Cell 2-representations and categorification at prime roots of unity

Abstract

Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2categories, 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 a class of 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 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.

Pretriangulated 2-representations via dg algebra 1-morphisms

“…where pairs (m, n) are ordered lexicographically; The above composition of 1-morphism gives a strict operation (for details, see [LM,Proposition 3.5]). We say that a dg 2-category is pretriangulated if the embedding C ֒→ C is part of a dg biequivalence.…”