A tensor 2-product of 2-representations of $\mathfrak{sl}_{2}^{+}$

We construct an explicit abelian model for the operation of 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 $\mathcal{ V}$ taken from the $2$-category of algebras. We study the case $\mathcal{ V}=\mathcal{ L}(1)$ in detail, and we show that the $2$-product in this case recovers the expected structure. Our construction partially verifies a conjecture of Rouquier from 2008.

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