On representation categories of wreath products in non-integral rank

For an arbitrary commutative ring kk and t∈kt∈k, we construct a 2-functor StSt which sends a tensor category to a new tensor category. By applying it to the representation category of a bialgebra we obtain a family of categories which interpolates the representation categories of the wreath products of the bialgebra. This generalizes the construction of Deligne’s category Rep(St,k) for representation categories of symmetric groups.