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In canonical gravity, covariance is implemented by brackets ofhypersurface-deformation generators forming a Lie algebroid. Lie algebroidmorphisms therefore allow one to relate different versions of the brackets thatcorrespond to the same space-time structure. An application to examples ofmodified brackets found mainly in models of loop quantum gravity can in somecases map the space-time structure back to the classical Riemannian form aftera field redefinition. For one type of quantum corrections (holonomies),signature change appears to be a generic feature of effective space-time, andis shown here to be a new quantum space-time phenomenon which cannot be mappedto an equivalent classical structure. In low-curvature regimes, ourconstructions prove the existence of classical space-time structures assumedelsewhere in models of loop quantum cosmology, but also shows the existence ofadditional quantum corrections that have not always been included.

Hypersurface-deformation algebroids and effective space-time models