G-Extensions of Quantum Group Categories and Functorial SPT

Understand

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.