# Cell 2-Representations and Categorification at Prime Roots of Unity

## TL;DR

This paper develops a theory of 2-representations for p-differential enriched 2-categories, with applications to categorifying quantum groups at prime roots of unity, advancing the understanding of their algebraic and categorical structures.

## Contribution

It introduces a new framework for 2-representations with p-differentials, constructs cell 2-representations, and applies these to categorify small quantum groups at prime roots of unity.

## Key findings

- Constructed cell 2-representations in p-dg 2-categories.
- Applied theory to cyclotomic quotients of quantum groups.
- Connected p-dg 2-representations to triangulated and stable 2-representations.

## Abstract

Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of type $\mathfrak{sl}_2$ at prime roots of unity by Elias-Qi [Advances in Mathematics 288 (2016)]. Passing to stable 2-representations gives a way to construct triangulated 2-representations, but our main focus is on working with p-dg enriched 2-representations that should be seen as a p-dg enhancement of these triangulated ones.