ScalingStacks

Let $A_{1}$ and $A_{2}$ be two differential algebras equipped with structures of $2$-representations $(E_{i},\tau_{i})$, $i=1,2$.

Let $B=A_{1}\otimes A_{2}$. It is endowed with commuting $2$-representations $(E_{1}\otimes A_{2},\tau_{1}\otimes 1)$ and $(A_{1}\otimes E_{2},1\otimes\tau_{2})$: the isomorphism $\sigma$ is induced by the swap map $E_{2}\otimes E_{1}\xrightarrow{\sim}E_{1}\otimes E_{2},\ a_{2}\otimes a_{1}\mapsto a_{1}\otimes a_{2}$. The tensor product identifies $(A_{1}\operatorname{\!-diff}\nolimits)\otimes(A_{2}\operatorname{\!-diff}\nolimits)$ with a full subcategory of $B\operatorname{\!-diff}\nolimits$.

Assume $E_{1}$ is right finite. The map $\lambda$ is an isomorphism. We put $A_{1}{\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}A_{2}=\Delta^{\prime}_{\lambda}(B)$. It is the quotient of the tensor algebra $T_{A_{1}\otimes A_{2}}(E_{1}^{\vee}\otimes E_{2})$ by the ideal generated by $p\tau_{2}(q)-\tau_{1}(p)q$ for $p\in(E_{1}^{\vee})^{\otimes 2}$ and $q\in(E_{2})^{\otimes 2}$. The underlying differential module is

The multiplication is defined by

We have

The right action of $A$ on $E$ is given by right multiplication, while the left action of $E_{1}^{\vee}\otimes E_{2}$ on $A_{1}\otimes E_{2}\oplus E_{1}\otimes A_{2}\subset E$ is given by

We have

The endomorphism $\tau$ of $E^{2}$ is given on

by

This construction provides the differential $2$-category of right finite $2$-representations on differential algebras with a monoidal structure.