Additive versus abelian $2$-representations of fiat $2$-categories

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 $\mathfrak{sl}_2$-categorification in the sense of Chuang and Rouquier, and $2$-categorical analogues of Schur algebras.