G-Extensions of Quantum Group Categories and Functorial SPT

In this short mostly expository note, we sketch a program for gauging fully extended topological field theories in 3 dimensions. One begins with the spherical fusion category with which one wants to do Levin-Wen or Turaev-Viro. One then computes a homotopic space with certain $\pi_\bullet$ given by autoequivalences, invertible objects and the ground (algebraic) field. Then for each desired symmetry group $G$ one looks at mapping $BG$ into that. This classifies equivalence classes of G-extended fusion categories. This is an equivalence at a fully extended level so will allow many defects rather than only evaluating partition functions on closed manifolds. We can now use these categories to build a new fully extended 3d topological field theory, but now possibly with extra data. This is the result of permeating defect walls and saying how that affects the assignment to the point strata.