Minimal nondegenerate extensions

We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories. Our proof relies on the new subject of fusion 2 2 -categories. We study in detail the Drinfel’d centre Z (M o d - B) Z (_{ Mod - B}) of the fusion 2 2 -category M o d - B _{ Mod - B} of module categories of a braided fusion 1 1 -category B B. We show that minimal nondegenerate extensions of B B correspond to certain trivializations of Z (M o d - B) Z (_{ Mod - B}). In the slightly degenerate case, such trivializations are obstructed by a class in H 5 (K (Z 2, 2) ; k