  We define a Galois structure on the category of pairs of equivalence
relations in an exact Mal'tsev category, and characterize central and double
central extensions in terms of higher commutator conditions. These results
generalize both the ones related to the abelianization functor in exact
Mal'tsev categories, and the ones corresponding to the reflection from the
category of internal reflexive graphs to the subcategory of internal groupoids.
Some examples and applications are given in the categories of groups,
precrossed modules, precrossed modules of Lie algebras, and compact groups.
