Algebra and Geometry Seminar

The categorification program for TQFTs has long sought a braided monoidal structure for the 2-category of 2-representations of Kac-Moody 2-algebras. Such structure requires a general construction for the tensor 2-product of 2-reps. Webster has given a diagrammatic categorification for products of simple 2-reps, and with Losev defined some axioms that determine these uniquely. Rouquier has formulated a general construction, but it returns $\mathcal{A}_\infty$-categories as the products of 2-reps given by (dg)-algebras. Manion-Rouquier applied this construction in the case of $\mathfrak{gl}(1|1)$ where homotopical complications disappear. We present a general construction preserving the 2-category of algebras for the case of $\mathfrak{sl}_2$, specifically the product of the fundamental rep $\mathcal{L}(1)$ and an arbitrary rep. We study the output for $\mathcal{L}(1)$ times $\mathcal{L}(n)$ and compare with the known categorification in this case.