We construct an explicit abelian model for the tensor 2-product of 2-representations of $\mathfrak{sl}_{2}$, specifically the product of a simple 2-representation $\mathcal{L}(1)$ with a given abelian 2-representation $\mathcal{V}$. Both are taken from the 2-category of algebras, and $\mathcal{V}$ is assumed to satisfy two further hypotheses.
The existence of an abelian model like this one, or a generalization of it, was conjectured by Rouquier in 2008.
We study the output of our construction in detail in the case $\mathcal{V}=\mathcal{L}(1)$, and we show that the 2-representation it determines recovers the expected structure of a categorification that is already known for that case.
We form the product construction first for 2-representations of the positive half $\mathcal{U}^{+}$ (a monoidal category) of the 2-category associated to the Lie algebra $\mathfrak{sl}_{2}$. In a subsequent chapter we show that the same construction gives a 2-representation of the full 2-category $\mathcal{U}$ when the inputs are also 2-representations of the full 2-category $\mathcal{U}$.
