  We define a modification of LQG in which graphs are required to consist in
piecewise linear edges, which we call piecewise linear LQG (plLQG). At the
diffeomorphism invariant level, we prove that plLQG is equivalent to standard
LQG, as long as one chooses the class of diffeomorphisms appropriately. That
is, we exhibit a unitary map between the diffeomorphism invariant Hilbert
spaces that maps physically equivalent operators into each other. In addition,
using the same ideas as in standard LQG, one can define a Hamiltonian and
Master constraint in plLQG, and the unitary map between plLQG and LQG then
provides an exact isomorphism of dynamics in the two frameworks.
  Furthermore, loop quantum cosmology (LQC) can be exactly embedded into plLQG.
This allows a prior program of the author to embed LQC into LQG at the
dynamical level to proceed. In particular, this allows a formal expression for
a physically motivated embedding of LQC into LQG at the diffeomorphism
invariant level to be given.
