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 $M^{\vee}$.