  In canonical gravity, covariance is implemented by brackets of
hypersurface-deformation generators forming a Lie algebroid. Lie algebroid
morphisms therefore allow one to relate different versions of the brackets that
correspond to the same space-time structure. An application to examples of
modified brackets found mainly in models of loop quantum gravity can in some
cases map the space-time structure back to the classical Riemannian form after
a field redefinition. For one type of quantum corrections (holonomies),
signature change appears to be a generic feature of effective space-time, and
is shown here to be a new quantum space-time phenomenon which cannot be mapped
to an equivalent classical structure. In low-curvature regimes, our
constructions prove the existence of classical space-time structures assumed
elsewhere in models of loop quantum cosmology, but also shows the existence of
additional quantum corrections that have not always been included.
