Topological manipulations, like gauging a finite symmetry, produce new quantum field theories from known ones. It is natural to ask how effective they are at moving one around theory space. I will sketch an appealing conjectural answer to this question in the context of 2d rational conformal field theories, which leverages ideas and techniques from 3d topological field theory. I will then present a variety of partial results in the direction of this conjecture, and physically motivate the discussion by situating it in broader quantum field theory lore.