Computing the Minimal Model for the Quantum Symmetric Algebra

In this note, we use some of the tensor categorial machinery developed by the quantum algebra community to study algebraic objects which appear in representation stability. In MR3430359, Sam and Snowden prove that the twisted commutative algebra Sym is Morita equivalent to the horizontal strip category. Their proof relies on a lemma proved by Olver in MR924166. We give a self contained proof that replaces Olver's lemma with information about the associator in the underlying category of polynomial GL(infty)-representations. In fact, we prove a quantum analogue of the theorem. The classical version follows by letting the parameter converge to 1.