Finitary 2-representation theory, pioneered by Mazorchuk and Miemietz in 2010, is a categorification of finite dimensional representations of finite dimensional algebras. It primarily studies the 2-representation theory of finitary 2-categories, which are additive, linear, Krull-Schmidt 2-categories with various finiteness conditions. Much progress has been made in the area since, including various results that fall under the conceptual banner of 'internal vs. external' - that is, finding equivalences between arbitrary 'external' 2-representations and 'internal' 2-representations whose data is fully encoded with the finitary 2-category itself.

In this talk, I will start by outlining the basic theory of finitary 2-categories and their finitary 2-representations, and I will discuss two examples of 'internal' 2-representations, namely cell 2-representations and 2-representations formed of comodule 1-morphisms over a coalgebra 1-morphism. I will then discuss relaxing the finiteness assumptions of finitary 2-categories, resulting in a type of 2-category called 'locally wide finitary 2-categories'. After discussing some of the difficulties this introduces, I will focus on a specific type of locally wide finitary 2-category, namely locally wide quasi-fiat 2-categories, and discuss what we know about coalgebra 1-morphisms and their associated 2-representations in this case.