Normalizers, Centralizers and Action Representability in Semi-Abelian Categories

We investigate the existence of normalizers of subobjects in pointed categories defined in the expected way, as motivated by the standard definition used in the category of groups. We show that, for a semi-abelian category C: (a) if the category C2 of morphisms in C is action representable, then so is the category Mon(C) of monomorphisms in C; (b) if Mon(C) is action representable, then normalizers and centralizers exist in C and, more generally, in the category of split epimorphisms in C with any fixed codomain.