Faithfulness of simple 2-representations of $\mathfrak{sl}_2$

Abstract

Let $\mathcal U$ be the 2-category associated with $\mathfrak{sl}_2$. We prove that a complex of 1-morphisms of $\mathcal U$ is null-homotopic if and only if its image in every simple 2-representation is null-homotopic. Under mild boundedness assumptions, we prove that it actually suffices for the image in the simple 2-representations to be acyclic. We apply this result to the study of the Rickard complex $\Theta$ categorifying the action of the simple reflection of $\mathrm{SL}_2$. We prove that $\Theta$ is invertible in the homotopy category of $\U$, and that there is a homotopy equivalence $\Theta E \simeq F\Theta[-1]$.