  Higher gauge theories play a prominent role in the construction of 4d
topological invariants and have been long ago proposed as a tool for 4d quantum
gravity. The Yetter lattice model and its continuum counterpart, the BFCG
theory, generalize BF theory to 2-gauge groups and -- when specialized to 4d
and the Poincar\'e 2-group -- they provide an exactly solvable
topologically-flat version of 4d general relativity. The 2-Poincar\'e Yetter
model was conjectured to be equivalent to a state sum model of quantum flat
spacetime developed by Baratin and Freidel after work by Korepanov (KBF model).
This conjecture was motivated by the origin of the KBF model in the theory of
2-representations of the Poincar\'e 2-group. Its proof, however, has remained
elusive due to the lack of a generalized Peter-Weyl theorem for 2-groups.
  In this work we prove this conjecture. Our proof avoids the Peter-Weyl
theorem and rather leverages the geometrical content of the Yetter model. Key
for the proof is the introduction of a kinematical boundary Hilbert space on
which 1- and 2-Lorentz invariance is imposed. Geometrically this allows the
identification of (quantum) tetrad variables and of the associated (quantum)
Levi-Civita connection. States in this Hilbert space are labelled by quantum
numbers that match the 2-group representation labels.
  Our results open exciting opportunities for the construction of new
representations of quantum geometries. Compared to loop quantum gravity, the
higher gauge theory framework provides a quantum representation of the
ADM-Regge initial data, including an identification of the intrinsic and
extrinsic curvature. Furthermore, it leads to a version of the diffeomorphism
and Hamiltonian constraints that acts on the vertices of the discretization,
thus providing a prospect for a quantum realization of the hypersurface
deformation algebra in 4d.
