  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-categories. We study in
detail the Drinfel'd centre Z(Mod-B) of the fusion 2-category Mod-B of module
categories of a braided fusion 1-category B. We show that minimal nondegenerate
extensions of B correspond to certain trivializations of Z(Mod-B). In the
slightly degenerate case, such trivializations are obstructed by a class in
$\mathrm{H}^5(K(\mathbb{Z}_2, 2); k^\times)$ and we use a numerical invariant
-- defined by evaluating a certain two-dimensional topological field theory on
a Klein bottle -- to prove that this obstruction always vanishes.
  Along the way, we develop techniques to explicitly compute in braided fusion
2-categories which we expect will be of independent interest. In addition to
the model of Z(Mod-B) in terms of braided B-module categories, we develop a
computationally useful model in terms of certain algebra objects in B. We
construct an S-matrix pairing for any braided fusion 2-category, and show that
it is nondegenerate for Z(Mod-B). As a corollary, we identify components of
Z(Mod-B) with blocks in the annular category of B and with the homomorphisms
from the Grothendieck ring of the M\"uger centre of B to the ground field.
