-bimodule to $M\otimes_{{\mathcal{C}}}-:{\mathcal{C}}\operatorname{\!-diff}\nolimits\to{\mathcal{C}}^{\prime}\operatorname{\!-diff}\nolimits$.

Composing the $2$-functor $\mathrm{Alg}\to\mathrm{Bimod}$ and the $2$-functor from $\mathrm{Bimod}$ to the $2$-category of differential categories, we obtain a differential $2$-functor from $\mathrm{Alg}$ to the $2$-category of differential categories: it sends $A$ to $A\operatorname{\!-diff}\nolimits$ and it sends an $(A^{\prime},A)$-bimodule $M$ to the functor $M\otimes_{A}-:A\operatorname{\!-diff}\nolimits\to A^{\prime}\operatorname{\!-diff}\nolimits$. Note that this $2$-functor is $2$-fully faithful.