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We introduce a notion of relative commutator -- an important special casebeing commutators twisted by an action -- as a straightforward modification ofthe definition of the Higgins commutator, establish its relation with a newnotion of commutativity -- also obtained as a modification of the usual notion-- and show how we can use it to characterise internal crossed modules in thecontext of a semi-abelian category.

Twisted Commutators and Internal Crossed Modules