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We describe a diagrammatic procedure which lifts strict monoidal actions from additive categories to categories of complexes avoiding any use of direct sums. As an application, we prove that every simple transitive $2$-representation of the $2$-category of projective bimodules over a finite dimensional algebra is equivalent to a cell $2$-representation.
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
The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $mathcal{J}$ of a fiat $2$-category $cC$ and the set of equivalence classes of faithful simple tra
In this article, we study short exact sequences of finitary 2-representations of a weakly fiat 2-category. We provide a correspondence between such short exact sequences with fixed middle term and coidempotent subcoalgebras of a coalgebra 1-morphism
Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitar
In this paper we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive $2$-representations and we complete their classification in all types but $H_{3}$ and $H_{4}$.