ScalingStacks

Let $B$ be a differential algebra endowed with two $2$-representations $(F_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ together with a closed morphism $\lambda:F_{1}E_{2}\to E_{2}F_{1}$ such that the diagrams (4.2.1) commute.

We define the algebra $A=\Delta^{\prime}_{\lambda}(B)$ as the quotient of the tensor algebra $T_{B}(F_{1}E_{2})$ by the two-sided ideal generated by the image of the composition

We have $A^{0}=B$ and $A^{1}=F_{1}E_{2}$.

Let $B^{\prime}$ be a differential algebra endowed with two $2$-representations $(F^{\prime}_{1},\tau^{\prime}_{1})$ and $(E^{\prime}_{2},\tau^{\prime}_{2})$ together with a closed morphism $\lambda^{\prime}:F^{\prime}_{1}E^{\prime}_{2}\to E^{\prime}_{2}F^{\prime}_{1}$ such that the analogs of the diagrams (4.2.1) commute. Let $A^{\prime}=\Delta^{\prime}_{\lambda^{\prime}}(B^{\prime})$. Let $P$ be a $(B^{\prime},B)$-bimodule and $\varphi_{1}:PF_{1}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}F^{\prime}_{1}P$ and $\varphi_{2}:PE_{2}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}_{2}P$ be two closed isomorphisms of bimodules such that $(P,\varphi_{1})$ and $(P,\varphi_{2})$ are morphisms of $2$-representations and such that

The isomorphism $F^{\prime}_{1}\varphi_{2}\circ\varphi_{1}E_{2}:PF_{1}E_{2}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}F^{\prime}_{1}E_{2}^{\prime}P$ induces an isomorphism of $(B^{\prime},B)$-bimodules $f:P\otimes_{B}T_{B}(F_{1}E_{2})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}T_{B^{\prime}}(F^{\prime}_{1}E^{\prime}_{2})\otimes_{B^{\prime}}P$. This isomorphism $f$ endows the right $T_{B}(F_{1}E_{2})$-module $P\otimes_{B}T_{B}(F_{1}E_{2})$ with a commuting left action of $T_{B^{\prime}}(F^{\prime}_{1}E^{\prime}_{2})$. The isomorphism $f$ induces an isomorphism

So, we obtain a structure of $(A^{\prime},A)$-bimodule on $P\otimes_{B}A$.