Additive versus abelian 2-Representations of Fiat 2-Categories

Abstract

We study connections between additive and abelian 2-representations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2-representation. We also describe, in detail, examples of fiat 2-categories associated to sl 2categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras.

Cell structure of bimodules over radical square zero Nakayama algebras

“…Note that S is an infinite set. For an A-A-bimodule X, we will denote by [X] the class of X in S. By [MM2,Section 3], the set S has the natural structure of a multisemigroup, cf. [KuM], defined as follows: for two indecomposable A-Abimodules X and Y , we have…”

Pyramids and 2-representations

“…Modern 2-representation theory originates from [BFK, CR, KL, Ro] which emphasize the use of 2-categories and 2-representations for solving various problems in algebra and topology. The series [MM1,MM2,MM3,MM4,MM5,MM6] of papers started a systematic study of 2-representations of so-called finitary 2-categories which are natural 2-analogues of finite dimensional algebras. The weak Jordan-Hölder theory developed in [MM5] motivates the study of so-called simple transitive 2-representations which are suitable 2-analogues of simple modules.…”

Cell 2-representations and categorification at prime roots of unity

“…[C ] i of [C] and the 2representation [P i ] (which is a 2-functor as applying [ ] is 2-functorial). Using the Yoneda Lemma from [MM2,Lemma 9] we can construct a morphism of additive 2representations η : P…”

“…For a strongly finitary p-dg 2-category C , we write S(C ) for the set of p-dg isomorphism classes of k-indecomposable 1-morphisms in C up to shift. This set forms a multi-semigroup and can be equipped with several natural preorders as in [MM2,Section 3]. Namely, given two k-indecomposable 1-morphisms F and G, we say G ≥ L F in the left preorder if there is a 1-morphism H such that [G] appears as a direct summand of…”