Interpolation of the oscillator representation and Azumaya algebras in tensor categories

Let C be a symmetric tensor category and let A be an Azumaya algebra in C. Assuming a certain invariant η(A) ∈ Pic(C)[2] vanishes, and fixing a certain choice of signs, we show that there is a universal tensor functor Φ C → D for which Φ(A) splits. We apply this when C=Rep(Spₜ(Fq)) is the interpolation category of finite symplectic groups and A is a certain twisted group algbera in C, and we show that the splitting category D is S. Kriz's interpolation category of the oscillator representation. This construction has a number advantages over previous ones; e.g., it works in non-semisimple cases. It also brings some conceptual clarity to the situation: the existence of Kriz's category is tied to the non-triviality of the Brauer group of C.