The Deligne category Rep(S_t) can be thought of as "interpolating" the representation categories of symmetric groups. After describing this category, I will introduce a "permutation module" module category, and explain how it categorifies certain stability patterns of Kronecker coefficients described by Stembridge. I will also explain how a calculation in the Deligne category can be used to prove stability properties of permutation patterns within conjugacy classes (joint with Christian Gaetz). Speaker(s): Christopher Ryba (MIT)