Additive versus abelian 2-representations of fiat 2-categories (Q2876654)

The authors, in the paper [Compos. Math. 147, No. 5, 1519--1545 (2011; Zbl 1232.17015)], defined a class of \textit{fiat} \(2\)-categories and then constructed, and studied their \textit{cell} \(2\)-representations. The present paper continues and extends the previous studies, and proposes a slightly different perspective. Connections between additive and abelian \(2\)-representations of fiat \(2\)-categories is discussed. Then, combinatorics of \(2\)-categories in language of multisemigroups are described and anihilators of \(2\)-representations, in particular, those of cell \(2\)-representations are determined.NEWLINENEWLINEAt the end, authors describe examples of fiat \(2\)-categories associated with \(\mathfrak{sl}_2\)-categorification in the sense of \textit{J. Chuang} and \textit{R. Rouquier} [Ann. Math. (2) 167, No. 1, 245--298 (2008; Zbl 1144.20001)] and \(2\)-categorical analogues of Schur algebras as well.