ScalingStacks

Let $B$ be a differential algebra. A lax bi-$2$-representation on $B$ is the data of

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  differential $(B,B)$-bimodules $E_{i,j}$ for $i,j\geq 0$

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  morphisms of differential algebras $H_{i}\otimes H_{j}\to\operatorname{End}\nolimits(E_{i,j})$

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  morphisms $\mu_{(i,j),(i^{\prime},j^{\prime})}:E_{i,j}E_{i^{\prime},j^{\prime}}\to E_{i+i^{\prime},j+j^{\prime}}$ satisfying properties (1) and (2) of §4.2.1.

Consider a lax bi-$2$-representation $E$. Note that the functors $(E_{i,j}\otimes_{B}-)$ provide a structure of lax bi-$2$-representation on $B\operatorname{\!-diff}\nolimits$.

We define the differential algebra $A=\Delta_{E}(B)$ as the quotient of the tensor algebra $T_{B}(E_{0,1}E_{1,0})$ by the two-sided ideal generated by $\bigoplus_{i\geq 0}K_{i}$, where $K_{i}$ is the kernel of the composition

We have $A^{0}=B$ and $A$ is generated by $A^{0}$ and $A^{1}=(E_{0,1}E_{1,0})/K_{1}$ as an algebra.

Let $(M,\varsigma)$ be an object of $\Delta_{E\otimes_{B}-}(B\operatorname{\!-diff}\nolimits)$. The action of $T_{B}(E_{0,1}E_{1,0})$ on $M$ vanishes on $K_{i}$ for all $i$, hence defines an action of $A$ on $M$. This gives a fully faithful differential functor $\Delta_{E\otimes_{B}-}(B\operatorname{\!-diff}\nolimits)\to(\Delta_{E}(B))\operatorname{\!-diff}\nolimits$. If the canonical injective morphism of differential $(B,B)$-bimodules

is a split injection for all $i\geq 1$, then the functor above is an isomorphism