Classification problems in 2-representation theory

Abstract

This article surveys recent advances and future challenges in the 2-representation theory of finitary 2-categories with a particular emphasis on problems related to classification of various classes of 2-representations.

Pyramids and 2-representations

“…The main result of the present paper, Theorem 12, asserts that the latter is the case for any finite dimensional algebra over an algebraically closed field. This answers (positively) [Maz2,Question 15]. Compared to all previous studies, our approach uses two crucial new ideas.…”

“…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. It turns out that, in many cases, simple transitive 2-representations can be explicitly classified, see [MM5,MM6,MaMa,KMMZ,MMMT,MMZ,MT,MZ1,MZ2,Zh,Zi1,Zi3] for the results and [Maz2] for a detailed survey on the subject. In many, but not all, cases, simple transitive 2-representations are exhausted by so-called cell 2-representations defined already in [MM1].…”

“…In this section we recall basic facts from the classical 2-representations theory developed in [MM1]- [MM6], see also [Maz2] for a survey and [Maz1] for more details.…”

On Simple Transitive 2-representations of Bimodules over the Dual Numbers

“…This motivated the natural problem of classification of simple transitive 2-representations for various classes of finitary 2-categories. This problem was considered and solved in a number of special cases, see [MM5, Zh2, Zi1, Zi2, MZ1, MZ2, MMZ1, MMZ2, MaMa, KMMZ, MT, MMMT, MMMZ, MMMTZ] and also [Ma2] for a slightly outdated overview on the status of that problem.…”

Monoidal categories, representation gap and cryptography

“…(ii) 2-representations of fiat 2-categories, see e.g. [Maz17] for a slightly outdated summary. For example, Soergel bimodules tend to have simple 2-representations of very big rank, see [MMM + 19] for a classification.…”