ScalingStacks

Definition 5.1.1. A $2$-representation on $A$ is the data of a differential $(A,A)$-bimodule $E$ and of an endomorphism $\tau$ of the $(A,A)$-bimodule $E\otimes_{A}E$ such that

$$\tau^{2}=0,\ d(\tau)=\operatorname{id}\nolimits\text{ and }(E\otimes\tau)\circ(\tau\otimes E)\circ(E\otimes\tau)=(\tau\otimes E)\circ(E\otimes\tau)\circ(\tau\otimes E).$$

We say that the $2$-representation is right finite if $E$ is finitely generated and projective as a (non-differential) $A^{\operatorname{opp}\nolimits}$-module.