Higher Tensor Product for sl2 and Webster algebras

We construct a model for the tensor product of the regular 2-representation of the enveloping algebra of ${\mathfrak{s}\mathfrak{l}}_2^{+}$ with the vector 2-representation, based on the $\mathrm{\infty }$-categorical definition of the second author. Our model contains McMillan’s minimal one. Our use of an infinite family of generators provides a simpler model that we prove is equivalent to Webster’s tensor product category.