ScalingStacks

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

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

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 $\Delta_{\sigma}(B)$.