Galois theory and commutators

We prove that the relative commutator with respect to a subvariety of a variety of Ω-groups introduced by the first author can be described in terms of categorical Galois theory. This extends the known correspondence between the Fröhlich–Lue and the Janelidze–Kelly notions of central extension. As an example outside the context of Ω-groups we study the reflection of the category of loops to the category of groups where we obtain an interpretation of the associator as a relative commutator.