ScalingStacks

Let $A^{\prime}$ be another differential algebra with a $2$-representation $(E^{\prime},\tau^{\prime})$. We define a morphism of $2$-representations from $(A,E,\tau)$ to $(A^{\prime},E^{\prime},\tau^{\prime})$ to be an $(A^{\prime},A)$-bimodule $P$ together with a closed isomorphism of $(A^{\prime},A)$-bimodules $\varphi:P\otimes_{A}E\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}\otimes_{A^{\prime}}P$ such that

(5.1.1)

$$\tau^{\prime}P\circ E^{\prime}\varphi\circ\varphi E=E^{\prime}\varphi\circ\varphi E\circ P\tau:PE^{2}\to E^{\prime 2}P.$$

Note that such a pair $(P,\varphi)$ gives rise to a morphism of $2$-representations $(P\otimes_{A}-,\varphi):(A\operatorname{\!-diff}\nolimits,E\otimes_{A}-,\tau)\to(A^{\prime}\operatorname{\!-diff}\nolimits,E^{\prime}\otimes_{A^{\prime}}-,\tau^{\prime})$.