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 differential algebra

$$A=\Delta_{\lambda}(B)=\bigoplus_{i\geq 0}(E_{2}^{i}F_{1}^{i})/((T_{r}\otimes 1)x-(1\otimes T_{r})x)_{x\in E_{2}^{i}F_{1}^{i},\ 1\leq r<i}.$$

Its multiplication is given by the maps $\mu_{i,j}=\mu_{(i,i),(j,j)}E_{2}^{i}F_{1}^{i}E_{2}^{j}F_{1}^{j}\to E_{2}^{i+j}F_{1}^{i+j}$ defined in §4.2.1.

Given $M$ a differential $A$-module and given $i\geq 1$, we have differential $B$-module maps $\varsigma_{i}:E_{2}^{i}F_{1}^{i}\otimes_{B}M\to M$. These make $(M,(\varsigma_{i})_{i})$ into an object of $\Delta_{\lambda}(B\operatorname{\!-diff}\nolimits)$ and provides an isomorphism of differential categories $\Delta_{\lambda}(B)\operatorname{\!-diff}\nolimits\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\lambda}(B\operatorname{\!-diff}\nolimits)$.

The map $\lambda$ extends (uniquely) to a morphism of algebras $\Delta^{\prime}_{\lambda}(B)\to\Delta_{\lambda}(B)$ that is the identity on $B$. If $\lambda$ is an isomorphism, then this map is an isomorphism $\Delta^{\prime}_{\lambda}(B)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\lambda}(B)$.

We assume now that $F_{1}$ is left finite and we put $E_{1}={{}^{\vee}F}_{1}$. Consider $\sigma\in Z\operatorname{Hom}\nolimits(E_{2}E_{1},E_{1}E_{2})$ defined as in (4.4.1).

Let $\pi:E_{2}\otimes_{B}A\to E_{1}\otimes_{B}A$ be the closed morphism of $(B,A)$-bimodules given as a composition

$$\pi:E_{2}\otimes_{B}A\xrightarrow{E_{2}\eta_{1}}E_{2}E_{1}F_{1}\otimes_{B}A\xrightarrow{\sigma F_{1}}E_{1}E_{2}E_{1}^{\vee}\otimes_{B}A\xrightarrow{E_{1}\mathrm{mult}}E_{1}\otimes_{B}A.$$

We put $E=\operatorname{cone}\nolimits(\pi)$. Given $i\geq 1$, we define a morphism of $(B,B)$-bimodules $\varsigma_{i}:E_{2}^{i}F_{1}^{i}E\to E$

$$\varsigma_{i}=\left(\begin{matrix}E_{2}\mathrm{mult}\circ\lambda_{(1\cdots 2i+1)}&\sum_{r=1}^{i}E_{2}\mathrm{mult}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\\ 0&E_{1}\mathrm{mult}\circ\lambda_{(1\cdots 2i+1)}\end{matrix}\right)$$

The following lemma is a consequence of Lemmas 4.3.5 and 4.3.7 applied to $m=A$.

Note that the isomorphism of differential categories $\Delta_{\lambda}(B)\operatorname{\!-diff}\nolimits\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\lambda}(B\operatorname{\!-diff}\nolimits)$ commutes with $E$.

Assume now $\sigma$ is an isomorphism. We define $\tau$ a $(B,A)$-bimodule endomorphism of $E^{2}$ as in (5.3.4).

Theorem 4.4.15 has the following consequence.

Note that we have an isomorphism of $2$-representations $\Delta_{\lambda}(B)\operatorname{\!-diff}\nolimits\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\lambda}(B\operatorname{\!-diff}\nolimits)$.

Consider the $(\Delta_{\sigma}(B),\Delta_{\lambda}(B))$-bimodule $\Delta_{\lambda}(B)$, where the right action is given by multiplication and the left action by multiplication preceded by the morphism of algebras $\Delta_{\sigma}(B)=\Delta^{\prime}_{\lambda}(B)\to\Delta_{\lambda}(B)$. It follows from Proposition 4.4.16 that this bimodule induces a morphism of $2$-representations from $\Delta_{\lambda}(B)$ to $\Delta_{\sigma}(B)$.