Central reflections and nilpotency in exact Mal’tsev categories

Abstract

We study nilpotency in the context of exact Mal’tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie’s functor calculus. We show in particular that the reflection into the subcategory of n-nilpotent objects is the universal endofunctor of degree n if and only if every n-nilpotent object is n-folded. In the special context of a semi-abelian category, an object is n-folded precisely when its Higgins commutator of length n+ 1 vanishes.