Isotypic faithful $2$-representations of $\mathcal{J}$-simple fiat   $2$-categories
  We introduce the class of isotypic $2$-representations for finitary $2$-categories and the notion of inflation of $2$-representations. Under some natural assumptions we show that isotypic $2$-representations are equivalent to inflations of cell $2$-representations. 
Volodymyr Mazorchuk and Vanessa Miemietz
math.RT math.CT