2-Representations and Associated Coalgebra 1-Morphisms for Locally Wide Finitary 2-Categories

TL;DR

This paper introduces the concept of locally wide finitary 2-categories, extending the framework to infinite objects and morphisms, and develops a method to construct coalgebra 1-morphisms for their transitive 2-representations.

Contribution

It defines locally wide finitary 2-categories and provides a new construction method for coalgebra 1-morphisms in this setting, linking transitive 2-representations to comodule categories.

Findings

Established the concept of locally wide finitary 2-categories.

Developed a new method to construct coalgebra 1-morphisms.

Applied the theory to examples involving infinite quivers and singular Soergel bimodules.

Abstract

We define locally wide finitary 2-categories by relaxing the definition of finitary 2-categories to allow infinitely many objects and isomorphism classes of 1-morphisms and infinite dimensional hom-spaces of 2-morphisms. After defining related concepts including transitive 2-representations in this setting, we provide a new method of constructing coalgebra 1-morphisms associated to transitive 2-representations of locally wide weakly fiat 2-categories, and demonstrate that any such transitive 2-representation is equivalent to a certain subcategory of the category of comodule 1-morphisms over the coalgebra 1-morphism. We finish the paper by examining two classes of examples of locally wide weakly fiat 2-categories: 2-categories associated to certain classes of infinite quivers, and singular Soergel bimodules associated to Coxeter groups with finitely many simple reflections.