Precrossed modules and Galois theory

The adjunction between crossed modules and precrossed modules over a fixed group can be seen as a special case of a more general adjunction between internal groupoids and internal reflexive graphs in a Malʹtsev variety. By using the categorical Galois theory, we characterize the central extensions with respect to this latter adjunction in terms of the universal algebraic commutator. In particular, we get a description of the central extensions of precrossed modules and of precrossed rings. This characterization provides a natural way to define a categorical notion of Peiffer commutator.