Twisted Commutators and Internal Crossed Modules

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