2-categories of symmetric bimodules and their 2-representations

In this article we analyze the structure of $2$-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting $2$-category is fiat (in the sense of cite{MM1}) and classify simple transitive $2$-representations of this $2$-category (under some mild technical assumption). We also study several classes of examples in detail.