In 2010, Mazorchuk and Miemietz laid the foundations of a systematic theory of finitary 2-representations of finitary 2-categories, which are the categorical analog of finite-dimensional representations of finite-dimensional algebras. In the last couple of years, this theory has been much further developed and has led to interesting classification results for e.g. certain finitary 2-representations of Soergel bimodules of finite Coxeter type, which form an important class of examples.

Together with Miemietz and Vaz, I've recently started to look at 2-representations of Soergel bimodules of affine type A, which form a 2-category that is no longer finitary but only locally wide finitary, a generalization which was introduced and studied by Macpherson. This has major consequences for their 2-representations, e.g. they now come in 3 different flavors: finitary, wide finitary and triangulated.

In my talk, I will first very briefly review finitary 2-representation theory of finitary 2-categories and recall the example of Soergel bimodules of finite Coxeter type. After that, I will zoom in on Soergel bimodules of affine type A and their three types of 2-representations. I will try to sketch some general features, but the talk will nevertheless be very example-based, since our research is still in its early stages.